text: − EDF calculations can be found in Ref. [18] and references M0ν, accounts to 10% 35% of M0ν [17, 19, 20, 22]. The F GT therein. SM details can be found in Refs [17, 32]. The tensor contribution, M0ν, gives a very small correction [17, T transition operator is discussed in depth in Refs. [17, 33]. 41]. Here we focus on the main NME component M0ν , GT 2.1. Energy density functional and discuss M0ν in the context of isospin conservation. F The initial and ﬁnal EDF states are found using the We assume the closure approximation, which has been Gogny D1S functional [35]. Beyond mean ﬁeld eﬀects shown to be a good approximation (up to 10%) [42, 43], such as particle number and angular momentum restora- suﬃcient for the purposes of this work. In this scheme the tion are included in addition to axial quadrupole conﬁg- GT and F parts of the NMEs follow the transition operator uration mixing within the generator coordinate method evaluated between the initial and ﬁnal states: (GCM) [34]. This method has been extensively used to (cid:104) | | (cid:105) study several nuclear structure properties throughout the MF0ν/GT = 0+f MˆF0ν/GT 0+i , (3) whole nuclear chart (see for instance Ref. [36] for its global with (cid:18) (cid:19) (cid:88) performance with an approximate GCM method). One of the advantages of this approach is the explicit calculation of the NMEs as a function of the quadrupole deformation Mˆ 0ν = (cid:88)gA 2 V (r , A, µ) τ−τ−, (4) of initial and ﬁnal nuclei. Hence, full shape-mixing and F gV F ij i j i<j spherical NMEs can be compared. Additional degrees of · − − Mˆ 0ν = V (r , A, µ) σ σ τ τ , (5) freedom such as ﬂuctuations in the pairing ﬁeld (already GT GT ij i j i j applied to NMEs in Ref. [19]), triaxiality or octupolar- i<j − ity are neglected due to prohibitive computational times. where τ is the isospin-lowering operator that trans- Nevertheless, we do not expect from these improvements forms neutrons into protons, and σ are the Pauli spin ma- any qualitative diﬀerence with respect to the analysis pre- trices. The neutrino potentials V depend on the rel- F/GT sented in this work. ative distance between the two decaying nucleons, r , the ij 2.2. Shell model mass number A, and the closure energy µ. Here we use SM calculations are performed in the valence space µ = 7.72 MeV for all decays, the standard value used for comprised by the 0f , 1p , 1p and 0f orbitals (pf 48Ca [16, 18]. A detailed form of the neutrino potentials 7/2 3/2 1/2 5/2 shell), on top of a 40Ca core. We employ the well-known can be found in Refs. [17, 33]. KB3G [37] and GXPF1A [38] eﬀective interactions, which In addition, short-range correlations are included within have been shown to describe well the nuclear structure the UCOM framework [44, 45]. Other prescriptions have of pf shell nuclei [32]. While the conﬁguration space is been recently proposed [46], but the diﬀerences are small, signiﬁcantly smaller in SM than in EDF calculations, the and not relevant for the purpose of this work. main advantage of the SM approach is that all possible Here we neglect two-body currents, related to the ef- correlations within this space are included. The eﬀect of fective quenching of the στ operator [47] in weak decays, the reduced SM valence space has been recently studied in and restrict to a purely two-body operator derived from the framework of many-body perturbation theory [39, 40], one-body currents only. While two-body current contri- with moderate increases in the NMEs. Here we neglect butions may be important for the absolute value of the these corrections. NMEs [47], including them would not alter the main con- Truncated calculations can be performed limiting the clusions of this study. number of neutrons and protons not coupled in J = 0 3. Results pairs (seniority truncations) [16], enabling the study of the correlations associated to high-seniority components. In With the transition operator described in Sec. d, iden- addition, SM states obtained in the full pf shell have good tical for SM and EDF calculations, we can make a di- isospin quantum number, and projection to good isospin rect comparison between the NMEs obtained by both ap- can be performed for truncated calculations. The SM code proaches. We have calculated the NMEs for the 0νββ NATHAN [32] has been used throughout this work. decay of the Ca, Ti and Cr isotopic chains. The GT 2.3. Nuclear Matrix Elements parts of the NMEs are compared in Fig. 1 for SM and With the initial and ﬁnal states obtained with the EDF EDF calculations. As in the case of actual 0νββ decay and SM methods, we calculate the 0νββ decay NMEs as candidates [17, 18], the SM NMEs are about half of the described in Refs. [17, 18].(cid:18)The(cid:19)se can be decomposed ac- EDF values. Moreover, this diﬀerence is independent on cording to spin structure into three diﬀerent terms the particular interaction used. For the SM, results with two eﬀective interactions, KB3G and GXPF1A, are shown, − g 2 − − M0ν = M0ν V M0ν M0ν, (2) with diﬀerences of around 10% 20%. This agrees with GT gA F T and extends previous studies in the pf shell restricted to where g = 1.0 and g = 1.25 are the vector and axial 48Ca [48, 49]. For the EDF, we have also calculated NMEs V A coupling constants, respectively. The Gamow-Teller (GT) with the Gogny D1M functional, which results in very part, M0ν , is dominant, and the Fermi (F) component, small diﬀerences with respect to the Gogny D1S. GT 2 ##LTLine## ##LTLine##

text: EDF

text: calculations

text: can

text: be

text: found

text: in

text: Ref.

text: and

text: references

text: accounts

text: to

text: of

text: The

text: F

text: GT

text: therein.

text: SM

text: details

text: can

text: be

text: found

text: in

text: Refs

text: The

text: tensor

text: contribution,

text: gives

text: a

text: very

text: small

text: correction

text: T

text: transition

text: operator

text: is

text: discussed

text: in

text: depth

text: in

text: Refs.

text: Here

text: we

text: focus

text: on

text: the

text: main

text: NME

text: component

text: GT

text: Energy

text: density

text: functional

text: and

text: discuss

text: in

text: the

text: context

text: of

text: isospin

text: conservation.

text: F

text: The

text: initial

text: and

text: ﬁnal

text: EDF

text: states

text: are

text: found

text: using

text: the

text: We

text: assume

text: the

text: closure

text: approximation,

text: which

text: has

text: been

text: Gogny

text: functional

text: Beyond

text: mean

text: ﬁeld

text: eﬀects

text: shown

text: to

text: be

text: a

text: good

text: approximation

text: (up

text: to

text: such

text: as

text: particle

text: number

text: and

text: angular

text: momentum

text: restora-

text: suﬃcient

text: for

text: the

text: purposes

text: of

text: this

text: work.

text: In

text: this

text: scheme

text: the

text: tion

text: are

text: included

text: in

text: addition

text: to

text: axial

text: quadrupole

text: conﬁg-

text: GT

text: and

text: F

text: parts

text: of

text: the

text: NMEs

text: follow

text: the

text: transition

text: operator

text: uration

text: mixing

text: within

text: the

text: generator

text: coordinate

text: method

text: evaluated

text: between

text: the

text: initial

text: and

text: ﬁnal

text: states:

text: (GCM)

text: This

text: method

text: has

text: been

text: extensively

text: used

text: to

text: study

text: several

text: nuclear

text: structure

text: properties

text: throughout

text: the

text: whole

text: nuclear

text: chart

text: (see

text: for

text: instance

text: Ref.

text: for

text: its

text: global

text: with

text: performance

text: with

text: an

text: approximate

text: GCM

text: method).

text: One

text: of

text: the

text: advantages

text: of

text: this

text: approach

text: is

text: the

text: explicit

text: calculation

text: of

text: the

text: NMEs

text: as

text: a

text: function

text: of

text: the

text: quadrupole

text: deformation

text: Mˆ

text: 0ν

text: V

text: (r

text: A,

text: µ)

text: τ−τ−,

text: of

text: initial

text: and

text: ﬁnal

text: nuclei.

text: Hence,

text: full

text: shape-mixing

text: and

text: F

text: gV

text: F

text: ij

text: i

text: j

text: i<j

text: spherical

text: NMEs

text: can

text: be

text: compared.

text: Additional

text: degrees

text: of

text: Mˆ

text: 0ν

text: V

text: (r

text: A,

text: µ)

text: σ

text: σ

text: τ

text: τ

text: freedom

text: such

text: as

text: ﬂuctuations

text: in

text: the

text: pairing

text: ﬁeld

text: (already

text: GT

text: GT

text: ij

text: i

text: j

text: i

text: j

text: applied

text: to

text: NMEs

text: in

text: Ref.

text: triaxiality

text: or

text: octupolar-

text: i<j

text: ity

text: are

text: neglected

text: due

text: to

text: prohibitive

text: computational

text: times.

text: where

text: τ

text: is

text: the

text: isospin-lowering

text: operator

text: that

text: trans-

text: Nevertheless,

text: we

text: do

text: not

text: expect

text: from

text: these

text: improvements

text: forms

text: neutrons

text: into

text: protons,

text: and

text: σ

text: are

text: the

text: Pauli

text: spin

text: ma-

text: any

text: qualitative

text: diﬀerence

text: with

text: respect

text: to

text: the

text: analysis

text: pre-

text: trices.

text: The

text: neutrino

text: potentials

text: V

text: depend

text: on

text: the

text: rel-

text: F/GT

text: sented

text: in

text: this

text: work.

text: ative

text: distance

text: between

text: the

text: two

text: decaying

text: nucleons,

text: r

text: the

text: ij

text: Shell

text: model

text: mass

text: number

text: A,

text: and

text: the

text: closure

text: energy

text: µ.

text: Here

text: we

text: use

text: SM

text: calculations

text: are

text: performed

text: in

text: the

text: valence

text: space

text: µ

text: MeV

text: for

text: all

text: decays,

text: the

text: standard

text: value

text: used

text: for

text: comprised

text: by

text: the

text: and

text: orbitals

text: (pf

text: A

text: detailed

text: form

text: of

text: the

text: neutrino

text: potentials

text: shell),

text: on

text: top

text: of

text: a

text: core.

text: We

text: employ

text: the

text: well-known

text: can

text: be

text: found

text: in

text: Refs.

text: and

text: eﬀective

text: interactions,

text: which

text: In

text: addition,

text: short-range

text: correlations

text: are

text: included

text: within

text: have

text: been

text: shown

text: to

text: describe

text: well

text: the

text: nuclear

text: structure

text: the

text: UCOM

text: framework

text: Other

text: prescriptions

text: have

text: of

text: pf

text: shell

text: nuclei

text: While

text: the

text: conﬁguration

text: space

text: is

text: been

text: recently

text: proposed

text: but

text: the

text: diﬀerences

text: are

text: small,

text: signiﬁcantly

text: smaller

text: in

text: SM

text: than

text: in

text: EDF

text: calculations,

text: the

text: and

text: not

text: relevant

text: for

text: the

text: purpose

text: of

text: this

text: work.

text: main

text: advantage

text: of

text: the

text: SM

text: approach

text: is

text: that

text: all

text: possible

text: Here

text: we

text: neglect

text: two-body

text: currents,

text: related

text: to

text: the

text: ef-

text: correlations

text: within

text: this

text: space

text: are

text: included.

text: The

text: eﬀect

text: of

text: fective

text: quenching

text: of

text: the

text: στ

text: operator

text: in

text: weak

text: decays,

text: the

text: reduced

text: SM

text: valence

text: space

text: has

text: been

text: recently

text: studied

text: in

text: and

text: restrict

text: to

text: a

text: purely

text: two-body

text: operator

text: derived

text: from

text: the

text: framework

text: of

text: many-body

text: perturbation

text: theory

text: one-body

text: currents

text: only.

text: While

text: two-body

text: current

text: contri-

text: with

text: moderate

text: increases

text: in

text: the

text: NMEs.

text: Here

text: we

text: neglect

text: butions

text: may

text: be

text: important

text: for

text: the

text: absolute

text: value

text: of

text: the

text: these

text: corrections.

text: NMEs

text: including

text: them

text: would

text: not

text: alter

text: the

text: main

text: con-

text: Truncated

text: calculations

text: can

text: be

text: performed

text: limiting

text: the

text: clusions

text: of

text: this

text: study.

text: number

text: of

text: neutrons

text: and

text: protons

text: not

text: coupled

text: in

text: J

text: Results

text: pairs

text: (seniority

text: truncations)

text: enabling

text: the

text: study

text: of

text: the

text: correlations

text: associated

text: to

text: high-seniority

text: components.

text: In

text: With

text: the

text: transition

text: operator

text: described

text: in

text: Sec.

text: d,

text: iden-

text: addition,

text: SM

text: states

text: obtained

text: in

text: the

text: full

text: pf

text: shell

text: have

text: good

text: tical

text: for

text: SM

text: and

text: EDF

text: calculations,

text: we

text: can

text: make

text: a

text: di-

text: isospin

text: quantum

text: number,

text: and

text: projection

text: to

text: good

text: isospin

text: rect

text: comparison

text: between

text: the

text: NMEs

text: obtained

text: by

text: both

text: ap-

text: can

text: be

text: performed

text: for

text: truncated

text: calculations.

text: The

text: SM

text: code

text: proaches.

text: We

text: have

text: calculated

text: the

text: NMEs

text: for

text: the

text: 0νββ

text: NATHAN

text: has

text: been

text: used

text: throughout

text: this

text: work.

text: decay

text: of

text: the

text: Ca,

text: Ti

text: and

text: Cr

text: isotopic

text: chains.

text: The

text: GT

text: Nuclear

text: Matrix

text: Elements

text: parts

text: of

text: the

text: NMEs

text: are

text: compared

text: in

text: Fig.

text: for

text: SM

text: and

text: With

text: the

text: initial

text: and

text: ﬁnal

text: states

text: obtained

text: with

text: the

text: EDF

text: EDF

text: calculations.

text: As

text: in

text: the

text: case

text: of

text: actual

text: 0νββ

text: decay

text: and

text: SM

text: methods,

text: we

text: calculate

text: the

text: 0νββ

text: decay

text: NMEs

text: as

text: candidates

text: the

text: SM

text: NMEs

text: are

text: about

text: half

text: of

text: the

text: described

text: in

text: Refs.

text: can

text: be

text: decomposed

text: ac-

text: EDF

text: values.

text: Moreover,

text: this

text: diﬀerence

text: is

text: independent

text: on

text: cording

text: to

text: spin

text: structure

text: into

text: three

text: diﬀerent

text: terms

text: the

text: particular

text: interaction

text: used.

text: For

text: the

text: SM,

text: results

text: with

text: two

text: eﬀective

text: interactions,

text: and

text: are

text: shown,

text: g

text: V

text: with

text: diﬀerences

text: of

text: around

text: This

text: agrees

text: with

text: GT

text: gA

text: F

text: T

text: and

text: extends

text: previous

text: studies

text: in

text: the

text: pf

text: shell

text: restricted

text: to

text: where

text: g

text: and

text: g

text: are

text: the

text: vector

text: and

text: axial

text: For

text: the

text: EDF,

text: we

text: have

text: also

text: calculated

text: NMEs

text: V

text: A

text: coupling

text: constants,

text: respectively.

text: The

text: Gamow-Teller

text: (GT)

text: with

text: the

text: Gogny

text: functional,

text: which

text: results

text: in

text: very

text: part,

text: is

text: dominant,

text: and

text: the

text: Fermi

text: (F)

text: component,

text: small

text: diﬀerences

text: with

text: respect

text: to

text: the

text: Gogny

text: GT

text: ##LTLine##

text: ##LTLine##


text: Example 3.3 We consider a network to illustrate the notion of contestability. Consider { } directed network D on node set N = 1, 2, 3, 4, 5, 6, 7 shown in Figure 1 on page 7. Table 1 below provides the predecessor and successor sets of all nodes in the network. Node Predecessor Set Successor Set { } 1 P (D) = ∅ S (D) = 2, 3, 4, 5, 6, 7 1 { } 1 { } 2 P (D) = 1 S (D) = 4, 5, 6, 7 2 { } 2 { } 3 P (D) = 1 S (D) = 5, 6, 7 3 { } 3 { } 4 P (D) = 1, 2 S (D) = 6, 7 4 { } 4 { } 5 P (D) = 1, 2, 3 S (D) = 6, 7 5 { } 5 { } 6 P (D) = 1, 2, 3, 4, 5 S (D) = 7 6 { } 6 7 P (D) = 1, 2, 3, 4, 5, 6 S (D) = ∅ 7 7 Table 1: Predecessor and successor sets of nodes in Figure 1 Using this information we deduce that intermediaries 3 and 4 are contested, whereas intermediaries 2, 5, and 6 are uncontested. { } ≡ Here, node 3 is contested by node 2: The predecessor set of node 3 is given by P (D) = 1 3 { } ⊂ { } P (D) and the successor set of node 3 is given by S (D) = 5, 6, 7 S (D) = 4, 5, 6, 7 . It 2 3 2 ⊆ − ∪{ } { } ⊆ − ∪{ } { } is also true that P (D) P (D 3) 2 = 1, 2 and S (D) S (D 3) 2 = 2, 4, 5, 6, 7 . 3 2 3 2 This case introduces what can be denoted as asymmetric contestability, meaning that although node i contests node j, it may not be true that node j contests node i. Here, node 2 directly contests by node 3, although node 2 is not directly contested by node 3. Only in rare cases will there exist symmetric contestability where node i contests node j and node j contests node i. (cid:7) The next example highlights group contestability where a highly connected node asym- metrically contests two others, while these two nodes in turn contest the highly connected node. (cid:48) { } Example 3.4 Consider directed network D on node set N = 1, 2, 3, 4, 5, 6 , shown in (cid:48) Figure 2, where M(D ) = ∅. Here, node 4 connects nodes 1 and 2 to nodes 5 and 6, and therefore directly contests node 2 while not being directly contested by any other individual { } node. However, node 4 is not a middleman and indeed the set C = 2, 3 contests node 4. Clearly, the coverage and reach of nodes 2 and 3 encapsulate the coverage of node 4. { } Therefore, although nodes 2 and 3 do not contest node 4 individually, the node set C = 2, 3 contests node 4. Indeed, the condition for group contestability holds: (cid:16) (cid:17) (cid:16) (cid:17) × ⊂ − × − ∪ − × − P (D) S (D) P (D 4) S (D 4) P (D 4) S (D 4) . (6) 4 4 2 2 3 3 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Example

label: paragraph

text: We

text: consider

text: a

text: network

text: to

text: illustrate

text: the

text: notion

text: of

text: contestability.

text: Consider

font: WFYKKP+txsy

text: directed

text: network

text: D

text: on

text: node

text: set

text: N

font: OOTUFF+rtxr

text: shown

text: in

text: Figure

text: on

text: page

text: Table

text: below

text: provides

text: the

text: predecessor

text: and

text: successor

text: sets

text: of

text: all

text: nodes

text: in

text: the

text: network.

text: Node

text: Predecessor

text: Set

text: Successor

text: Set

text: P

text: (D)

font: LFFLPR+txsyb

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: P

text: (D)

text: S

text: (D)

text: Table

text: Predecessor

text: and

text: successor

text: sets

text: of

text: nodes

text: in

text: Figure

text: Using

text: this

text: information

text: we

text: deduce

text: that

text: intermediaries

text: and

text: are

text: contested,

text: whereas

text: intermediaries

text: and

text: are

text: uncontested.

text: Here,

text: node

text: is

text: contested

text: by

text: node

text: The

text: predecessor

text: set

text: of

text: node

text: is

text: given

text: by

text: P

text: (D)

text: P

text: (D)

text: and

text: the

text: successor

text: set

text: of

text: node

text: is

text: given

text: by

text: S

text: (D)

text: S

text: (D)

text: It

text: is

text: also

text: true

text: that

text: P

text: (D)

text: P

text: (D

text: and

text: S

text: (D)

text: S

text: (D

text: This

text: case

text: introduces

text: what

text: can

text: be

text: denoted

text: as

text: asymmetric

text: contestability,

text: meaning

text: that

text: although

text: node

text: i

text: contests

text: node

text: j,

text: it

text: may

text: not

text: be

text: true

text: that

text: node

text: j

text: contests

text: node

text: i.

text: Here,

text: node

text: directly

text: contests

text: by

text: node

text: although

text: node

text: is

text: not

text: directly

text: contested

text: by

text: node

text: Only

text: in

text: rare

text: cases

text: will

text: there

text: exist

text: symmetric

text: contestability

text: where

text: node

text: i

text: contests

text: node

text: j

text: and

text: node

text: j

text: contests

text: node

text: i.

text: The

text: next

text: example

text: highlights

text: group

text: contestability

text: where

text: a

text: highly

text: connected

text: node

text: asym-

text: metrically

text: contests

text: two

text: others,

text: while

text: these

text: two

text: nodes

text: in

text: turn

text: contest

text: the

text: highly

text: connected

text: node.

text: Example

text: Consider

text: directed

text: network

text: D

text: on

text: node

text: set

text: N

text: shown

text: in

text: Figure

text: where

text: M(D

text: Here,

text: node

text: connects

text: nodes

text: and

text: to

text: nodes

text: and

text: and

text: therefore

text: directly

text: contests

text: node

text: while

text: not

text: being

text: directly

text: contested

text: by

text: any

text: other

text: individual

text: node.

text: However,

text: node

text: is

text: not

text: a

text: middleman

text: and

text: indeed

text: the

text: set

text: C

text: contests

text: node

text: Clearly,

text: the

text: coverage

text: and

text: reach

text: of

text: nodes

text: and

text: encapsulate

text: the

text: coverage

text: of

text: node

text: Therefore,

text: although

text: nodes

text: and

text: do

text: not

text: contest

text: node

text: individually,

text: the

text: node

text: set

text: C

text: contests

text: node

text: Indeed,

text: the

text: condition

text: for

text: group

text: contestability

text: holds:

text: P

text: (D)

text: S

text: (D)

text: P

text: (D

text: S

text: (D

text: P

text: (D

text: S

text: (D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Node Degree PageRank Betweenness Closeness Bonacich β-Measure 1 1 0.112 0.000 0.360 0.328 0.333 2 1 0.112 0.000 0.360 0.328 0.333 3 1 0.112 0.000 0.360 0.328 0.333 4 3 0.330 0.456 0.474 1.047 1.400 5 3 0.330 0.456 0.474 1.047 1.400 6 3 0.330 0.456 0.474 1.047 1.400 7 4 0.480 0.694 0.692 1.565 2.000 8 4 0.480 0.694 0.692 1.565 2.000 9 2 0.297 0.011 0.474 0.863 0.400 10 2 0.297 0.011 0.474 0.863 0.400 (cid:48)(cid:48) Table 2: Centrality results for the undirected network U We provide a mechanism to identify middlemen and quantify their brokerage power in a meaningful way from basic information regarding the topology of the network. The proposed measure counts the disconnections that emerge when a given node is removed from the network. { } ∈ Let D be a directed network on node set N = 1, . . . , n and let i N be an arbitrary node. The brokerage of node i is quantiﬁed as (cid:88) (cid:88) − − − − b (D) = #S (D) #S (D i) #S (D) #P (D). (8) i j j i i ∈ j N j(cid:44)i (cid:80) Note that the successor set of a node contains all other nodes that can be reached by a directed walk from the node. The ﬁrst part of Equation 8, ∈ #S (D), counts the total j N j number of successors of all n nodes in the network. Hence, it provides an indication of the t(cid:80)otal connectivity of the network as a whole. Given th(cid:80)e intuition of the(cid:80)ﬁrst part of the equation, it can b(cid:80)e implied that t(cid:80)he second part, − #S (D i), refers to the total connectivity of the network when node i is removed. We j(cid:44)i j − − remark tha(cid:80)t j∈N #S j(D) >(cid:80) j(cid:44)i #S j(D i) if di(D) (cid:62) 1, and j∈N #S j(D) = j(cid:44)i #S j(D i) if d = 0. i − − Thus, ∈ #S (D) #S (D i) expresses the connectivity diﬀerential between j N j j(cid:44)i j − the full network D and the subnetwork D i. The connectivity diﬀerential captures two features: (1) The direct connectivity of node i in terms of its successor and predecessor set. Indeed, the larger the predecessor and successor sets of node i the larger the diﬀerential will be regardless of whether i is a middleman or not. (2) The lost connectivity to other nodes not including i. 15 ##LTLine## ##LTLine##

text: Node

text: Degree

text: PageRank

text: Betweenness

text: Closeness

text: Bonacich

text: β-Measure

label: caption

text: Table

text: Centrality

text: results

text: for

text: the

text: undirected

text: network

text: U

text: We

text: provide

text: a

text: mechanism

text: to

text: identify

text: middlemen

text: and

text: quantify

text: their

text: brokerage

text: power

text: in

text: a

text: meaningful

text: way

text: from

text: basic

text: information

text: regarding

text: the

text: topology

text: of

text: the

text: network.

text: The

text: proposed

text: measure

text: counts

text: the

text: disconnections

text: that

text: emerge

text: when

text: a

text: given

text: node

text: is

text: removed

text: from

text: the

text: network.

text: Let

text: D

text: be

text: a

text: directed

text: network

text: on

text: node

text: set

text: N

text: n

text: and

text: let

text: i

text: N

text: be

text: an

text: arbitrary

text: node.

text: The

text: brokerage

text: of

text: node

text: i

text: is

text: quantiﬁed

text: as

text: b

text: (D)

text: #S

text: (D)

text: #S

text: (D

text: i)

text: #S

text: (D)

text: #P

text: (D).

text: i

text: j

text: j

text: i

text: i

text: j

text: N

text: Note

text: that

text: the

text: successor

text: set

text: of

text: a

text: node

text: contains

text: all

text: other

text: nodes

text: that

text: can

text: be

text: reached

text: by

text: a

text: directed

text: walk

text: from

text: the

text: node.

text: The

text: ﬁrst

text: part

text: of

text: Equation

text: #S

text: (D),

text: counts

text: the

text: total

text: j

text: N

text: j

text: number

text: of

text: successors

text: of

text: all

text: n

text: nodes

text: in

text: the

text: network.

text: Hence,

text: it

text: provides

text: an

text: indication

text: of

text: the

text: connectivity

text: of

text: the

text: network

text: as

text: a

text: whole.

text: Given

text: intuition

text: of

text: part

text: of

text: the

text: equation,

text: it

text: can

text: implied

text: that

text: second

text: part,

text: #S

text: (D

text: i),

text: refers

text: to

text: the

text: total

text: connectivity

text: of

text: the

text: network

text: when

text: node

text: i

text: is

text: removed.

text: We

text: j

text: remark

text: j∈N

text: #S

text: j(D)

text: #S

text: j(D

text: i)

text: if

text: di(D)

text: and

text: j∈N

text: #S

text: j(D)

text: #S

text: j(D

text: i)

text: if

text: d

text: i

text: Thus,

text: #S

text: (D)

text: #S

text: (D

text: i)

text: expresses

text: the

text: connectivity

text: diﬀerential

text: between

text: j

text: N

text: j

text: j

text: the

text: full

text: network

text: D

text: and

text: the

text: subnetwork

text: D

text: i.

text: The

text: connectivity

text: diﬀerential

text: captures

text: two

text: features:

text: The

text: direct

text: connectivity

text: of

text: node

text: i

text: in

text: terms

text: of

text: its

text: successor

text: and

text: predecessor

text: set.

text: Indeed,

text: the

text: larger

text: the

text: predecessor

text: and

text: successor

text: sets

text: of

text: node

text: i

text: the

text: larger

text: the

text: diﬀerential

text: will

text: be

text: regardless

text: of

text: whether

text: i

text: is

text: a

text: middleman

text: or

text: not.

text: The

text: lost

text: connectivity

text: to

text: other

text: nodes

text: not

text: including

text: i.

text: ##LTLine##

text: ##LTLine##


text: Plurals: individuals and sets in a richly typed semantics (cid:63) Bruno Mery, Richard Moot, and Christian Retor´e (cid:63)(cid:63) LaBRI-CNRS & Universit´e de Bordeaux Abstract We developed a type-theoretical framework for natural lan- guage semantics that, in addition to the usual Montagovian treatment of compositional semantics, includes a treatment of some phenomena of lex- ical semantic: coercions, meaning, transfers, (in)felicitous co-predication. In this setting we see how the various readings of plurals (collective, dis- tributive, coverings,. . . ) can be modelled. 1 Introduction In this paper, we incorporate the treatment of plurals into the uniﬁed type- theoretical framework of [4,18], using polymorphism and type-coercions to ac- count for the basic facts of plurals. We must warn the reader that the purpose of the present paper is neither to assert acceptability judgements nor to to provide linguistic criteria that pre- dict the correctness of readings. Our paper provides a formal and computational account of linguistic analyses and theories of plurals, which is fully compatible with the type theoretical framework in which we studied other lexical or compos- itional phenomena. As such, a prototype implementation has been incorporated in our syntactical and semantical parser Grail [17]. The rest of this paper is structured as follows. Section 1 will give a brief overview of the problems a theory of plurals faces and some of the solutions proposed for treating them. Section (2) introduces our type-theoretic framework ΛTy and compares it to other approaches. Section (2) introduces the basic n elements of our theory of plurals, which will be developed with a fully work out lexicon and several examples in the following section. 2 Plurals in formal semantics: known diﬃculties, typical examples and classical models 2.1 Plural readings Plurals are one of the most ubiquitous of semantic phenomena. Though not explicitly treated by Montague himself, a large body of research has been de- veloped since then and we cannot hope to treat all the subtleties in a single (cid:63) Work supported by ANR project Polymnie (cid:63)(cid:63) On CNRS sabbatical at IRIT (Toulouse) from September 2012 to August 2013 ##LTLine##

text: Plurals:

label: title

text: individuals

text: and

text: sets

text: in

text: a

text: richly

text: typed

text: semantics

text: Bruno

label: author

text: Mery,

text: Richard

text: Moot,

text: and

text: Christian

text: Retor´e

text: LaBRI-CNRS

text: Universit´e

text: de

text: Bordeaux

text: Abstract

text: We

label: abstract

text: developed

text: a

text: type-theoretical

text: framework

text: for

text: natural

text: lan-

text: guage

text: semantics

text: that,

text: in

text: addition

text: to

text: the

text: usual

text: Montagovian

text: treatment

text: of

text: compositional

text: semantics,

text: includes

text: a

text: treatment

text: of

text: some

text: phenomena

text: of

text: lex-

text: ical

text: semantic:

text: coercions,

text: meaning,

text: transfers,

text: (in)felicitous

text: co-predication.

text: In

text: this

text: setting

text: we

text: see

text: how

text: the

text: various

text: readings

text: of

text: plurals

text: (collective,

text: dis-

text: tributive,

text: coverings,.

text: can

text: be

text: modelled.

label: section

text: Introduction

text: In

text: this

text: paper,

text: we

text: incorporate

text: the

text: treatment

text: of

text: plurals

text: into

text: the

text: uniﬁed

text: type-

text: theoretical

text: framework

text: of

text: using

text: polymorphism

text: and

text: type-coercions

text: to

text: ac-

text: count

text: for

text: the

text: basic

text: facts

text: of

text: plurals.

text: We

text: must

text: warn

text: the

text: reader

text: that

text: the

text: purpose

text: of

text: the

text: present

text: paper

text: is

text: neither

text: to

text: assert

text: acceptability

text: judgements

text: nor

text: to

text: to

text: provide

text: linguistic

text: criteria

text: that

text: pre-

text: dict

text: the

text: correctness

text: of

text: readings.

text: Our

text: paper

text: provides

text: a

text: formal

text: and

text: computational

text: account

text: of

text: linguistic

text: analyses

text: and

text: theories

text: of

text: plurals,

text: which

text: is

text: fully

text: compatible

text: with

text: the

text: type

text: theoretical

text: framework

text: in

text: which

text: we

text: studied

text: other

text: lexical

text: or

text: compos-

text: itional

text: phenomena.

text: As

text: such,

text: a

text: prototype

text: implementation

text: has

text: been

text: incorporated

text: in

text: our

text: syntactical

text: and

text: semantical

text: parser

text: Grail

text: The

text: rest

text: of

text: this

text: paper

text: is

text: structured

text: as

text: follows.

text: Section

text: will

text: give

text: a

text: brief

text: overview

text: of

text: the

text: problems

text: a

text: theory

text: of

text: plurals

text: faces

text: and

text: some

text: of

text: the

text: solutions

text: proposed

text: for

text: treating

text: them.

text: Section

text: introduces

text: our

text: type-theoretic

text: framework

text: ΛTy

text: and

text: compares

text: it

text: to

text: other

text: approaches.

text: Section

text: introduces

text: the

text: basic

text: n

text: elements

text: of

text: our

text: theory

text: of

text: plurals,

text: which

text: will

text: be

text: developed

text: with

text: a

text: fully

text: work

text: out

text: lexicon

text: and

text: several

text: examples

text: in

text: the

text: following

text: section.

text: Plurals

text: in

text: formal

text: semantics:

text: known

text: diﬃculties,

text: typical

text: examples

text: and

text: classical

text: models

text: Plural

text: readings

text: Plurals

text: are

text: one

text: of

text: the

text: most

text: ubiquitous

text: of

text: semantic

text: phenomena.

text: Though

text: not

text: explicitly

text: treated

text: by

text: Montague

text: himself,

text: a

text: large

text: body

text: of

text: research

text: has

text: been

text: de-

text: veloped

text: since

text: then

text: and

text: we

text: cannot

text: hope

text: to

text: treat

text: all

text: the

text: subtleties

text: in

text: a

text: single

text: Work

text: supported

text: by

text: ANR

text: project

text: Polymnie

text: On

text: CNRS

text: sabbatical

text: at

text: IRIT

text: (Toulouse)

text: from

text: September

text: to

text: August

text: ##LTLine##


text: ¬ → ⊥ – Although there is a negation A = A , negation does not implement set theoretic complements. Indeed, there cannot be both terms of type A → ⊥ and terms of type A . One of these two types must be empty. When → ¬ → considering the associated predicate A˜ : e t, there is a complement: t tA˜ : → e t. – Disjunctive types are often left out of the type system, since to obtain nor- mal form one needs to consider unpleasant commutative conversions Fur- ∨ thermore, the terms in a disjunctive type A A are not the unions of 1 2(cid:104) (cid:105) terms of type A and of terms of type A , but pairs i, t of an integer i and 1 2 a term of type A , hence they never contain any common term to types A i ∪ 1 and A — even if one pure term is common to both, as a proof of A A , 2 1 2 it is a diﬀerent typed term, and the integer in front event prevent to have ∪ proofs of A as proof of A A. ∧ – There is no intersection of types. A conjunctive type A A contains pairs 1 2 of terms of respective types A and A , but no common term to A and A . 1 2 1 2 Nevertheless some types are closer to sets, namely data types, see e.g. [8], deﬁned from constants and operators like, integers, lists of objects of type A, ﬁnite trees with or without labels etc. A data type T resembles a set in the sense that there is a bijective correspondence between normal terms of type T and data of type T. . Data types can either be deﬁned internally as in plain F or by constants and operators with speciﬁc reduction rules as in G¨odel’s system T or Martin L¨of type theory. The easiest (cid:98)account of a set in our Montagovian setting is a predicate which → is of type X t. In order to view the sorts e as sets, for each of them we → i have predicate e ( ) : e t that encodes the property of being of type e . The i i choice of the domain of the predicate here e, the sort into which base types e i maps, is rather free, since the domain of a predicate can be both restricted (as a function can) and extended (by saying it is false elsewhere). Once we have the base predicates, with their types, and the predicate associated with base types, plus our system of coercions, that include ontological inclusions, we can then deﬁne sets as one usually does. For instance, black applies to physical objects, cats may be viewed as physical objects (meaning transfer), hence “the black cats in the garden” is a well deﬁned set. This is the way we modelled, predicates, determiners and quantiﬁers. [24] 4.2 Operators for handling sets We can deﬁne operators that represent operations on sets, that can be as- → similated with predicates, of type e t. As integers can be deﬁned, we shall | | have a constant for counting elements satisfying a predicate. Inclusion is easily deﬁned as a function that maps pairs of predicates to a proposition: ⊆ → → ⇒ : ΛαλPα tλQα tλxαP(x) = Q(x) — α usually is e but this is not man- ∧ ∨ datory. The logical operators such as and between predicates correspond respectively to union and intersection of sets of terms, as expected (while oper- ations on types like conjunction and disjunction do not, cf. Section (2)). ##LTLine## ##LTLine##

label: list

text: Although

text: there

text: is

text: a

text: negation

text: A

text: A

text: negation

text: does

text: not

text: implement

text: set

text: theoretic

text: complements.

text: Indeed,

text: there

text: cannot

text: be

text: both

text: terms

text: of

text: type

text: A

text: and

text: terms

text: of

text: type

text: A

text: One

text: of

text: these

text: two

text: types

text: must

text: be

text: empty.

text: When

text: considering

text: the

text: associated

text: predicate

text: A˜

text: e

text: t,

text: there

text: is

text: a

text: complement:

text: t

text: tA˜

text: e

text: t.

text: Disjunctive

text: types

text: are

text: often

text: left

text: out

text: of

text: the

text: type

text: system,

text: since

text: to

text: obtain

text: nor-

text: mal

text: form

text: one

text: needs

text: to

text: consider

text: unpleasant

text: commutative

text: conversions

text: Fur-

text: thermore,

text: the

text: terms

text: in

text: a

text: disjunctive

text: type

text: A

text: A

text: are

text: not

text: the

text: unions

text: of

text: terms

text: of

text: type

text: A

text: and

text: of

text: terms

text: of

text: type

text: A

text: but

text: pairs

text: i,

text: t

text: of

text: an

text: integer

text: i

text: and

text: a

text: term

text: of

text: type

text: A

text: hence

text: they

text: never

text: contain

text: any

text: common

text: term

text: to

text: types

text: A

text: i

text: and

text: A

text: even

text: if

text: one

text: pure

text: term

text: is

text: common

text: to

text: both,

text: as

text: a

text: proof

text: of

text: A

text: A

text: it

text: is

text: a

text: diﬀerent

text: typed

text: term,

text: and

text: the

text: integer

text: in

text: front

text: event

text: prevent

text: to

text: have

text: proofs

text: of

text: A

text: as

text: proof

text: of

text: A

text: A.

text: There

text: is

text: no

text: intersection

text: of

text: types.

text: A

text: conjunctive

text: type

text: A

text: A

text: contains

text: pairs

text: of

text: terms

text: of

text: respective

text: types

text: A

text: and

text: A

text: but

text: no

text: common

text: term

text: to

text: A

text: and

text: A

text: Nevertheless

text: some

text: types

text: are

text: closer

text: to

text: sets,

text: namely

text: data

text: types,

text: see

text: e.g.

text: deﬁned

text: from

text: constants

text: and

text: operators

text: like,

text: integers,

text: lists

text: of

text: objects

text: of

text: type

text: A,

text: ﬁnite

text: trees

text: with

text: or

text: without

text: labels

text: etc.

text: A

text: data

text: type

text: T

text: resembles

text: a

text: set

text: in

text: the

text: sense

text: that

text: there

text: is

text: a

text: bijective

text: correspondence

text: between

text: normal

text: terms

text: of

text: type

text: T

text: and

text: data

text: of

text: type

text: T.

text: Data

text: types

text: can

text: either

text: be

text: deﬁned

text: internally

text: as

text: in

text: plain

text: F

text: or

text: by

text: constants

text: and

text: operators

text: with

text: speciﬁc

text: reduction

text: rules

text: as

text: in

text: G¨odel’s

text: system

text: T

text: or

text: Martin

text: L¨of

text: type

text: theory.

text: The

text: easiest

text: of

text: a

text: set

text: in

text: our

text: Montagovian

text: setting

text: is

text: a

text: predicate

text: which

text: is

text: of

text: type

text: X

text: t.

text: In

text: order

text: to

text: view

text: the

text: sorts

text: e

text: as

text: sets,

text: for

text: each

text: of

text: them

text: we

text: i

text: have

text: predicate

text: e

text: e

text: t

text: that

text: encodes

text: the

text: property

text: of

text: being

text: of

text: type

text: e

text: The

text: i

text: i

text: choice

text: of

text: the

text: domain

text: of

text: the

text: predicate

text: here

text: e,

text: the

text: sort

text: into

text: which

text: base

text: types

text: e

text: i

text: maps,

text: is

text: rather

text: free,

text: since

text: the

text: domain

text: of

text: a

text: predicate

text: can

text: be

text: both

text: restricted

text: (as

text: a

text: function

text: can)

text: and

text: extended

text: (by

text: saying

text: it

text: is

text: false

text: elsewhere).

text: Once

text: we

text: have

text: the

text: base

text: predicates,

text: with

text: their

text: types,

text: and

text: the

text: predicate

text: associated

text: with

text: base

text: types,

text: plus

text: our

text: system

text: of

text: coercions,

text: that

text: include

text: ontological

text: inclusions,

text: we

text: can

text: then

text: deﬁne

text: sets

text: as

text: one

text: usually

text: does.

text: For

text: instance,

text: black

text: applies

text: to

text: physical

text: objects,

text: cats

text: may

text: be

text: viewed

text: as

text: physical

text: objects

text: (meaning

text: transfer),

text: hence

text: “the

text: black

text: cats

text: in

text: the

text: garden”

text: is

text: a

text: well

text: deﬁned

text: set.

text: This

text: is

text: the

text: way

text: we

text: modelled,

text: predicates,

text: determiners

text: and

text: quantiﬁers.

text: Operators

text: for

text: handling

text: sets

text: We

text: can

text: deﬁne

text: operators

text: that

text: represent

text: operations

text: on

text: sets,

text: that

text: can

text: be

text: as-

text: similated

text: with

text: predicates,

text: of

text: type

text: e

text: t.

text: As

text: integers

text: can

text: be

text: deﬁned,

text: we

text: shall

text: have

text: a

text: constant

text: for

text: counting

text: elements

text: satisfying

text: a

text: predicate.

text: Inclusion

text: is

text: easily

text: deﬁned

text: as

text: a

text: function

text: that

text: maps

text: pairs

text: of

text: predicates

text: to

text: a

text: proposition:

text: ΛαλPα

text: tλQα

text: tλxαP(x)

text: Q(x)

text: α

text: usually

text: is

text: e

text: but

text: this

text: is

text: not

text: man-

text: datory.

text: The

text: logical

text: operators

text: such

text: as

text: and

text: between

text: predicates

text: correspond

text: respectively

text: to

text: union

text: and

text: intersection

text: of

text: sets

text: of

text: terms,

text: as

text: expected

text: (while

text: oper-

text: ations

text: on

text: types

text: like

text: conjunction

text: and

text: disjunction

text: do

text: not,

text: cf.

text: Section

text: ##LTLine##

text: ##LTLine##


text: 2.3 Further comments: lifted shape spaces and multishapes In this section we provide one last way to study shape spaces and describe two interesting and important variants of shape spaces, namely lifted shape spaces and multishapes. We show that a slightly diﬀerent optimal control problem can model the shape deformation problem in these spaces. Lifted shape spaces. Lifted shapes can be used to keep track of additional parameters when studying the deformation of a shape. For example, when studying n landmarks (x , . . . , x ) in IRd, 1 n it can be interesting to keep track of how another point x is moved by the deformation. Let M and Mˆ be two shape spaces, open subsets of two Banach spaces X and Xˆ respectively, on which the group of diﬀeomorphisms of IRd acts smoothly with respective orders (cid:96) and (cid:96)ˆ. Let V be an RKHS of vector ﬁelds in IRd of class Cmax((cid:96),(cid:96)ˆ). We denote by ξ (respectively ξ ) the 0 q C qˆ inﬁnitesimal action of V on M (respectively Mˆ ). We assume that there exists a 1 equivariant → submersion P : Mˆ M. · · ∈ By equivariant, we mean that P(ϕ qˆ) = ϕ P(qˆ), for every diﬀeomorphism ϕ Diﬀ max((cid:96),(cid:96)ˆ),1 ∈ ∗ ∗ ∗ and every qˆ Mˆ . Note that this implies that dP .ξ = ξ and ξ dP = ξ . qˆ qˆ q→ qˆ qˆ q For example, for n < nˆ, the projection P : Lmk (nˆ) Lmk (n) deﬁned by P(x , . . . , x ) = C d d 1 nˆ (x , . . . , x ) is a 1 equivariant submersion. More generally, for a compact Riemannian manifold 1 n ⊂ → Sˆ and a submanifold S Sˆ, the restriction mapping P : Emb(Sˆ, IRd) Emb(S, IRd) deﬁned by C P(q) = q| is a 1 equivariant submersion for the action by composition of Diﬀ 1(IRd). S The constructions and results of Section can be applied to this setting, and in particular the deformation evolution induces a control system on Mˆ , as investigated previously. Remark 7. Let V be an RKHS of bounded vector ﬁelds of class Cmax((cid:96),(cid:96)ˆ)+1. Let g be a data 0 ◦ attachment function on M and let C be a mapping of constraints. We set gˆ = g P and Cˆ = C . qˆ P(qˆ) Then a time-dependent vector ﬁeld v in V is a solution of Problem 1 for M with constraints C and data attachment g if and only if it is also a solution of Problem 1 for Mˆ with constraints Cˆ and data attachment gˆ. This remark will be used for ﬁnite-dimensional approximations in Section . One can however deﬁne a control system of a diﬀerent form, by lifting the control applied on the smaller shape space M to the bigger shape space Mˆ . ∈ ∈ − The method goes as follows. Let q M and qˆ P 1(q ). Consider a measurable map → ∗ 0 · 0 0 u : [0, 1] X and the corresponding curve q( ) deﬁned by q(0) = q and q˙(t) = K u(t) for ∈ ∗ 0 q(t) almost every t [0, 1], where K = ξ K ξ . This curve is the same as the one induced by the · q q ∗ V q· time-dependent vector ﬁeld v( ) = K ξ · u( ). The deformation ϕ corresponding to the ﬂow of v V· q( ) deﬁnes on Mˆ a new curve qˆ(t) = ϕ(t) qˆ with speed 0 ∗ ∗ qˆ˙(t) = ξ K ξ u(t) = K dP u(t), qˆ V q(t) qˆ(t) qˆ(t) ∗ ∈ with K = ξ K ξ . Note that P(qˆ(t)) = q(t) for every t [0, 1]. We have thus obtained a new qˆ qˆ V qˆ class of control problems. ∈ × → Problem 3. Let qˆ Mˆ , and let C : Mˆ V Y be continuous and linear with respect to the 0 → second variable, with Y a Banach s(cid:90)pace. Let g : Mˆ IR be a real function on Mˆ . We consider the problem of minimizing the functional 1 1(cid:104) (cid:105) J (qˆ, u) = u(t), K u(t) ∗ dt + g(qˆ(1)) 3 2 P(qˆ(t)) X ,X 0 · · → ∗ · ∈ over all (qˆ( ), u( )), where u : [0, 1] X is a measurable function and qˆ( ) W1,1(0, 1; Mˆ ) are ∗ ∗ ∈ qˆ0 such that qˆ˙(t) = K dP u(t) and C K ξ u(t) = 0 for almost every t [0, 1]. qˆ(t) qˆ(t) qˆ(t) V P(qˆ(t)) 14 ##LTLine##

text: Further

text: comments:

text: lifted

text: shape

text: spaces

text: and

text: multishapes

text: In

text: this

text: section

text: we

text: provide

text: one

text: last

text: way

text: to

text: study

text: shape

text: spaces

text: and

text: describe

text: two

text: interesting

text: and

text: important

text: variants

text: of

text: shape

text: spaces,

text: namely

text: lifted

text: shape

text: spaces

text: and

text: multishapes.

text: We

text: show

text: that

text: a

text: slightly

text: diﬀerent

text: optimal

text: control

text: problem

text: can

text: model

text: the

text: shape

text: deformation

text: problem

text: in

text: these

text: spaces.

text: Lifted

text: shape

text: spaces.

text: Lifted

text: shapes

text: can

text: be

text: used

text: to

text: keep

text: track

text: of

text: additional

text: parameters

text: when

text: studying

text: the

text: deformation

text: of

text: a

text: shape.

text: For

text: example,

text: when

text: studying

text: n

text: landmarks

text: (x

text: x

text: in

text: IRd,

text: n

text: it

text: can

text: be

text: interesting

text: to

text: keep

text: track

text: of

text: how

text: another

text: point

text: x

text: is

text: moved

text: by

text: the

text: deformation.

text: Let

text: M

text: and

text: Mˆ

text: be

text: two

text: shape

text: spaces,

text: open

text: subsets

text: of

text: two

text: Banach

text: spaces

text: X

text: and

text: Xˆ

text: respectively,

text: on

text: which

text: the

text: group

text: of

text: diﬀeomorphisms

text: of

text: IRd

text: acts

text: smoothly

text: with

text: respective

text: orders

text: and

text: Let

text: V

text: be

text: an

text: RKHS

text: of

text: vector

text: ﬁelds

text: in

text: IRd

text: of

text: class

text: We

text: denote

text: by

text: ξ

text: (respectively

text: ξ

text: the

text: q

text: C

text: qˆ

text: inﬁnitesimal

text: action

text: of

text: V

text: on

text: M

text: (respectively

text: Mˆ

text: We

text: assume

text: that

text: there

text: exists

text: a

text: equivariant

text: submersion

text: P

text: Mˆ

text: M.

text: By

text: equivariant,

text: we

text: mean

text: that

text: P(ϕ

text: qˆ)

text: ϕ

text: P(qˆ),

text: for

text: every

text: diﬀeomorphism

text: ϕ

text: Diﬀ

text: and

text: every

text: qˆ

text: Mˆ

text: Note

text: that

text: this

text: implies

text: that

text: dP

text: .ξ

text: ξ

text: and

text: ξ

text: dP

text: ξ

text: qˆ

text: qˆ

text: q→

text: qˆ

text: qˆ

text: q

text: For

text: example,

text: for

text: n

text: nˆ,

text: the

text: projection

text: P

text: Lmk

text: (nˆ)

text: Lmk

text: (n)

text: deﬁned

text: by

text: P(x

text: x

text: C

text: d

text: d

text: nˆ

text: (x

text: x

text: is

text: a

text: equivariant

text: submersion.

text: More

text: generally,

text: for

text: a

text: compact

text: Riemannian

text: manifold

text: n

text: Sˆ

text: and

text: a

text: submanifold

text: S

text: Sˆ,

text: the

text: restriction

text: mapping

text: P

text: Emb(Sˆ,

text: IRd)

text: Emb(S,

text: IRd)

text: deﬁned

text: by

text: C

text: P(q)

text: q|

text: is

text: a

text: equivariant

text: submersion

text: for

text: the

text: action

text: by

text: composition

text: of

text: Diﬀ

text: S

text: The

text: constructions

text: and

text: results

text: of

text: Section

text: can

text: be

text: applied

text: to

text: this

text: setting,

text: and

text: in

text: particular

text: the

text: deformation

text: evolution

text: induces

text: a

text: control

text: system

text: on

text: Mˆ

text: as

text: investigated

text: previously.

text: Remark

text: Let

text: V

text: be

text: an

text: RKHS

text: of

text: bounded

text: vector

text: ﬁelds

text: of

text: class

text: Let

text: g

text: be

text: a

text: data

text: attachment

text: function

text: on

text: M

text: and

text: let

text: C

text: be

text: a

text: mapping

text: of

text: constraints.

text: We

text: set

text: gˆ

text: g

text: P

text: and

text: Cˆ

text: C

text: qˆ

text: P(qˆ)

text: Then

text: a

text: time-dependent

text: vector

text: ﬁeld

text: v

text: in

text: V

text: is

text: a

text: solution

text: of

text: Problem

text: for

text: M

text: with

text: constraints

text: C

text: and

text: data

text: attachment

text: g

text: if

text: and

text: only

text: if

text: it

text: is

text: also

text: a

text: solution

text: of

text: Problem

text: for

text: Mˆ

text: with

text: constraints

text: Cˆ

text: and

text: data

text: attachment

text: gˆ.

text: This

text: remark

text: will

text: be

text: used

text: for

text: ﬁnite-dimensional

text: approximations

text: in

text: Section

text: One

text: can

text: however

text: deﬁne

text: a

text: control

text: system

text: of

text: a

text: diﬀerent

text: form,

text: by

text: lifting

text: the

text: control

text: applied

text: on

text: the

text: smaller

text: shape

text: space

text: M

text: to

text: the

text: bigger

text: shape

text: space

text: Mˆ

text: The

text: method

text: goes

text: as

text: follows.

text: Let

text: q

text: M

text: and

text: qˆ

text: P

text: Consider

text: a

text: measurable

text: map

text: u

text: X

text: and

text: the

text: corresponding

text: curve

text: q(

text: deﬁned

text: by

text: q

text: and

text: q˙(t)

text: K

text: u(t)

text: for

text: q(t)

text: almost

text: every

text: t

text: where

text: K

text: ξ

text: K

text: ξ

text: This

text: curve

text: is

text: the

text: same

text: as

text: the

text: one

text: induced

text: by

text: the

text: q

text: q

text: V

text: q·

text: time-dependent

text: vector

text: ﬁeld

text: v(

text: K

text: ξ

text: u(

text: The

text: deformation

text: ϕ

text: corresponding

text: to

text: the

text: ﬂow

text: of

text: v

text: V·

text: q(

text: deﬁnes

text: on

text: Mˆ

text: a

text: new

text: curve

text: qˆ(t)

text: ϕ(t)

text: qˆ

text: with

text: speed

text: qˆ˙(t)

text: ξ

text: K

text: ξ

text: u(t)

text: K

text: dP

text: u(t),

text: qˆ

text: V

text: q(t)

text: qˆ(t)

text: qˆ(t)

text: with

text: K

text: ξ

text: K

text: ξ

text: Note

text: that

text: P(qˆ(t))

text: q(t)

text: for

text: every

text: t

text: We

text: have

text: thus

text: obtained

text: a

text: new

text: qˆ

text: qˆ

text: V

text: qˆ

text: class

text: of

text: control

text: problems.

text: Problem

text: Let

text: qˆ

text: Mˆ

text: and

text: let

text: C

text: Mˆ

text: V

text: Y

text: be

text: continuous

text: and

text: linear

text: with

text: respect

text: to

text: the

text: second

text: variable,

text: with

text: Y

text: a

text: Banach

text: Let

text: g

text: Mˆ

text: IR

text: be

text: a

text: real

text: function

text: on

text: Mˆ

text: We

text: consider

text: the

text: problem

text: of

text: minimizing

text: the

text: functional

text: J

text: (qˆ,

text: u)

text: u(t),

text: K

text: u(t)

text: dt

text: P(qˆ(t))

text: X

text: ,X

text: over

text: all

text: (qˆ(

text: u(

text: where

text: u

text: X

text: is

text: a

text: measurable

text: function

text: and

text: qˆ(

text: Mˆ

text: are

text: such

text: that

text: qˆ˙(t)

text: K

text: dP

text: u(t)

text: and

text: C

text: K

text: ξ

text: u(t)

text: for

text: almost

text: every

text: t

text: qˆ(t)

text: qˆ(t)

text: qˆ(t)

text: V

text: P(qˆ(t))

text: ##LTLine##


text: 1 Introduction The mathematical analysis of shapes has become a subject of growing interest in the past few decades, and has motivated the development of eﬃcient image acquisition and segmentation meth- ods, with applications to many domains, including computational anatomy and object recognition. The general purpose of shape analysis is to compare two (or more) shapes in a way that takes into account their geometric properties. Two shapes can be very similar from a human’s point of view, like a circle and an ellipse, but very diﬀerent from a computer’s automated perspective. In Shape Deformation Analysis, one optimizes a deformation mapping one shape onto the other and bases the analysis on its properties. This of course implies that a cost has been assigned to every possible deformation of a shape, the design of this cost function being a crucial step in the method. This approach has been used extensively in the analysis of anatomical organs from medical images (see [15]). In this framework, a powerful and convenient approach represents deformations as ﬂows of diﬀeomorphisms generated by time-dependent vector ﬁelds [12, 28, 29]. Indeed, when considering the studied shapes as embedded in a real vector space IRd, deformations of the whole space, like diﬀeomorphisms, induce deformations of the shape itself. The set of all possible deformations is then deﬁned as the set of ﬂows of time-dependent vector ﬁelds of a Hilbert space V , called space of ”inﬁnitesimal transformations”, which is a subset of the space of all smooth bounded vector ﬁelds on IRd. This point of view has several interesting features, not the least of which being that the space of possible deformations is a well-deﬁned subgroup of the group of diﬀeomorphisms, equipped with a structure similar to the one of a right-invariant sub-Riemannian metric [8, 24]. This framework has led to the development of a family of registration algorithms called Large Deformation Dif- feomorphic Metric Mapping (LDDMM), in which the correspondence between two shapes comes from the minimization of an objective functional deﬁned as a sum of two terms [6, 7, 19, 22, 23]. The ﬁrst term takes into account the cost of the deformation, deﬁned as the integral of the squared norm of the time-dependent vector ﬁeld from which it arises. In a way, it is the total kinetic energy of the deformation. The second term is a data attachment penalizing the diﬀerence between the deformed shape and a target. An appropriate class of Hilbert spaces of vector ﬁelds for V is the one of reproducing kernel Hilbert spaces (in short, RKHS) [5], because they provide very simple solutions to the spline interpolation problem when the shape is given by a set of landmarks [32, 34], which is an important special case since it includes most practical situations after discretization. This framework allows one to use tools from Riemannian geometry [32], along with classical results from the theory of Lie groups equipped with right-invariant metrics [3, 4, 17, 21, 34]. These existing approaches provide an account for some of the geometric information in the shape, like singularities for example. However, they do not consider other intrinsic properties of the studied shape, which can also depend on the nature of the object represented by the shape. For example, for landmarks representing articulations of a robotic arm, the deformation can be searched so as to preserve the distance between certain landmarks. For cardiac motions, it may be relevant to consider deformations of the shape assuming that the movement only comes from a force applied only along the ﬁber structure of the muscle. In other words, it may be interesting to constrain the possible deformations (by considering non-holonomic constraints) in order to better ﬁt the model. In order to take into account such constraints in shape deformation problems, we propose to model these problems within the framework of optimal control theory, where the control system would model the evolution of the deformation and the control would be the time-dependent vector ﬁeld (see preliminary ideas in [32]). The purpose of this paper is to develop the point of view of optimal control for shape deformation analysis as comprehensively as possible. We will show the relevance of this framework, in particular 3

text: Introduction

text: The

text: mathematical

text: analysis

text: of

text: shapes

text: has

text: become

text: a

text: subject

text: of

text: growing

text: interest

text: in

text: the

text: past

text: few

text: decades,

text: and

text: has

text: motivated

text: the

text: development

text: of

text: eﬃcient

text: image

text: acquisition

text: and

text: segmentation

text: meth-

text: ods,

text: with

text: applications

text: to

text: many

text: domains,

text: including

text: computational

text: anatomy

text: and

text: object

text: recognition.

text: The

text: general

text: purpose

text: of

text: shape

text: analysis

text: is

text: to

text: compare

text: two

text: (or

text: more)

text: shapes

text: in

text: a

text: way

text: that

text: takes

text: into

text: account

text: their

text: geometric

text: properties.

text: Two

text: shapes

text: can

text: be

text: very

text: similar

text: from

text: a

text: human’s

text: point

text: of

text: view,

text: like

text: a

text: circle

text: and

text: an

text: ellipse,

text: but

text: very

text: diﬀerent

text: from

text: a

text: computer’s

text: automated

text: perspective.

text: In

text: Shape

text: Deformation

text: Analysis,

text: one

text: optimizes

text: a

text: deformation

text: mapping

text: one

text: shape

text: onto

text: the

text: other

text: and

text: bases

text: the

text: analysis

text: on

text: its

text: properties.

text: This

text: of

text: course

text: implies

text: that

text: a

text: cost

text: has

text: been

text: assigned

text: to

text: every

text: possible

text: deformation

text: of

text: a

text: shape,

text: the

text: design

text: of

text: this

text: cost

text: function

text: being

text: a

text: crucial

text: step

text: in

text: the

text: method.

text: This

text: approach

text: has

text: been

text: used

text: extensively

text: in

text: the

text: analysis

text: of

text: anatomical

text: organs

text: from

text: medical

text: images

text: (see

text: In

text: this

text: framework,

text: a

text: powerful

text: and

text: convenient

text: approach

text: represents

text: deformations

text: as

text: ﬂows

text: of

text: diﬀeomorphisms

text: generated

text: by

text: time-dependent

text: vector

text: ﬁelds

text: Indeed,

text: when

text: considering

text: the

text: studied

text: shapes

text: as

text: embedded

text: in

text: a

text: real

text: vector

text: space

text: IRd,

text: deformations

text: of

text: the

text: whole

text: space,

text: like

text: diﬀeomorphisms,

text: induce

text: deformations

text: of

text: the

text: shape

text: itself.

text: The

text: set

text: of

text: all

text: possible

text: deformations

text: is

text: then

text: deﬁned

text: as

text: the

text: set

text: of

text: ﬂows

text: of

text: time-dependent

text: vector

text: ﬁelds

text: of

text: a

text: Hilbert

text: space

text: V

text: called

text: space

text: of

text: ”inﬁnitesimal

text: transformations”,

text: which

text: is

text: a

text: subset

text: of

text: the

text: space

text: of

text: all

text: smooth

text: bounded

text: vector

text: ﬁelds

text: on

text: IRd.

text: This

text: point

text: of

text: view

text: has

text: several

text: interesting

text: features,

text: not

text: the

text: least

text: of

text: which

text: being

text: that

text: the

text: space

text: of

text: possible

text: deformations

text: is

text: a

text: well-deﬁned

text: subgroup

text: of

text: the

text: group

text: of

text: diﬀeomorphisms,

text: equipped

text: with

text: a

text: structure

text: similar

text: to

text: the

text: one

text: of

text: a

text: right-invariant

text: sub-Riemannian

text: metric

text: This

text: framework

text: has

text: led

text: to

text: the

text: development

text: of

text: a

text: family

text: of

text: registration

text: algorithms

text: called

text: Large

text: Deformation

text: Dif-

text: feomorphic

text: Metric

text: Mapping

text: (LDDMM),

text: in

text: which

text: the

text: correspondence

text: between

text: two

text: shapes

text: comes

text: from

text: the

text: minimization

text: of

text: an

text: objective

text: functional

text: deﬁned

text: as

text: a

text: sum

text: of

text: two

text: terms

text: The

text: ﬁrst

text: term

text: takes

text: into

text: account

text: the

text: cost

text: of

text: the

text: deformation,

text: deﬁned

text: as

text: the

text: integral

text: of

text: the

text: squared

text: norm

text: of

text: the

text: time-dependent

text: vector

text: ﬁeld

text: from

text: which

text: it

text: arises.

text: In

text: a

text: way,

text: it

text: is

text: the

text: total

text: kinetic

text: energy

text: of

text: the

text: deformation.

text: The

text: second

text: term

text: is

text: a

text: data

text: attachment

text: penalizing

text: the

text: diﬀerence

text: between

text: the

text: deformed

text: shape

text: and

text: a

text: target.

text: An

text: appropriate

text: class

text: of

text: Hilbert

text: spaces

text: of

text: vector

text: ﬁelds

text: for

text: V

text: is

text: the

text: one

text: of

text: reproducing

text: kernel

text: Hilbert

text: spaces

text: (in

text: short,

text: RKHS)

text: because

text: they

text: provide

text: very

text: simple

text: solutions

text: to

text: the

text: spline

text: interpolation

text: problem

text: when

text: the

text: shape

text: is

text: given

text: by

text: a

text: set

text: of

text: landmarks

text: which

text: is

text: an

text: important

text: special

text: case

text: since

text: it

text: includes

text: most

text: practical

text: situations

text: after

text: discretization.

text: This

text: framework

text: allows

text: one

text: to

text: use

text: tools

text: from

text: Riemannian

text: geometry

text: along

text: with

text: classical

text: results

text: from

text: the

text: theory

text: of

text: Lie

text: groups

text: equipped

text: with

text: right-invariant

text: metrics

text: These

text: existing

text: approaches

text: provide

text: an

text: account

text: for

text: some

text: of

text: the

text: geometric

text: information

text: in

text: the

text: shape,

text: like

text: singularities

text: for

text: example.

text: However,

text: they

text: do

text: not

text: consider

text: other

text: intrinsic

text: properties

text: of

text: the

text: studied

text: shape,

text: which

text: can

text: also

text: depend

text: on

text: the

text: nature

text: of

text: the

text: object

text: represented

text: by

text: the

text: shape.

text: For

text: example,

text: for

text: landmarks

text: representing

text: articulations

text: of

text: a

text: robotic

text: arm,

text: the

text: deformation

text: can

text: be

text: searched

text: so

text: as

text: to

text: preserve

text: the

text: distance

text: between

text: certain

text: landmarks.

text: For

text: cardiac

text: motions,

text: it

text: may

text: be

text: relevant

text: to

text: consider

text: deformations

text: of

text: the

text: shape

text: assuming

text: that

text: the

text: movement

text: only

text: comes

text: from

text: a

text: force

text: applied

text: only

text: along

text: the

text: ﬁber

text: structure

text: of

text: the

text: muscle.

text: In

text: other

text: words,

text: it

text: may

text: be

text: interesting

text: to

text: constrain

text: the

text: possible

text: deformations

text: (by

text: considering

text: non-holonomic

text: constraints)

text: in

text: order

text: to

text: better

text: ﬁt

text: the

text: model.

text: In

text: order

text: to

text: take

text: into

text: account

text: such

text: constraints

text: in

text: shape

text: deformation

text: problems,

text: we

text: propose

text: to

text: model

text: these

text: problems

text: within

text: the

text: framework

text: of

text: optimal

text: control

text: theory,

text: where

text: the

text: control

text: system

text: would

text: model

text: the

text: evolution

text: of

text: the

text: deformation

text: and

text: the

text: control

text: would

text: be

text: the

text: time-dependent

text: vector

text: ﬁeld

text: (see

text: preliminary

text: ideas

text: in

text: The

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: develop

text: the

text: point

text: of

text: view

text: of

text: optimal

text: control

text: for

text: shape

text: deformation

text: analysis

text: as

text: comprehensively

text: as

text: possible.

text: We

text: will

text: show

text: the

text: relevance

text: of

text: this

text: framework,

text: in

text: particular


text: C V be an RKHS of vector ﬁelds of class 1, acting on M as described in Section . Let T be C 0C a submanifold of S of class 1. Set Y = 0(T, IRd), and consider the kinetic constraints C : × → ◦ M V Y deﬁned by C v = v q| . If q is diﬀerentiable along T, then no nondiﬀerentiable map ∈ q T f Y is in Range(C ). q ⊥ Remark 13. It is possible to replace C with the orthogonal projection on Null(C ) , which is q (cid:83) q automatically surjective. However in this case the constraints become ﬁber-valued, and for the proof of our theorem to remain valid, one needs to assume that there exists a Hilbert space V such ∈ ∼ 1× that, for every q M, there exists a neighborhood U of q in M such that ∈ Null(C ) U V . 0 0 q U q 1 Remark 14. An important consequence of the surjectivity of C is that the norm on Y is equivalent ∈ q ∗ ∗ → to the Hilbert norm induced from V by C , for every q M. The operator C K C : Y Y is q q V q ∗ the isometry associated with this norm. In particular Y must be reﬂexive, and hence L2(0, 1; Y ) = ∗ L2(0, 1; Y ) (whereas we only have an inclusion for general Banach spaces). Remark 15. Theorem 2 withstands several generalizations. For instance it remains valid whenever C we consider nonlinear constraints C(q, v) = 0 and a general Lagrangian L of class 1 without any C estimates and a nonlinear control system q˙ = ξ(q, v), provided that ξ is of class 1 and that ∈ ∈ ∞ ∂ C(q(t), v(t)) is surjective for every t [0, 1]. However, this requires to consider v L (0, 1; V ) v and makes the proof of the regularity of the Lagrange multipliers slightly more involved. Before proving Theorem 2, it can be noted that many versions of the PMP can be found in the existing literature for inﬁnite-dimensional optimal control problems – for instance, with dynamics consisting of partial diﬀerential equations, and however, most of the time, without constraint on the state. Versions of the PMP with state constraints can also be found in the literature (see the survey [16]), most of the time in ﬁnite dimension, and, for the very few of them existing in inﬁnite dimension, under the additional assumption that the constraints are of ﬁnite codimension. To the best of our knowledge, no version does exist that would cover our speciﬁc framework, concerning shape spaces, group actions, with an inﬁnite number of constraints on the acting diﬀeomorphisms. The proof that we provide hereafter contains some subtleties such as Lemma 4, and hence Theorem 2 is a nontrivial extension of the usual PMP. × → × Proof of Theorem 2. We deﬁne the mapping Γ : H1 (0, 1; M) L2(0, 1; V ) L2(0, 1; X) L2(0, 1; Y ) q0 − by Γ(q, v) = (Γ (q, v), Γ (q, v)) with Γ (q, v)(t) = q˙(t) ξ v(t) and Γ (q, v)(t) = C v(t) for ∈1 2 1 q(t) 2 q(t) almost every t [0, 1]. The mapping Γ stands for the constraints imposed to the unknowns of the optimal control problem. × → C The functional J : Hq10(cid:90)(0, 1; M) L2(0, 1; V ) IR and the mapping Γ are of class 1, and their respective diﬀerentials at some point (q, v) are given by 1 (cid:104) (cid:105) (cid:104) (cid:105) dJ .(δq, δv) = ( ∂ L(q(t), v(t)), δq(t) ∗ + ∂ L(q(t), v(t)), δv(t) ∗ ) dt (q,v) q X ,X v V ,V 0 (cid:104) (cid:105) + dg , δq(1) ∗ , q(1) X ,X ∈ × for all (δq, δv) H1(0, 1; X) L2(0, 1; V ), and dΓ = (dΓ , dΓ ) with 0 (q,v) 1(q,v) 2(q,v) − − (dΓ .(δq, δv))(t) = δ˙q(t) ∂ ξ v(t).δq(t) ξ δv(t), 1(q,v) q q(t) q(t) (dΓ .(δq, δv))(t) = ∂ C v(t).δq(t) + C δv(t), 2(q,v) q q(t) q(t) ∈ for almost every t [0, 1]. ∈ × Lemma 3. For every (q, v) H1 (0, 1; M) L2(0, 1; V ), the linear continuous mapping dΓ : × → q0 × (q,v) H1(0, 1; X) L2(0, 1; V ) L2(0, 1; X) L2(0, 1; Y ) is surjective. Moreover the mapping ∂ Γ : 0 → q 1(q,v) H1(0, 1; M) L2(0, 1; X) is an isomorphism. 0 21

text: C

text: V

text: be

text: an

text: RKHS

text: of

text: vector

text: ﬁelds

text: of

text: class

text: acting

text: on

text: M

text: as

text: described

text: in

text: Section

text: Let

text: T

text: be

text: C

text: a

text: submanifold

text: of

text: S

text: of

text: class

text: Set

text: Y

text: IRd),

text: and

text: consider

text: the

text: kinetic

text: constraints

text: C

text: M

text: V

text: Y

text: deﬁned

text: by

text: C

text: v

text: v

text: q|

text: If

text: q

text: is

text: diﬀerentiable

text: along

text: T,

text: then

text: no

text: nondiﬀerentiable

text: map

text: q

text: T

text: f

text: Y

text: is

text: in

text: Range(C

text: q

text: Remark

text: It

text: is

text: possible

text: to

text: replace

text: C

text: with

text: the

text: orthogonal

text: projection

text: on

text: Null(C

text: which

text: is

text: q

text: q

text: automatically

text: surjective.

text: However

text: in

text: this

text: case

text: the

text: constraints

text: become

text: ﬁber-valued,

text: and

text: for

text: the

text: proof

text: of

text: our

text: theorem

text: to

text: remain

text: valid,

text: one

text: needs

text: to

text: assume

text: that

text: there

text: exists

text: a

text: Hilbert

text: space

text: V

text: such

text: 1×

text: that,

text: for

text: every

text: q

text: M,

text: there

text: exists

text: a

text: neighborhood

text: U

text: of

text: q

text: in

text: M

text: such

text: that

text: Null(C

text: U

text: V

text: q

text: U

text: q

text: Remark

text: An

text: important

text: consequence

text: of

text: the

text: surjectivity

text: of

text: C

text: is

text: that

text: the

text: norm

text: on

text: Y

text: is

text: equivalent

text: q

text: to

text: the

text: Hilbert

text: norm

text: induced

text: from

text: V

text: by

text: C

text: for

text: every

text: q

text: M.

text: The

text: operator

text: C

text: K

text: C

text: Y

text: Y

text: is

text: q

text: q

text: V

text: q

text: the

text: isometry

text: associated

text: with

text: this

text: norm.

text: In

text: particular

text: Y

text: must

text: be

text: reﬂexive,

text: and

text: hence

text: Y

text: Y

text: (whereas

text: we

text: only

text: have

text: an

text: inclusion

text: for

text: general

text: Banach

text: spaces).

text: Remark

text: Theorem

text: withstands

text: several

text: generalizations.

text: For

text: instance

text: it

text: remains

text: valid

text: whenever

text: C

text: we

text: consider

text: nonlinear

text: constraints

text: C(q,

text: v)

text: and

text: a

text: general

text: Lagrangian

text: L

text: of

text: class

text: without

text: any

text: C

text: estimates

text: and

text: a

text: nonlinear

text: control

text: system

text: q˙

text: ξ(q,

text: v),

text: provided

text: that

text: ξ

text: is

text: of

text: class

text: and

text: that

text: C(q(t),

text: v(t))

text: is

text: surjective

text: for

text: every

text: t

text: However,

text: this

text: requires

text: to

text: consider

text: v

text: L

text: V

text: v

text: and

text: makes

text: the

text: proof

text: of

text: the

text: regularity

text: of

text: the

text: Lagrange

text: multipliers

text: slightly

text: more

text: involved.

text: Before

text: proving

text: Theorem

text: it

text: can

text: be

text: noted

text: that

text: many

text: versions

text: of

text: the

text: PMP

text: can

text: be

text: found

text: in

text: the

text: existing

text: literature

text: for

text: inﬁnite-dimensional

text: optimal

text: control

text: problems

text: for

text: instance,

text: with

text: dynamics

text: consisting

text: of

text: partial

text: diﬀerential

text: equations,

text: and

text: however,

text: most

text: of

text: the

text: time,

text: without

text: constraint

text: on

text: the

text: state.

text: Versions

text: of

text: the

text: PMP

text: with

text: state

text: constraints

text: can

text: also

text: be

text: found

text: in

text: the

text: literature

text: (see

text: the

text: survey

text: most

text: of

text: the

text: time

text: in

text: ﬁnite

text: dimension,

text: and,

text: for

text: the

text: very

text: few

text: of

text: them

text: existing

text: in

text: inﬁnite

text: dimension,

text: under

text: the

text: additional

text: assumption

text: that

text: the

text: constraints

text: are

text: of

text: ﬁnite

text: codimension.

text: To

text: the

text: best

text: of

text: our

text: knowledge,

text: no

text: version

text: does

text: exist

text: that

text: would

text: cover

text: our

text: speciﬁc

text: framework,

text: concerning

text: shape

text: spaces,

text: group

text: actions,

text: with

text: an

text: inﬁnite

text: number

text: of

text: constraints

text: on

text: the

text: acting

text: diﬀeomorphisms.

text: The

text: proof

text: that

text: we

text: provide

text: hereafter

text: contains

text: some

text: subtleties

text: such

text: as

text: Lemma

text: and

text: hence

text: Theorem

text: is

text: a

text: nontrivial

text: extension

text: of

text: the

text: usual

text: PMP.

text: Proof

text: of

text: Theorem

text: We

text: deﬁne

text: the

text: mapping

text: Γ

text: M)

text: V

text: X)

text: Y

text: by

text: Γ(q,

text: v)

text: (Γ

text: (q,

text: v),

text: Γ

text: (q,

text: v))

text: with

text: Γ

text: (q,

text: v)(t)

text: q˙(t)

text: ξ

text: v(t)

text: and

text: Γ

text: (q,

text: v)(t)

text: C

text: v(t)

text: for

text: ∈1

text: q(t)

text: q(t)

text: almost

text: every

text: t

text: The

text: mapping

text: Γ

text: stands

text: for

text: the

text: constraints

text: imposed

text: to

text: the

text: unknowns

text: of

text: the

text: optimal

text: control

text: problem.

text: C

text: The

text: functional

text: J

text: M)

text: V

text: IR

text: and

text: the

text: mapping

text: Γ

text: are

text: of

text: class

text: and

text: their

text: respective

text: diﬀerentials

text: at

text: some

text: point

text: (q,

text: v)

text: are

text: given

text: by

text: dJ

text: .(δq,

text: δv)

text: L(q(t),

text: v(t)),

text: δq(t)

text: L(q(t),

text: v(t)),

text: δv(t)

text: dt

text: (q,v)

text: q

text: X

text: ,X

text: v

text: V

text: ,V

text: dg

text: X

text: ,X

text: for

text: all

text: (δq,

text: δv)

text: X)

text: V

text: and

text: dΓ

text: (dΓ

text: dΓ

text: with

text: (q,v)

text: (dΓ

text: .(δq,

text: δv))(t)

text: δ˙q(t)

text: ξ

text: v(t).δq(t)

text: ξ

text: δv(t),

text: q

text: q(t)

text: q(t)

text: (dΓ

text: .(δq,

text: δv))(t)

text: C

text: v(t).δq(t)

text: C

text: δv(t),

text: q

text: q(t)

text: q(t)

text: for

text: almost

text: every

text: t

text: Lemma

text: For

text: every

text: (q,

text: v)

text: M)

text: V

text: the

text: linear

text: continuous

text: mapping

text: dΓ

text: (q,v)

text: X)

text: V

text: X)

text: Y

text: is

text: surjective.

text: Moreover

text: the

text: mapping

text: Γ

text: q

text: M)

text: X)

text: is

text: an

text: isomorphism.


text: (cid:90) ∗ Lemma 4. We can iden(cid:10)tify (cid:11)p with an element of L2([(cid:10)0, 1], X ),(cid:11)so that 1 p, r ∗ = p(t), r(t) ∗ dt, L2(0,1;X ),L2(0,1;X) X ,X 0 (cid:82) ∈ for every r L2(0, 1; X). ∈ ∈ Proof of Lemma 4. For every r L2(0, 1; X), we deﬁne δq H1(0, 1; X) by δq(s) = s r(t) dt for ∈ 0 ∈ 0 ∗ every s [0, 1] (Bochner integral in the Banach space X), so that r = δ˙q. Deﬁning α L2(0, 1; X) by (cid:10) (cid:11) (cid:10) (cid:11) (cid:90) (cid:10) (cid:11) α, f L2(0,1;X)∗,L2(0,1;X) = p, ∂qξqv.f L2(0,1;X)∗,L2(0,1;X) − 1 ∗ (23) ∂qL(q(t), v(t)) + (∂qCq(t)v(t)) λ(t), f(t) X∗,X dt, 0 (cid:10) ∈(cid:11) (cid:10) (cid:11) (cid:10) (cid:11) for every f L2(0, 1; X), and taking δv = 0 in (22), we get − p, r L2(0,1;X)∗,L2(0,1;X) = α, δq L2(0,1;X)∗,L2(0,1;X) dgq(1), δq(1) X∗,X. (cid:82) (24) (cid:82) Let us express δq in another way with respect to r. By deﬁn(cid:82)ition, one has δq(s) = s r(t) dt ∈ 0 1 for every s [0, 1], and this can be also written as δq(s) = χ (s)r(t) dt, with χ (s) = 1 ∈ (cid:82) 0 1[t,1] [t,1] whenever s [t, 1] and 0 otherwise. In other words, one has δq = χ r(t) dt (Bochner integral). ∈ → 0 [t,1] For every t [0, 1], we deﬁne the operator A : X L2(0, 1; X) by A x = χ x. It is clearly t t [t,1] linea(cid:10)r an(cid:11)d continuous. Then, w(cid:68)e h(cid:90)ave δq = 1 A(cid:69)(r(t)) dt, and therefor(cid:10)e, using(cid:90)(24), (cid:11) 0 t 1 − 1 p, r L2(0,1;X)∗,L2(0,1;X) = α, 0 At(r(t)) dt L2(0,1;X)∗,L2(0,1;X) dgq(1), 0 r(t) dt X∗,X. (cid:90) (cid:90) Now(cid:10), int(cid:11)erchanging the Bochner in(cid:10)tegrals and(cid:11)the linear forms, we infer t(cid:10)hat (cid:11) 1 − 1 p, r L2(0,1;X)∗,L2(0,1;X) = α, At(r(t)) L2(0,1;X)∗,L2(0,1;X) dt dgq(1), r(t) X∗,X dt, 0 0 ∗ ∗ →(cid:90) ∗ and then, using the (cid:10)adjo(cid:11)int A : L2(0, 1; X) X(cid:10), we get (cid:11) t 1 ∗ − p, r L2(0,1;X)∗,L2(0,1;X) = At α dgq(1), r(t) X∗,X dt. 0 ∈ Since this identity holds true for every r L2(0, 1; X), it follows that p can be identiﬁed with an ∗ ∗ − ∈ element of L2(0, 1; X ), still denoted by p, with p(t) = A α dg for almost every t [0, 1]. t q(1) ∈ Still using the notation introduced in the proof of Lemma 4, now that we know that p ∗ ∗ L2(0, 1; X ), we infer from (23) that α can as well be identiﬁed with an element of L2([0, 1], X ), with ∗ − − ∗ α(t) = (∂ ξ v(t)) p(t) ∂ L(q(t), v(t)) (∂ C v(t)) λ(t), q q(t) q q q(t) ∈ for almost every t [0, 1]. Note that α(t) = ∂ H(q(t), p(t), v(t), λ(t)), where the Hamiltonian H is q deﬁned by (18). 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: We

text: can

text: with

text: an

text: element

text: of

text: X

text: that

text: p,

text: r

text: p(t),

text: r(t)

text: dt,

text: X

text: ,X

text: for

text: every

text: r

text: X).

text: Proof

text: of

text: Lemma

text: For

text: every

text: r

text: X),

text: we

text: deﬁne

text: δq

text: X)

text: by

text: δq(s)

text: s

text: r(t)

text: dt

text: for

text: every

text: s

text: (Bochner

text: integral

text: in

text: the

text: Banach

text: space

text: X),

text: so

text: that

text: r

text: δ˙q.

text: Deﬁning

text: α

text: X)

text: by

label: equation

text: α,

text: f

text: p,

text: ∂qξqv.f

text: ∂qL(q(t),

text: v(t))

text: (∂qCq(t)v(t))

text: λ(t),

text: f(t)

text: X∗,X

text: dt,

text: for

text: every

text: f

text: X),

text: and

text: taking

text: δv

text: in

text: we

text: get

text: p,

text: r

text: α,

text: δq

text: X∗,X.

text: Let

text: us

text: express

text: δq

text: in

text: another

text: way

text: with

text: respect

text: to

text: r.

text: By

text: one

text: has

text: δq(s)

text: s

text: r(t)

text: dt

text: for

text: every

text: s

text: and

text: this

text: can

text: be

text: also

text: written

text: as

text: δq(s)

text: χ

text: (s)r(t)

text: dt,

text: with

text: χ

text: (s)

text: whenever

text: s

text: [t,

text: and

text: otherwise.

text: In

text: other

text: words,

text: one

text: has

text: δq

text: χ

text: r(t)

text: dt

text: (Bochner

text: integral).

text: For

text: every

text: t

text: we

text: deﬁne

text: the

text: operator

text: A

text: X

text: X)

text: by

text: A

text: x

text: χ

text: x.

text: It

text: is

text: clearly

text: t

text: t

text: continuous.

text: Then,

text: δq

text: dt,

text: and

text: t

text: p,

text: r

text: α,

text: At(r(t))

text: dt

text: r(t)

text: dt

text: X∗,X.

text: the

text: Bochner

text: linear

text: forms,

text: we

text: infer

text: p,

text: r

text: α,

text: At(r(t))

text: dt

text: r(t)

text: X∗,X

text: dt,

text: and

text: then,

text: using

text: the

text: A

text: X)

text: we

text: get

text: t

text: p,

text: r

text: At

text: α

text: r(t)

text: X∗,X

text: dt.

text: Since

text: this

text: identity

text: holds

text: true

text: for

text: every

text: r

text: X),

text: it

text: follows

text: that

text: p

text: can

text: be

text: identiﬁed

text: with

text: an

text: element

text: of

text: X

text: still

text: denoted

text: by

text: p,

text: with

text: p(t)

text: A

text: α

text: dg

text: for

text: almost

text: every

text: t

text: t

text: Still

text: using

text: the

text: notation

text: introduced

text: in

text: the

text: proof

text: of

text: Lemma

text: now

text: that

text: we

text: know

text: that

text: p

text: X

text: we

text: infer

text: from

text: that

text: α

text: can

text: as

text: well

text: be

text: identiﬁed

text: with

text: an

text: element

text: of

text: X

text: with

text: α(t)

text: ξ

text: v(t))

text: p(t)

text: L(q(t),

text: v(t))

text: C

text: v(t))

text: λ(t),

text: q

text: q(t)

text: q

text: q

text: q(t)

text: for

text: almost

text: every

text: t

text: Note

text: that

text: α(t)

text: H(q(t),

text: p(t),

text: v(t),

text: λ(t)),

text: where

text: the

text: Hamiltonian

text: H

text: is

text: q

text: deﬁned

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: because it can be used to model constrained shapes among many other applications. Indeed, a lot of tools have been developed in control theory for solving optimal control problems with or without constraints. The well-known Pontryagin maximum principle (in short PMP, see [26]) provides ﬁrst-order conditions for optimality in the form of Hamiltonian extremal equations with a maximization condition permitting the computation of the optimal control. It has been generalized in many ways, and a large number of variants or improvements have been made over the past decades, with particular eﬀorts in order to be able to address optimal control problems involving general state/control constraints (see the survey article [16] and the many references therein). The analysis is, however, mainly done in ﬁnite dimension. Since shape analysis has a natural setting in inﬁnite dimension (indeed, in 2D, the shape space is typically a space of smooth curves in IR2), we need to derive an appropriate inﬁnite-dimensional variant of the PMP for constrained problems. Such a variant is nontrivial and nonstandard, given that our constrained shape analysis problems generally involve an inﬁnite number of equality constraints. Such a PMP will allow us to derive in a rigorous geometric setting the (constrained) geodesic equations that must be satisﬁed by the optimal deformations. Moreover, modeling shape deformation problems within the framework of optimal control the- ory can inherit from the many numerical methods in optimal control and thus lead to new algo- rithms in shape analysis. The paper is organized as follows. Section is devoted to modeling shape deformation problems with optimal control. We ﬁrst brieﬂy describe, in Section , the framework of diﬀeomorphic deformations arising from the inte- gration of time-dependent vector ﬁelds belonging to a given RKHS, and recall some properties of RKHS’s of vector ﬁelds. In Section we introduce the action of diﬀeomorphisms on a shape space, and we model and deﬁne the optimal control problem on diﬀeomorphisms which is at the heart of the present study, where the control system stands for the evolving deformation and the minimization runs over all possible time-dependent vector ﬁelds attached to a given RKHS and satisfying some constraints. We prove that, under weak assumptions, this problem is well posed and has at least one solution (Theorem 1). Since the RKHS is in general only known through its kernel, we then provide a kernel formulation of the optimal control problem and we analyze the equivalence between both problems. In Section we investigate in our framework two important variants of shape spaces, which are lifted shapes and multi-shapes. Section is devoted to the study of ﬁnite-dimensional approximations of the optimal control problem. Section contains a proof of Theorem 1. Section is dedicated to the derivation of the constrained geodesic equations in shape spaces, that must be satisﬁed by optimal deformations. We ﬁrst establish in Section an inﬁnite dimensional variant of the PMP which is adapted to our setting (Theorem 2). As an application, we derive in Section the geodesic equations in shape spaces (Theorem 3), in a geometric setting, and show that they can be written as a Hamiltonian system. In Section , we design some algorithms in order to solve the optimal control problem modeling the shape deformation problem. Problems without constraints are ﬁrst analyzed in Section , and we recover some already known algorithms used in unconstrained shape spaces, however with a more general point of view. We are thus able to extend and generalize existing methods. Problems with constraints are investigated in Section in view of solving constrained matching problems. We analyze in particular the augmented Lagrangian algorithm, and we also design a method based on shooting. In Section we provide numerical examples, investigating ﬁrst a matching problem with constant total volume, and then a multishape matching problem. 4

text: because

text: it

text: can

text: be

text: used

text: to

text: model

text: constrained

text: shapes

text: among

text: many

text: other

text: applications.

text: Indeed,

text: a

text: lot

text: of

text: tools

text: have

text: been

text: developed

text: in

text: control

text: theory

text: for

text: solving

text: optimal

text: control

text: problems

text: with

text: or

text: without

text: constraints.

text: The

text: well-known

text: Pontryagin

text: maximum

text: principle

text: (in

text: short

text: PMP,

text: see

text: provides

text: ﬁrst-order

text: conditions

text: for

text: optimality

text: in

text: the

text: form

text: of

text: Hamiltonian

text: extremal

text: equations

text: with

text: a

text: maximization

text: condition

text: permitting

text: the

text: computation

text: of

text: the

text: optimal

text: control.

text: It

text: has

text: been

text: generalized

text: in

text: many

text: ways,

text: and

text: a

text: large

text: number

text: of

text: variants

text: or

text: improvements

text: have

text: been

text: made

text: over

text: the

text: past

text: decades,

text: with

text: particular

text: eﬀorts

text: in

text: order

text: to

text: be

text: able

text: to

text: address

text: optimal

text: control

text: problems

text: involving

text: general

text: state/control

text: constraints

text: (see

text: the

text: survey

text: article

text: and

text: the

text: many

text: references

text: therein).

text: The

text: analysis

text: is,

text: however,

text: mainly

text: done

text: in

text: ﬁnite

text: dimension.

text: Since

text: shape

text: analysis

text: has

text: a

text: natural

text: setting

text: in

text: inﬁnite

text: dimension

text: (indeed,

text: in

text: the

text: shape

text: space

text: is

text: typically

text: a

text: space

text: of

text: smooth

text: curves

text: in

text: we

text: need

text: to

text: derive

text: an

text: appropriate

text: inﬁnite-dimensional

text: variant

text: of

text: the

text: PMP

text: for

text: constrained

text: problems.

text: Such

text: a

text: variant

text: is

text: nontrivial

text: and

text: nonstandard,

text: given

text: that

text: our

text: constrained

text: shape

text: analysis

text: problems

text: generally

text: involve

text: an

text: inﬁnite

text: number

text: of

text: equality

text: constraints.

text: Such

text: a

text: PMP

text: will

text: allow

text: us

text: to

text: derive

text: in

text: a

text: rigorous

text: geometric

text: setting

text: the

text: (constrained)

text: geodesic

text: equations

text: that

text: must

text: be

text: satisﬁed

text: by

text: the

text: optimal

text: deformations.

text: Moreover,

text: modeling

text: shape

text: deformation

text: problems

text: within

text: the

text: framework

text: of

text: optimal

text: control

text: the-

text: ory

text: can

text: inherit

text: from

text: the

text: many

text: numerical

text: methods

text: in

text: optimal

text: control

text: and

text: thus

text: lead

text: to

text: new

text: algo-

text: rithms

text: in

text: shape

text: analysis.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: is

text: devoted

text: to

text: modeling

text: shape

text: deformation

text: problems

text: with

text: optimal

text: control.

text: We

text: ﬁrst

text: brieﬂy

text: describe,

text: in

text: Section

text: the

text: framework

text: of

text: diﬀeomorphic

text: deformations

text: arising

text: from

text: the

text: inte-

text: gration

text: of

text: time-dependent

text: vector

text: ﬁelds

text: belonging

text: to

text: a

text: given

text: RKHS,

text: and

text: recall

text: some

text: properties

text: of

text: RKHS’s

text: of

text: vector

text: ﬁelds.

text: In

text: Section

text: we

text: introduce

text: the

text: action

text: of

text: diﬀeomorphisms

text: on

text: a

text: shape

text: space,

text: and

text: we

text: model

text: and

text: deﬁne

text: the

text: optimal

text: control

text: problem

text: on

text: diﬀeomorphisms

text: which

text: is

text: at

text: the

text: heart

text: of

text: the

text: present

text: study,

text: where

text: the

text: control

text: system

text: stands

text: for

text: the

text: evolving

text: deformation

text: and

text: the

text: minimization

text: runs

text: over

text: all

text: possible

text: time-dependent

text: vector

text: ﬁelds

text: attached

text: to

text: a

text: given

text: RKHS

text: and

text: satisfying

text: some

text: constraints.

text: We

text: prove

text: that,

text: under

text: weak

text: assumptions,

text: this

text: problem

text: is

text: well

text: posed

text: and

text: has

text: at

text: least

text: one

text: solution

text: (Theorem

text: Since

text: the

text: RKHS

text: is

text: in

text: general

text: only

text: known

text: through

text: its

text: kernel,

text: we

text: then

text: provide

text: a

text: kernel

text: formulation

text: of

text: the

text: optimal

text: control

text: problem

text: and

text: we

text: analyze

text: the

text: equivalence

text: between

text: both

text: problems.

text: In

text: Section

text: we

text: investigate

text: in

text: our

text: framework

text: two

text: important

text: variants

text: of

text: shape

text: spaces,

text: which

text: are

text: lifted

text: shapes

text: and

text: multi-shapes.

text: Section

text: is

text: devoted

text: to

text: the

text: study

text: of

text: ﬁnite-dimensional

text: approximations

text: of

text: the

text: optimal

text: control

text: problem.

text: Section

text: contains

text: a

text: proof

text: of

text: Theorem

text: Section

text: is

text: dedicated

text: to

text: the

text: derivation

text: of

text: the

text: constrained

text: geodesic

text: equations

text: in

text: shape

text: spaces,

text: that

text: must

text: be

text: satisﬁed

text: by

text: optimal

text: deformations.

text: We

text: ﬁrst

text: establish

text: in

text: Section

text: an

text: inﬁnite

text: dimensional

text: variant

text: of

text: the

text: PMP

text: which

text: is

text: adapted

text: to

text: our

text: setting

text: (Theorem

text: As

text: an

text: application,

text: we

text: derive

text: in

text: Section

text: the

text: geodesic

text: equations

text: in

text: shape

text: spaces

text: (Theorem

text: in

text: a

text: geometric

text: setting,

text: and

text: show

text: that

text: they

text: can

text: be

text: written

text: as

text: a

text: Hamiltonian

text: system.

text: In

text: Section

text: we

text: design

text: some

text: algorithms

text: in

text: order

text: to

text: solve

text: the

text: optimal

text: control

text: problem

text: modeling

text: the

text: shape

text: deformation

text: problem.

text: Problems

text: without

text: constraints

text: are

text: ﬁrst

text: analyzed

text: in

text: Section

text: and

text: we

text: recover

text: some

text: already

text: known

text: algorithms

text: used

text: in

text: unconstrained

text: shape

text: spaces,

text: however

text: with

text: a

text: more

text: general

text: point

text: of

text: view.

text: We

text: are

text: thus

text: able

text: to

text: extend

text: and

text: generalize

text: existing

text: methods.

text: Problems

text: with

text: constraints

text: are

text: investigated

text: in

text: Section

text: in

text: view

text: of

text: solving

text: constrained

text: matching

text: problems.

text: We

text: analyze

text: in

text: particular

text: the

text: augmented

text: Lagrangian

text: algorithm,

text: and

text: we

text: also

text: design

text: a

text: method

text: based

text: on

text: shooting.

text: In

text: Section

text: we

text: provide

text: numerical

text: examples,

text: investigating

text: ﬁrst

text: a

text: matching

text: problem

text: with

text: constant

text: total

text: volume,

text: and

text: then

text: a

text: multishape

text: matching

text: problem.