text: (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) − αβ − β2σ2 − − ln(x) √µ + βσ2 exp βµ + xβ 1 1 erf 2(α + β) 2 σ 2 (9) If the assumptions of Hypothesis 2 are correct, it should be expected that the double Pareto log-normal distribution provides a good ﬁt to the data. However, if one or more of the assumptions in Hypothesis 2 are not valid, the ﬁt should not be a satisfactory one. The distribution of population sizes of the system of Bulgarian cities in 2004 and in 2011 are presented in Fig. 4. The dash line represents the double Pareto log-normal distribution that corresponds to the distribution of the Bulgarian cities in 2011. From the values of α and β and from the properties of the double ≈ − Pareto log-normal distribution it follows that p(x) x 0.8 , for the case of small ≈ − city sizes and p(x) x 3 for the large cities. Similar results are found for the years for which the data are available. Thus, the double Pareto log-normal distribution ﬁts well the city population size distributions for the Bulgarian city system whatever the year in this century. Thus Hypothesis 2 is conﬁrmed. 4. Concluding remarks The study of the size of populations in cities of a country has many economic applications. Apart on the decision taking about placing industrial plants or building trade centers, the city population sizes inﬂuence the eﬃcient use of resources or the possibilities of economic growth (Henderson (1986); Henderson (2001)). Thus, the investigation of the evolution of the populations in a system of cities is an important research topic. Our study reported here above has shown that for the case of the Bulgarian city system Hypothesis 1 is not conﬁrmed but Hypothesis 2 is conﬁrmed. The validity of Hypothesis 2 even gives some information about the history of settlement formations in Bulgaria. It seems that the distribution of the initial populations of the settlements was indeed log-normal and that the formation happened according to the Yule process, i.e. the new settlements have been formed on the basis of the existing older settlements nearby. Acknowledgment This work has been performed in the framework of COST Action IS1104 ”The EU in the new economic complex geography: models, tools and policy evaluation”. We ac- knowledge some support through the project ’Evolution spatiale et temporelle d’infrastructures r´egionales et ´economiques en Bulgarie et en F´ed´eration Wallonie-Bruxelles’ within the intergovernmental agreement for cooperation between the Republic of Bulgaria and la Communaut´e Fran¸caise de Belgique. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: αβ

text: β2σ2

text: ln(x)

text: √µ

text: βσ2

text: exp

text: βµ

text: xβ

text: erf

text: β)

text: σ

text: If

text: the

text: assumptions

text: of

text: Hypothesis

text: are

text: correct,

text: it

text: should

text: be

text: expected

text: that

text: the

text: double

text: Pareto

text: log-normal

text: distribution

text: provides

text: a

text: good

text: ﬁt

text: to

text: the

text: data.

text: However,

text: if

text: one

text: or

text: more

text: of

text: the

text: assumptions

text: in

text: Hypothesis

text: are

text: not

text: valid,

text: the

text: ﬁt

text: should

text: not

text: be

text: a

text: satisfactory

text: one.

text: The

text: distribution

text: of

text: population

text: sizes

text: of

text: the

text: system

text: of

text: Bulgarian

text: cities

text: in

text: and

text: in

text: are

text: presented

text: in

text: Fig.

text: The

text: dash

text: line

text: represents

text: the

text: double

text: Pareto

text: log-normal

text: distribution

text: that

text: corresponds

text: to

text: the

text: distribution

text: of

text: the

text: Bulgarian

text: cities

text: in

text: From

text: the

text: values

text: of

text: α

text: and

text: β

text: and

text: from

text: the

text: properties

text: of

text: the

text: double

text: Pareto

text: log-normal

text: distribution

text: it

text: follows

text: that

text: p(x)

text: x

text: for

text: the

text: case

text: of

text: small

text: city

text: sizes

text: and

text: p(x)

text: x

text: for

text: the

text: large

text: cities.

text: Similar

text: results

text: are

text: found

text: for

text: the

text: years

text: for

text: which

text: the

text: data

text: are

text: available.

text: Thus,

text: the

text: double

text: Pareto

text: log-normal

text: distribution

text: ﬁts

text: well

text: the

text: city

text: population

text: size

text: distributions

text: for

text: the

text: Bulgarian

text: city

text: system

text: whatever

text: the

text: year

text: in

text: this

text: century.

text: Thus

text: Hypothesis

text: is

text: conﬁrmed.

text: Concluding

text: remarks

text: The

text: study

text: of

text: the

text: size

text: of

text: populations

text: in

text: cities

text: of

text: a

text: country

text: has

text: many

text: economic

text: applications.

text: Apart

text: on

text: the

text: decision

text: taking

text: about

text: placing

text: industrial

text: plants

text: or

text: building

text: trade

text: centers,

text: the

text: city

text: population

text: sizes

text: inﬂuence

text: the

text: eﬃcient

text: use

text: of

text: resources

text: or

text: the

text: possibilities

text: of

text: economic

text: growth

text: (Henderson

text: Henderson

text: Thus,

text: the

text: investigation

text: of

text: the

text: evolution

text: of

text: the

text: populations

text: in

text: a

text: system

text: of

text: cities

text: is

text: an

text: important

text: research

text: topic.

text: Our

text: study

text: reported

text: here

text: above

text: has

text: shown

text: that

text: for

text: the

text: case

text: of

text: the

text: Bulgarian

text: city

text: system

text: Hypothesis

text: is

text: not

text: conﬁrmed

text: but

text: Hypothesis

text: is

text: conﬁrmed.

text: The

text: validity

text: of

text: Hypothesis

text: even

text: gives

text: some

text: information

text: about

text: the

text: history

text: of

text: settlement

text: formations

text: in

text: Bulgaria.

text: It

text: seems

text: that

text: the

text: distribution

text: of

text: the

text: initial

text: populations

text: of

text: the

text: settlements

text: was

text: indeed

text: log-normal

text: and

text: that

text: the

text: formation

text: happened

text: according

text: to

text: the

text: Yule

text: process,

text: i.e.

text: the

text: new

text: settlements

text: have

text: been

text: formed

text: on

text: the

text: basis

text: of

text: the

text: existing

text: older

text: settlements

text: nearby.

text: Acknowledgment

text: This

text: work

text: has

text: been

text: performed

text: in

text: the

text: framework

text: of

text: COST

text: Action

text: ”The

text: EU

text: in

text: the

text: new

text: economic

text: complex

text: geography:

text: models,

text: tools

text: and

text: policy

text: evaluation”.

text: We

text: ac-

text: knowledge

text: some

text: support

text: through

text: the

text: project

text: ’Evolution

text: spatiale

text: et

text: temporelle

text: d’infrastructures

text: r´egionales

text: et

text: ´economiques

text: en

text: Bulgarie

text: et

text: en

text: F´ed´eration

text: Wallonie-Bruxelles’

text: within

text: the

text: intergovernmental

text: agreement

text: for

text: cooperation

text: between

text: the

text: Republic

text: of

text: Bulgaria

text: and

text: la

text: Communaut´e

text: Fran¸caise

text: de

text: Belgique.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Jiang B., Tia J., 2011. Zipf’s law for all the natural cities in the United States: a geospatial perspective. Int. J. Geogr. Inf. Sci. 25, 1269 - 1281. Knudsen T., 2004. Zipf’s law for cities and beyond: The case of Denmark. Am. J. Econ. Soc. 60, 123-146. Krugman P., 1996. Confronting the mistery of urban hierarchy. Journal of Japanese and International Economics 10, 399-418. Madden C.H., 1956. Some Temporal Aspects of the Growth of Cities in the United States. Economic Development and Cultural Change 4, 236-252. Moura N.J. Jr., Ribeiro M.B. 2006. Zipf law for Brasilian cities. Physica A 367, 441-448. Newman M.E.J., 2005. Power laws, Pareto distributions and Zipf’s law. Con- temporary Physics 46, 323-351. Pareto V., 1896. Cours d’Economie Politique Droz, Geneva, Switzerland. Peng C.-H. 2010. Zipf’s law for Chinese cities: rolling sample regressions. Phys- ica A 389, 3804-3813. Puu T., Panchuk A., 1991. Nonlinear economic dynamics. Springer, Berlin. Reed W.J., 2001. The Pareto, Zipf and other power laws. Economics Letters 74, 15-19. Reed W.J., Jorgensen M., 2004. The double Pareto-log-normal distribution: a new parametric model for size distributions. Communications in Statistics- Theory and Methods 33, 1733-1753. Rubenstein J.M., 2007. The Cultural Landscape: An Introduction to Human Geography, ninth ed. Prentice Hall, Englewood Cliﬀs, NJ. Seto K.C., Fragkias M., 2005. Quantifying spatiotemporal patterns of urban land-use change in four cities of China with time series landscape metrics. Landscape Ecology 20, 871-888. Sheppard E., 1982. City size distributions and spatial economic change. WP-82- 31, Working papers of the International Institute for Applied System Analysis, Laxenburg, Austria. Vespignani A., 2009. Predicting the behavior of techno-social systems. Science 325, 425-428. Vitanov N.K., Sakai K., Jordanov I.P., Managi S., Demura K., 2007. Analysis of a Japan government intervention on the domestic agriculture market. Physica A 382, 330-335. Zipf G.K., 1949. Human Behavior and the Principle of Least Eﬀort : An Intro- duction to Human Ecology. Addison Wesley, Cambridge, Mass. 8

text: Jiang

label: reference

text: B.,

text: Tia

text: J.,

text: Zipf’s

text: law

text: for

text: all

text: the

text: natural

text: cities

text: in

text: the

text: United

text: States:

text: a

text: geospatial

text: perspective.

text: Int.

text: J.

text: Geogr.

text: Inf.

text: Sci.

text: Knudsen

text: T.,

text: Zipf’s

text: law

text: for

text: cities

text: and

text: beyond:

text: The

text: case

text: of

text: Denmark.

text: Am.

text: J.

text: Econ.

text: Soc.

text: Krugman

text: P.,

text: Confronting

text: the

text: mistery

text: of

text: urban

text: hierarchy.

text: Journal

text: of

text: Japanese

text: and

text: International

text: Economics

text: Madden

text: C.H.,

text: Some

text: Temporal

text: Aspects

text: of

text: the

text: Growth

text: of

text: Cities

text: in

text: the

text: United

text: States.

text: Economic

text: Development

text: and

text: Cultural

text: Change

text: Moura

text: N.J.

text: Jr.,

text: Ribeiro

text: M.B.

text: Zipf

text: law

text: for

text: Brasilian

text: cities.

text: Physica

text: A

text: Newman

text: M.E.J.,

text: Power

text: laws,

text: Pareto

text: distributions

text: and

text: Zipf’s

text: law.

text: Con-

text: temporary

text: Physics

text: Pareto

text: V.,

text: Cours

text: d’Economie

text: Politique

text: Droz,

text: Geneva,

text: Switzerland.

text: Peng

text: C.-H.

text: Zipf’s

text: law

text: for

text: Chinese

text: cities:

text: rolling

text: sample

text: regressions.

text: Phys-

text: ica

text: A

text: Puu

text: T.,

text: Panchuk

text: A.,

text: Nonlinear

text: economic

text: dynamics.

text: Springer,

text: Berlin.

text: Reed

text: W.J.,

text: The

text: Pareto,

text: Zipf

text: and

text: other

text: power

text: laws.

text: Economics

text: Letters

text: Reed

text: W.J.,

text: Jorgensen

text: M.,

text: The

text: double

text: Pareto-log-normal

text: distribution:

text: a

text: new

text: parametric

text: model

text: for

text: size

text: distributions.

text: Communications

text: in

text: Statistics-

text: Theory

text: and

text: Methods

text: Rubenstein

text: J.M.,

text: The

text: Cultural

text: Landscape:

text: An

text: Introduction

text: to

text: Human

text: Geography,

text: ninth

text: ed.

text: Prentice

text: Hall,

text: Englewood

text: Cliﬀs,

text: NJ.

text: Seto

text: K.C.,

text: Fragkias

text: M.,

text: Quantifying

text: spatiotemporal

text: patterns

text: of

text: urban

text: land-use

text: change

text: in

text: four

text: cities

text: of

text: China

text: with

text: time

text: series

text: landscape

text: metrics.

text: Landscape

text: Ecology

text: Sheppard

text: E.,

text: City

text: size

text: distributions

text: and

text: spatial

text: economic

text: change.

text: Working

text: papers

text: of

text: the

text: International

text: Institute

text: for

text: Applied

text: System

text: Analysis,

text: Laxenburg,

text: Austria.

text: Vespignani

text: A.,

text: Predicting

text: the

text: behavior

text: of

text: techno-social

text: systems.

text: Science

text: Vitanov

text: N.K.,

text: Sakai

text: K.,

text: Jordanov

text: I.P.,

text: Managi

text: S.,

text: Demura

text: K.,

text: Analysis

text: of

text: a

text: Japan

text: government

text: intervention

text: on

text: the

text: domestic

text: agriculture

text: market.

text: Physica

text: A

text: Zipf

text: G.K.,

text: Human

text: Behavior

text: and

text: the

text: Principle

text: of

text: Least

text: Eﬀort

text: An

text: Intro-

text: duction

text: to

text: Human

text: Ecology.

text: Addison

text: Wesley,

text: Cambridge,

text: Mass.


text: (cid:16) (cid:17) Using the Gronwall inequality and imposing N large enough so that ≥ ≤ N (cid:60) 8C ≤ N C0, 1 + C 1 + (cid:90)p λ0, θ0(cid:16)0 (θ0/(cid:17)2)2pN , there holds (cid:107)g(s)(cid:107)2 + (cid:107)(E, B)(s)(cid:107)2 ≤ 2C (cid:16) s e2(1+(cid:17)C)(s−τ) εpe(cid:60)λ0τ 2(1+Np ) dτ Hk Hk 0 3 0 ≤ (cid:60)(cid:16) 2(1+(cid:17)N ) 2C εpe λ0s p 0 ≤ 2(cid:16)C0 εp(cid:17)e(cid:60)λ0s(cid:18)2 (cid:19) 2N (θ0/2) p ≤ 1 (cid:60) 2 θ (cid:60) 2 θ2 εpe λ0s = 0 εpe λ0s 4 0 2 θ (cid:60) < 0 εpe λ0s. 2 | | ∈ Finally, deﬁning Tε := (cid:60)1λ0 lo≥g(εp θ20 ) , it follows that for all s [0, T˜ε], (3.33) is satisﬁed. This proves that we have Tε T˜ε. ∈ Finally, recall that f = f + g. Thus, by the triangle inequality, as long as s [0, T˜ε], we get app (cid:107) (cid:107) ≥ (cid:107) (cid:107) − (cid:107) (cid:107) f f g L2 app L2 L2 ≥ (cid:107) (cid:107) − θ (cid:60) (3.34) fapp L2 20 εpe λ0s. We therefore need to get a lower bound on the L2 norm f . First, we have app (cid:107) (cid:107) ≥ (cid:107) (cid:107) (cid:82)− (cid:107) · ∇ (cid:107) f g (cid:15) A µ . 1 L2 1 L2 1 v L2 (cid:90) (cid:90) (cid:90) It follows from the construction that the average of vg dv is equal to zero. Indeed, by Lemma 2.2, we 1 have ∇ · · 1 µ k vgˆ dvdy = eik0 y dy | | ·v − 0 v dv = 0. 1 k 2 k (v ω) 0 0 (cid:104) (cid:105) (cid:90) From the proof of Lemma 3.2, we know that A = 0, so that 1 (cid:107) · ∇ (cid:107) ≤ s (cid:107) (cid:107) (cid:15) A µ (cid:15)C f(τ) dτ. 1 v L2 0 Hn 3 0 Recalling (3.30), we end up with (cid:107) · ∇ (cid:107) ≤ (cid:60) (cid:15) A1 vµ L2 (cid:15)C0e λ0s. By (3.29), we deduce (at least for (cid:15) > 0 small enough), (cid:107) (cid:107) ≥ (cid:60) (cid:15)pf1 L2 θ0εpe λ0s. (cid:16) (cid:17) Finally using (3.31), we obtain (cid:107) (cid:107) ≥ (cid:60) − (cid:60) 1+1 fapp L2 θ0εpe λ0s CN εpe λ0s p and thus 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Using

text: the

text: Gronwall

text: inequality

text: and

text: imposing

text: N

text: large

text: enough

text: so

text: that

text: N

text: N

text: C

text: there

text: holds

text: s

text: dτ

text: Hk

text: Hk

text: εpe

text: p

text: p

text: θ

text: θ2

text: εpe

text: εpe

text: θ

text: εpe

text: Finally,

text: deﬁning

text: Tε

text: lo≥g(εp

text: it

text: follows

text: that

text: for

text: all

text: s

text: T˜ε],

text: is

text: satisﬁed.

text: This

text: proves

text: that

text: we

text: have

text: Tε

text: T˜ε.

text: Finally,

text: recall

text: that

text: f

text: f

text: g.

text: Thus,

text: by

text: the

text: triangle

text: inequality,

text: as

text: long

text: as

text: s

text: T˜ε],

text: we

text: get

text: app

text: f

text: f

text: g

text: app

text: θ

text: fapp

text: εpe

text: We

text: therefore

text: need

text: to

text: get

text: a

text: lower

text: bound

text: on

text: the

text: norm

text: f

text: First,

text: we

text: have

text: app

text: f

text: g

text: A

text: µ

text: v

text: It

text: follows

text: from

text: the

text: construction

text: that

text: the

text: average

text: of

text: vg

text: dv

text: is

text: equal

text: to

text: zero.

text: Indeed,

text: by

text: Lemma

text: we

text: have

text: µ

text: k

text: vgˆ

text: dvdy

text: y

text: dy

text: ·v

text: v

text: dv

text: k

text: k

text: (v

text: ω)

text: From

text: the

text: proof

text: of

text: Lemma

text: we

text: know

text: that

text: A

text: so

text: that

text: s

text: A

text: µ

text: f(τ)

text: dτ.

text: v

text: Hn

text: Recalling

text: we

text: end

text: up

text: with

text: vµ

text: By

text: we

text: deduce

text: (at

text: least

text: for

text: small

text: enough),

text: Finally

text: using

text: we

text: obtain

text: fapp

text: CN

text: εpe

text: p

text: and

text: thus

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Anomalous recurrence properties of many-dimensional zero-drift random walks ∗† ∗ Nicholas Georgiou Mikhail V. Menshikov ‡ ∗ Aleksandar Mijatovi´c Andrew R. Wade 29 June 2015 Abstract Famously, a d-dimensional, spatially homogeneous random walk whose incre- ments are non-degenerate, have ﬁnite second moments, and have zero mean is re- ∈ { } ≥ current if d 1, 2 but transient if d 3. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous ran- dom walks (Markov processes) whose increment covariance matrix is asymptotically ≥ constant along rays from the origin, and which, in any ambient dimension d 2, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk’s current position; these elliptic random walks generalize the classical homogeneous Pearson–Rayleigh walk (the spherical case). Our proof of the recurrence classiﬁcation is based on fundamental work of Lamperti. Key words: Non-homogeneous random walk; elliptic random walk; zero drift; recurrence; transience. AMS Subject Classiﬁcation: 60J05 (Primary) 60J10, 60G42, 60G50 (Secondary) 1 Introduction A d-dimensional random walk that proceeds via a sequence of unit-length steps, each in an independent and uniformly random direction, is sometimes called a Pearson– Rayleigh random walk (PRRW), after the exchange in the letters pages of Nature between Karl Pearson and Lord Rayleigh in 1905 [16]. Pearson was interested in two dimensions and questions of migration of species (such as mosquitoes) [17], although Carazza has speculated that Pearson was a golfer [3, p. 419]; Rayleigh had earlier considered the acoustic ‘random walks’ in phase space produced by combinations of sound waves of the same amplitude and random phases. ∗ Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. † Heilbronn Institute for Mathematical Research, School of Mathematics, University of Bristol, Uni- versity Walk, Bristol, BS8 1TW. ‡ Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK. 1 ##LTLine##

text: Anomalous

label: title

text: recurrence

text: properties

text: of

text: many-dimensional

text: zero-drift

text: random

text: walks

label: author

text: Nicholas

text: Georgiou

text: Mikhail

text: V.

text: Menshikov

text: Aleksandar

text: Mijatovi´c

text: Andrew

text: R.

text: Wade

label: date

text: June

text: Abstract

text: Famously,

label: abstract

text: a

text: d-dimensional,

text: spatially

text: homogeneous

text: random

text: walk

text: whose

text: incre-

text: ments

text: are

text: non-degenerate,

text: have

text: ﬁnite

text: second

text: moments,

text: and

text: have

text: zero

text: mean

text: is

text: re-

text: current

text: if

text: d

text: but

text: transient

text: if

text: d

text: Once

text: spatial

text: homogeneity

text: is

text: relaxed,

text: this

text: is

text: no

text: longer

text: true.

text: We

text: study

text: a

text: family

text: of

text: zero-drift

text: spatially

text: non-homogeneous

text: ran-

text: dom

text: walks

text: (Markov

text: processes)

text: whose

text: increment

text: covariance

text: matrix

text: is

text: asymptotically

text: constant

text: along

text: rays

text: from

text: the

text: origin,

text: and

text: which,

text: in

text: any

text: ambient

text: dimension

text: d

text: can

text: be

text: adjusted

text: so

text: that

text: the

text: walk

text: is

text: either

text: transient

text: or

text: recurrent.

text: Natural

text: examples

text: are

text: provided

text: by

text: random

text: walks

text: whose

text: increments

text: are

text: supported

text: on

text: ellipsoids

text: that

text: are

text: symmetric

text: about

text: the

text: ray

text: from

text: the

text: origin

text: through

text: the

text: walk’s

text: current

text: position;

text: these

text: elliptic

text: random

text: walks

text: generalize

text: the

text: classical

text: homogeneous

text: Pearson–Rayleigh

text: walk

text: (the

text: spherical

text: case).

text: Our

text: proof

text: of

text: the

text: recurrence

text: classiﬁcation

text: is

text: based

text: on

text: fundamental

text: work

text: of

text: Lamperti.

text: Key

text: words:

text: Non-homogeneous

text: random

text: walk;

text: elliptic

text: random

text: walk;

text: zero

text: drift;

text: recurrence;

text: transience.

text: AMS

text: Subject

text: Classiﬁcation:

text: (Primary)

text: (Secondary)

text: Introduction

text: A

text: d-dimensional

text: random

text: walk

text: that

text: proceeds

text: via

text: a

text: sequence

text: of

text: unit-length

text: steps,

text: each

text: in

text: an

text: independent

text: and

text: uniformly

text: random

text: direction,

text: is

text: sometimes

text: called

text: a

text: Pearson–

text: Rayleigh

text: random

text: walk

text: (PRRW),

text: after

text: the

text: exchange

text: in

text: the

text: letters

text: pages

text: of

text: Nature

text: between

text: Karl

text: Pearson

text: and

text: Lord

text: Rayleigh

text: in

text: Pearson

text: was

text: interested

text: in

text: two

text: dimensions

text: and

text: questions

text: of

text: migration

text: of

text: species

text: (such

text: as

text: mosquitoes)

text: although

text: Carazza

text: has

text: speculated

text: that

text: Pearson

text: was

text: a

text: golfer

text: p.

text: Rayleigh

text: had

text: earlier

text: considered

text: the

text: acoustic

text: ‘random

text: walks’

text: in

text: phase

text: space

text: produced

text: by

text: combinations

text: of

text: sound

text: waves

text: of

text: the

text: same

text: amplitude

text: and

text: random

text: phases.

text: Department

text: of

text: Mathematical

text: Sciences,

text: Durham

text: University,

text: South

text: Road,

text: Durham

text: UK.

text: Heilbronn

text: Institute

text: for

text: Mathematical

text: Research,

text: School

text: of

text: Mathematics,

text: University

text: of

text: Bristol,

text: Uni-

text: versity

text: Walk,

text: Bristol,

text: Department

text: of

text: Mathematics,

text: Imperial

text: College

text: London,

text: Queen’s

text: Gate,

text: London

text: UK.

text: ##LTLine##


text: ≥ Lemma 6.2. In cases (ii) and (iii) of Theorem 2.3, there exists a ﬁnite r r such 1 0 that for any r > r and R > r there exists a ﬁnite positive c such that 1 P ≥ | ≥ − { } [τ n + m X , . . . , X ] cm 1/2, on n < τ , (6.1) r 0 n R for all suﬃciently large m. Assuming this result, we can complete the proof of Theorem 2.4. Proof of Theorem 2.4. In case (i), the result is contained in Lemma 6.1. So consider cases (ii) and (iii). Fix r and R with R > r > r , with r as in Lemma 6.2. Note that (cid:107) (cid:107) ≤ ≤ 1 1 lim inf →∞ X r r , a.s. n n 0 1 ∈ N Set γ := 0 and then deﬁne recursively, for (cid:96) , the stopping times 1 { ≥ ∈ } { ≥ ∈ } η := min n γ : X / B , γ := min n η : X B , (cid:96) (cid:96) n R (cid:96)+1 (cid:96) n r ∅ ∞ (cid:107) (cid:107) ∞ with the convention that min := . Since r > r and lim sup →∞ X = (by ∈ N ∞ 0 ∞ n n Proposition 2.1), for all (cid:96) we have η < and γ < , a.s., and (cid:96) (cid:96) · · · 0 = γ < η < γ < η < . 1 1 2 2 ∞ In particular, lim →∞ γ = lim →∞ η = , a.s. (cid:96)F (cid:96) (cid:96) (cid:96) We now write := σ(X , . . . , X ). We use Lemma 4.1 to show that the process must n 0 n − exit from B rapidly enough. Indeed, if κ is any ﬁnite stopping time, set Y = X X G FR n κ+n κ and = . Then the assumptions (A0), (A1) and (A2) show that the hypotheses of n κ+n Lemma 4.1 are satisﬁed, since, for example, E (cid:107) − (cid:107) | G E (cid:107) − (cid:107) | F E (cid:107) (cid:107) [ Y Y p ] = [ X X p ] = [ ∆ p], n+1 n n κ+n+1 κ+n κ+n Xκ+n by the strong Markov property for X at the ﬁnite stopping time κ + n. In particular, another application of Lemma 4.1, similarly to (4.6), shows that we may choose n = ∈ N (cid:104) (cid:12) (cid:105) (cid:12) n(R) suﬃciently large so that (cid:12) P (cid:107) − (cid:107) ≥ F ≥ 1 max X X 2R , a.s., (6.2) 0≤(cid:96)≤n(R) κ+(cid:96) κ κ 2 ∈ an event whose occurrence ensures that if X B , then X exits B before time ∈ N (cid:104) κ R (cid:12) (cid:105) R κ + n(R). Fix k . Then, an application of (6.2) at stopping ti(cid:12)me κ = γ shows that (cid:12) k P − | F ≤ P (cid:107) − (cid:107) F ≤ 1 [η γ > n(R) ] max X X < 2R , a.s. k k γk 0≤(cid:96)≤n(R) γk+(cid:96) γk γk 2 (cid:2) (cid:12) (cid:3) (cid:12) Similarly, P − | F E { − } E { − } | F F [η γ > 2n(R) ] = 1 η γ > n(R) [1 η γ > 2n(R) ] k k γk k k k k γk+n(R) γk ≤ 1P − | F ≤ 1 [η γ > n(R) ] , k k γ 2 k 4 this time applying (6.2) at stopping time κ = γ + n(R) as well. Iterating this argument, P − · | F ≤ k− ∈ N it follows that [η γ > m n(R) ] 2 m, a.s., for all m . From here, it is k k γk ∞ ∈ N straightforward to deduce that, for some constant C < , for any k , 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: In

text: cases

text: (ii)

text: and

text: (iii)

text: of

text: Theorem

text: there

text: exists

text: a

text: ﬁnite

text: r

text: r

text: such

text: that

text: for

text: any

text: r

text: r

text: and

text: R

text: r

text: there

text: exists

text: a

text: ﬁnite

text: positive

text: c

text: such

text: that

text: P

text: [τ

text: n

text: m

text: X

text: X

text: cm

text: on

text: n

text: τ

text: r

text: n

text: R

text: for

text: all

text: suﬃciently

text: large

text: m.

text: Assuming

text: this

text: result,

text: we

text: can

text: complete

text: the

text: proof

text: of

text: Theorem

text: Proof

text: of

text: Theorem

text: In

text: case

text: (i),

text: the

text: result

text: is

text: contained

text: in

text: Lemma

text: So

text: consider

text: cases

text: (ii)

text: and

text: (iii).

text: Fix

text: r

text: and

text: R

text: with

text: R

text: r

text: r

text: with

text: r

text: as

text: in

text: Lemma

text: Note

text: that

text: lim

text: inf

text: X

text: r

text: r

text: a.s.

text: n

text: n

text: N

text: Set

text: γ

text: and

text: then

text: deﬁne

text: recursively,

text: for

text: the

text: stopping

text: times

text: η

text: min

text: n

text: γ

text: X

text: B

text: γ

text: min

text: n

text: η

text: X

text: B

text: n

text: R

text: n

text: r

text: with

text: the

text: convention

text: that

text: min

text: Since

text: r

text: r

text: and

text: lim

text: sup

text: X

text: (by

text: N

text: n

text: n

text: Proposition

text: for

text: all

text: we

text: have

text: η

text: and

text: γ

text: a.s.,

text: and

text: γ

text: η

text: γ

text: η

text: In

text: particular,

text: lim

text: γ

text: lim

text: η

text: a.s.

text: We

text: now

text: write

text: σ(X

text: X

text: We

text: use

text: Lemma

text: to

text: show

text: that

text: the

text: process

text: must

text: n

text: n

text: exit

text: from

text: B

text: rapidly

text: enough.

text: Indeed,

text: if

text: κ

text: is

text: any

text: ﬁnite

text: stopping

text: time,

text: set

text: Y

text: X

text: X

text: G

text: FR

text: n

text: κ+n

text: κ

text: and

text: Then

text: the

text: assumptions

text: and

text: show

text: that

text: the

text: hypotheses

text: of

text: n

text: κ+n

text: Lemma

text: are

text: satisﬁed,

text: since,

text: for

text: example,

text: E

text: G

text: E

text: F

text: E

text: Y

text: Y

text: p

text: X

text: X

text: p

text: p],

text: n

text: n

text: κ+n

text: κ+n

text: Xκ+n

text: by

text: the

text: strong

text: Markov

text: property

text: for

text: X

text: at

text: the

text: ﬁnite

text: stopping

text: time

text: κ

text: n.

text: In

text: particular,

text: another

text: application

text: of

text: Lemma

text: similarly

text: to

text: shows

text: that

text: we

text: may

text: choose

text: n

text: N

text: n(R)

text: suﬃciently

text: large

text: so

text: that

text: P

text: F

text: max

text: X

text: X

text: a.s.,

text: κ

text: κ

text: an

text: event

text: whose

text: occurrence

text: ensures

text: that

text: if

text: X

text: B

text: then

text: X

text: exits

text: B

text: before

text: time

text: N

text: κ

text: R

text: R

text: κ

text: n(R).

text: Fix

text: k

text: Then,

text: an

text: application

text: of

text: at

text: stopping

text: κ

text: γ

text: shows

text: that

text: k

text: P

text: F

text: P

text: F

text: [η

text: γ

text: n(R)

text: max

text: X

text: X

text: a.s.

text: k

text: k

text: γk

text: γk

text: γk

text: Similarly,

text: P

text: F

text: E

text: E

text: F

text: F

text: [η

text: γ

text: η

text: γ

text: n(R)

text: η

text: γ

text: k

text: k

text: γk

text: k

text: k

text: k

text: k

text: γk+n(R)

text: γk

text: F

text: [η

text: γ

text: n(R)

text: k

text: k

text: γ

text: k

text: this

text: time

text: applying

text: at

text: stopping

text: time

text: κ

text: γ

text: n(R)

text: as

text: well.

text: Iterating

text: this

text: argument,

text: P

text: F

text: k−

text: N

text: it

text: follows

text: that

text: [η

text: γ

text: m

text: n(R)

text: m,

text: a.s.,

text: for

text: all

text: m

text: From

text: here,

text: it

text: is

text: k

text: k

text: γk

text: N

text: straightforward

text: to

text: deduce

text: that,

text: for

text: some

text: constant

text: C

text: for

text: any

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: = ∈ D normalized to 1 for g 0. As there are D quartic melonic interactions (one for each c ), one must introduce D intermediate matrix ﬁelds Hc. The original tensor T and T¯ can be integrated out to obtain: ( ) = ∫ (∏[ ]) ( ) = ∑ [ ] − [ ( )] − ( ) − D D= 1 D= Zsym. g dHc e S H , S H ND 1 Tr HcHc Tr lnR H , (4) c 2 c 1 c 1 D ( ) = where Tr is the trace over the index of color c, Tr is the trace over all the indices, and: c − ∑ ⊗ D D∖{ } = 1 R H I I = ⊗, (5) g D Hc C c c 1 ∈C I I I where we denoted c the identity matrix with indices of color c and c. c In this paper we will discuss a quartic melonic model with only one interaction. In this case one uses only one intermediate matrix ﬁeld and (dropping the index c) the partition function simpliﬁes to: ( ) = ∫ [ ] ( ) = [ ] − [ ( )] − ( ) − − 1 1 Z g dH e ND 1S H , S H Tr H2 Tr ln . (6) × 2 1 gH − Observe that in this form a tensor models in rank D looks very similar to a matrix model for a N N matrix. The only diﬀerence comes from the modiﬁed scaling factor ND 1 instead of N for the action. This apparently innocuous modiﬁcation leads to numerous consequences which we will be exploring in detail in this paper. 3 Vacua of the theory − = The equations of motion of the model deﬁned in Eq−. (6) are: g H 0 . (7) 1 gH − = = Due to the unitary invariance, Eq. (7) ﬁxes on−ly the eigenvalues λ of H: upon diagonalization, the equations of i motion become: g √ λ 0 for i 1,...,N , (8) i ± − 1 gλ ≡i whose two solutions are: ± = / 1 1 4g2 a . (9) 2g = = + ( ) − − We have a critical value of the coupling at g 1 2, where we meet a square root singularity and the two solutions + c merge. The solution a has a regular Taylor expansion around g 0 (starting at linear order, a g O g3 ), while the solution a has a simple pole at the origin. Up to conjugation by an arbitrary unitary matrix, any solution of the equations of motion (i.e. a vacuum of the theory) is of the form: = ( ) = ( ) + ( ) ≡ + + + + ( ) + − + + − − I − − I − + − a 0 0 0 0 H¯ I a × a I a ×P a P , + = (10) N ,N 0 a 0 0 0 + − + + + − − + − I I [ ] where and are the identity matrices of size N N and respectively N N , with N N N, otherwise arbitrary. There are N 1 possible solutions of this type. Observe that from the onset we have neglected the contribution of the measure dH to the equations of motion. = ≥ In fact this contribution can profoundly alter the equations of motion and it is possible that the “vacua” we identiﬁed / by just considering the action are nowhere near the true vacua of the theory. As we will see below, this is in fact U( ) × U( ) what happens in the case of matrices D 2. However, for tensors (D 3) the contribution of the measure is ( ) + − subleading in 1 N and the solutions in Eq. (10) are true vacuum states in the large N limit. + − Each H¯ is invariant=u(nder the)residual sy(mmetry )group∀ N∈ U( )N∀ : ∈ U( ) N ,N + + ( ) ( ) + + − − + − U 0− + − U† 0− H¯ N ,N 0 U H¯ N ,N 0 U† , U N , U N , (11) 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D

text: normalized

text: to

text: for

text: g

text: As

text: there

text: are

text: D

text: quartic

text: melonic

text: interactions

text: (one

text: for

text: each

text: c

text: one

text: must

text: introduce

text: D

text: intermediate

text: matrix

text: ﬁelds

text: Hc.

text: The

text: original

text: tensor

text: T

text: and

text: T¯

text: can

text: be

text: integrated

text: out

text: to

text: obtain:

text: D

text: D=

text: D=

text: Zsym.

text: g

text: dHc

text: e

text: S

text: H

text: S

text: H

text: ND

text: Tr

text: HcHc

text: Tr

text: lnR

text: H

text: c

text: c

text: c

text: D

text: where

text: Tr

text: is

text: the

text: trace

text: over

text: the

text: index

text: of

text: color

text: c,

text: Tr

text: is

text: the

text: trace

text: over

text: all

text: the

text: indices,

text: and:

text: c

text: D

text: D∖{

text: R

text: H

text: I

text: I

text: g

text: D

text: Hc

text: C

text: c

text: c

text: ∈C

text: I

text: I

text: I

text: where

text: we

text: denoted

text: c

text: the

text: identity

text: matrix

text: with

text: indices

text: of

text: color

text: c

text: and

text: c.

text: c

text: In

text: this

text: paper

text: we

text: will

text: discuss

text: a

text: quartic

text: melonic

text: model

text: with

text: only

text: one

text: interaction.

text: In

text: this

text: case

text: one

text: uses

text: only

text: one

text: intermediate

text: matrix

text: ﬁeld

text: and

text: (dropping

text: the

text: index

text: c)

text: the

text: partition

text: function

text: simpliﬁes

text: to:

text: Z

text: g

text: dH

text: e

text: ND

text: H

text: S

text: H

text: Tr

text: Tr

text: ln

text: gH

text: Observe

text: that

text: in

text: this

text: form

text: a

text: tensor

text: models

text: in

text: rank

text: D

text: looks

text: very

text: similar

text: to

text: a

text: matrix

text: model

text: for

text: a

text: N

text: N

text: matrix.

text: The

text: only

text: diﬀerence

text: comes

text: from

text: the

text: modiﬁed

text: scaling

text: factor

text: ND

text: instead

text: of

text: N

text: for

text: the

text: action.

text: This

text: apparently

text: innocuous

text: modiﬁcation

text: leads

text: to

text: numerous

text: consequences

text: which

text: we

text: will

text: be

text: exploring

text: in

text: detail

text: in

text: this

text: paper.

text: Vacua

text: of

text: the

text: theory

text: The

text: equations

text: of

text: motion

text: of

text: the

text: model

text: deﬁned

text: in

text: Eq−.

text: are:

text: g

text: H

text: gH

text: Due

text: to

text: the

text: unitary

text: invariance,

text: Eq.

text: ﬁxes

text: on−ly

text: the

text: eigenvalues

text: λ

text: of

text: H:

text: upon

text: diagonalization,

text: the

text: equations

text: of

text: i

text: motion

text: become:

text: g

text: λ

text: for

text: i

text: i

text: gλ

text: ≡i

text: whose

text: two

text: solutions

text: are:

text: a

text: We

text: have

text: a

text: critical

text: value

text: of

text: the

text: coupling

text: at

text: g

text: where

text: we

text: meet

text: a

text: square

text: root

text: singularity

text: and

text: the

text: two

text: solutions

text: c

text: merge.

text: The

text: solution

text: a

text: has

text: a

text: regular

text: Taylor

text: expansion

text: around

text: g

text: (starting

text: at

text: linear

text: order,

text: a

text: g

text: O

text: while

text: the

text: solution

text: a

text: has

text: a

text: simple

text: pole

text: at

text: the

text: origin.

text: Up

text: to

text: conjugation

text: by

text: an

text: arbitrary

text: unitary

text: matrix,

text: any

text: solution

text: of

text: the

text: equations

text: of

text: motion

text: (i.e.

text: a

text: vacuum

text: of

text: the

text: theory)

text: is

text: of

text: the

text: form:

text: I

text: I

text: a

text: H¯

text: I

text: a

text: a

text: I

text: a

text: ×P

text: a

text: P

text: N

text: ,N

text: a

text: I

text: I

text: where

text: and

text: are

text: the

text: identity

text: matrices

text: of

text: size

text: N

text: N

text: and

text: respectively

text: N

text: N

text: with

text: N

text: N

text: N,

text: otherwise

text: arbitrary.

text: There

text: are

text: N

text: possible

text: solutions

text: of

text: this

text: type.

text: Observe

text: that

text: from

text: the

text: onset

text: we

text: have

text: neglected

text: the

text: contribution

text: of

text: the

text: measure

text: dH

text: to

text: the

text: equations

text: of

text: motion.

text: In

text: fact

text: this

text: contribution

text: can

text: profoundly

text: alter

text: the

text: equations

text: of

text: motion

text: and

text: it

text: is

text: possible

text: that

text: the

text: “vacua”

text: we

text: identiﬁed

text: by

text: just

text: considering

text: the

text: action

text: are

text: nowhere

text: near

text: the

text: true

text: vacua

text: of

text: the

text: theory.

text: As

text: we

text: will

text: see

text: below,

text: this

text: is

text: in

text: fact

text: U(

text: U(

text: what

text: happens

text: in

text: the

text: case

text: of

text: matrices

text: D

text: However,

text: for

text: tensors

text: (D

text: the

text: contribution

text: of

text: the

text: measure

text: is

text: subleading

text: in

text: N

text: and

text: the

text: solutions

text: in

text: Eq.

text: are

text: true

text: vacuum

text: states

text: in

text: the

text: large

text: N

text: limit.

text: Each

text: H¯

text: is

text: invariant=u(nder

text: the)residual

text: sy(mmetry

text: )group∀

text: N∈

text: U(

text: )N∀

text: U(

text: N

text: ,N

text: U

text: 0−

text: U†

text: 0−

text: H¯

text: N

text: ,N

text: U

text: H¯

text: N

text: ,N

text: U†

text: U

text: N

text: U

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: + + + −− + − + − + − vertices have a joint coordination q q at least one, but can have coordination 0 in one of the sectors. These vertices represent a pair of a and a polygons connected at a nodal point. The polygons (resp. the polygons) will glue +into −(respectively ) surfaces, and the +−and surfaces will be connected at a mixe+d vertex by a nodal−point. We can represent the and vertices as solid vertices, and the vertices as two solid vertices (one for the + + sector and one for the sector) connected by a dashed edge, see Figure 1. Each graph represents a nodal surface (for deﬁnitions and properties of nodal surfaces see for example [42]). The E edges connecting two M half edges + − +− [ ] [ ] [ ] [ ] + − + − Figure 1: An example of a graph generated by the eﬀective action in Eq. (34). Here the two dashed edges represent two vertices of the type Tr M2 Tr M2 and Tr M2 Tr M . + − ( − ) − − − + (which we call the edges) bring( −a fa)ctor 1 . The E edges connecting two M half edges (which we call − ND 1 1 a2 − + the edges) bring a factor 1 . + ND 1 1 a2 + − The graph is connected but, by erasing the dashed edges, it might disconnect into several connected components. + Let us call C the number of the connected components of the graph made exclusively of the edges , g the sum of the genera of these components and F the number of faces of these components (and similarly for the components). Up to irrelevant combinatorial factors the amplitude of a graph is: ( ) ( ) ( − ) + ( − ) − ( − ) ∑ ( − ) ∑ − − + + +≥ + + − − −≥ − − + E − E × (− 1)∑ ( + ) 1 N D 1 V a p 3 p vp N D 1 V a p 3 p vp ND 1 1 +a+2 −≥ ND 1 1 a2 ( − ) + −≥ + + − + − + 1 − qq∑,qq+++ 0−1−≥≥q( +1+vq−)q + − NF+NF− . (35) a a qq ,qq 01 q q vq q< = > > + + Note that in the symmetric phase (i.e. g g and N 0) the amplitudes are positively deﬁnite, while in the c broken phase (i.e. g g and N 0) they are complex. In the following we will not keep track of global minus c signs for individual amplitudes. The various numbers of vertices, edges and faces are related by the following topological relations: = ∑ + +∑≥ = ∑ + +∑≥ + + + − + − − + − − ≥ + q q ≥1 + − ≥ − q q ≥1 + − 2E + p v + − q v , 2E − p v + − q v , (36) p q q p q q p 3 q ,q 0 p 3 q ,q 0 + − + = − + − + = − + +− + + + + − +− − − − − V V E F 2C 2g , V V E F 2C 2g , (37) − − + ≥ − − + ≥ + + + − +−− +− + ≥ − − +− − E+ −V − −V + +C−≥ 0+−, E V V C 0 , +− + − E E V V V 1 0 , V C C 1 0 . (38) 8 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: vertices

text: have

text: a

text: joint

text: coordination

text: q

text: q

text: at

text: least

text: one,

text: but

text: can

text: have

text: coordination

text: in

text: one

text: of

text: the

text: sectors.

text: These

text: vertices

text: represent

text: a

text: pair

text: of

text: a

text: and

text: a

text: polygons

text: connected

text: at

text: a

text: nodal

text: point.

text: The

text: polygons

text: (resp.

text: the

text: polygons)

text: will

text: glue

text: +into

text: −(respectively

text: surfaces,

text: and

text: the

text: +−and

text: surfaces

text: will

text: be

text: connected

text: at

text: a

text: mixe+d

text: vertex

text: by

text: a

text: nodal−point.

text: We

text: can

text: represent

text: the

text: and

text: vertices

text: as

text: solid

text: vertices,

text: and

text: the

text: vertices

text: as

text: two

text: solid

text: vertices

text: (one

text: for

text: the

text: sector

text: and

text: one

text: for

text: the

text: sector)

text: connected

text: by

text: a

text: dashed

text: edge,

text: see

text: Figure

text: Each

text: graph

text: represents

text: a

text: nodal

text: surface

text: (for

text: deﬁnitions

text: and

text: properties

text: of

text: nodal

text: surfaces

text: see

text: for

text: example

text: The

text: E

text: edges

text: connecting

text: two

text: M

text: half

text: edges

text: Figure

text: An

text: example

text: of

text: a

text: graph

text: generated

text: by

text: the

text: eﬀective

text: action

text: in

text: Eq.

text: Here

text: the

text: two

text: dashed

text: edges

text: represent

text: two

text: vertices

text: of

text: the

text: type

text: Tr

text: Tr

text: and

text: Tr

text: Tr

text: M

text: (which

text: we

text: call

text: the

text: edges)

text: bring(

text: −a

text: fa)ctor

text: The

text: E

text: edges

text: connecting

text: two

text: M

text: half

text: edges

text: (which

text: we

text: call

text: ND

text: the

text: edges)

text: bring

text: a

text: factor

text: ND

text: The

text: graph

text: is

text: connected

text: but,

text: by

text: erasing

text: the

text: dashed

text: edges,

text: it

text: might

text: disconnect

text: into

text: several

text: connected

text: components.

text: Let

text: us

text: call

text: C

text: the

text: number

text: of

text: the

text: connected

text: components

text: of

text: the

text: graph

text: made

text: exclusively

text: of

text: the

text: edges

text: g

text: the

text: sum

text: of

text: the

text: genera

text: of

text: these

text: components

text: and

text: F

text: the

text: number

text: of

text: faces

text: of

text: these

text: components

text: (and

text: similarly

text: for

text: the

text: components).

text: Up

text: to

text: irrelevant

text: combinatorial

text: factors

text: the

text: amplitude

text: of

text: a

text: graph

text: is:

text: E

text: E

text: N

text: D

text: V

text: a

text: p

text: p

text: vp

text: N

text: D

text: V

text: a

text: p

text: p

text: vp

text: ND

text: ND

text: qq∑,qq+++

text: 0−1−≥≥q(

text: NF+NF−

text: a

text: a

text: qq

text: ,qq

text: q

text: q

text: vq

text: q<

text: Note

text: that

text: in

text: the

text: symmetric

text: phase

text: (i.e.

text: g

text: g

text: and

text: N

text: the

text: amplitudes

text: are

text: positively

text: deﬁnite,

text: while

text: in

text: the

text: c

text: broken

text: phase

text: (i.e.

text: g

text: g

text: and

text: N

text: they

text: are

text: complex.

text: In

text: the

text: following

text: we

text: will

text: not

text: keep

text: track

text: of

text: global

text: minus

text: c

text: signs

text: for

text: individual

text: amplitudes.

text: The

text: various

text: numbers

text: of

text: vertices,

text: edges

text: and

text: faces

text: are

text: related

text: by

text: the

text: following

text: topological

text: relations:

text: q

text: q

text: ≥1

text: q

text: q

text: ≥1

text: p

text: v

text: q

text: v

text: p

text: v

text: q

text: v

text: p

text: q

text: q

text: p

text: q

text: q

text: p

text: q

text: ,q

text: p

text: q

text: ,q

text: V

text: V

text: E

text: F

text: V

text: V

text: E

text: F

text: E+

text: −V

text: −V

text: +C−≥

text: E

text: V

text: V

text: C

text: E

text: E

text: V

text: V

text: V

text: V

text: C

text: C

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • select a vertex with minimum degree in the current graph ; • select a vertex with minimum number of ﬁll-in edges in the current graph. After each vertex selection, the current graph is modiﬁed by removing the selected vertex and making a clique on its neighbors. The new edges added by this clique are ﬁll-in edges. A ver- tex with no ﬁll-in edges is called a simplicial vertex. Fast implementations of minimum degree algorithms have been developed, see e.g., AMD (Amestoy et al., 1996) with time complexity in O(nm) (Heggernes et al., 2001) for an input graph G with n vertices and m edges. The minimum ﬁll-in heuristic tends to be slower to compute but yields slightly better treewidth approximations in practice. Moreover, it will ﬁnd a perfect elimination ordering (ı.e., adding no ﬁll-in edges) if it exists, thus recognizing chordal graphs and it returns the optimal treewidth in this particular case ((this can be easily established from results in Bodlaender et al., 2005). Notice that there exists linear time O(n + m) algorithms to detect chordal graphs as the Maxi- mum Cardinality Search (MCS) greedy algorithm (Tarjan and Yannakakis, 1984) but the treewidth approximation they return is usually worse than the previous heuristic approaches. A simple way to improve the treewidth bound found by these greedy algorithms is to break ties for the selected criterion using a second criterion, such as minimum ﬁll-in ﬁrst and then maximum degree, or to break ties at random and to iterate on the resulting randomized algorithms as done in Kask et al. (2011). We compared the mean treewidth bound found by these four approaches (minimum degree, minimum ﬁll-in, MCS and randomized iterative minimum ﬁll-in) on a set of ﬁve CSP and MRF benchmarks used as combinatorial optimization problems in various solver competitions. Par- ityLearning is an optimization variant of the minimal disagreement parity CSP problem originally contributed to the DIMACS benchmark and used in the Minizinc challenge (Optimization Re- search Group, 2012). Linkage is a genetic linkage analysis benchmark (Elidan and Globerson, 2010). GeomSurf and SceneDecomp are respectively geometric surface labeling and scene de- composition problems in computer vision (Andres et al., 2013). The number of instances per problem as well as their mean characteristics are given in Table 2. Results are reported in Figure 6 (Left).The randomized iterative minimum ﬁll-in algorithm used a maximum of 30, 000 iterations or 180 seconds (respectively 10, 000 iterations and 60 seconds for ParityLearning and Linkage), compared to a maximum of 0.37 second used by the non-iterative approaches. The minimum ﬁll- in algorithm (using maximum degree for ties breaking) performed better than the other greedy approaches, being slightly improved by its randomized iterative version. On the same benchmark, we also compared three exact methods for the task of mode evaluation that exploit either minimum ﬁll-in ordering or its randomized iterative version: variable elimina- tion (ELIM), BTD (de Givry et al., 2006) and AND/OR search (Marinescu and Dechter, 2006). Elim and BTD exploit the minimum ﬁll-in ordering while AND/OR search used its randomized iterative version. In addition, BTD and AND/OR Search exploit a tree decomposition during a Depth First Branch and Bound method in order to get a good trade-off between memory space and search effort. As variable elimination, they have a worst-case time complexity exponential in the treewidth. All methods were allocated a maximum of 1 hour and 4 GB of RAM on an AMD Operon 6176 at 2.3 GHz. The results, as reported in Figure 6 (Right) shown that BTD was able to solve more problems than the two other methods for a ﬁxed CPU time. However, on a given 18

text: select

text: a

text: vertex

text: with

text: minimum

text: degree

text: in

text: the

text: current

text: graph

text: select

text: a

text: vertex

text: with

text: minimum

text: number

text: of

text: ﬁll-in

text: edges

text: in

text: the

text: current

text: graph.

text: After

text: each

text: vertex

text: selection,

text: the

text: current

text: graph

text: is

text: modiﬁed

text: by

text: removing

text: the

text: selected

text: vertex

text: and

text: making

text: a

text: clique

text: on

text: its

text: neighbors.

text: The

text: new

text: edges

text: added

text: by

text: this

text: clique

text: are

text: ﬁll-in

text: edges.

text: A

text: ver-

text: tex

text: with

text: no

text: ﬁll-in

text: edges

text: is

text: called

text: a

text: simplicial

text: vertex.

text: Fast

text: implementations

text: of

text: minimum

text: degree

text: algorithms

text: have

text: been

text: developed,

text: see

text: e.g.,

text: AMD

text: (Amestoy

text: et

text: al.,

text: with

text: time

text: complexity

text: in

text: O(nm)

text: (Heggernes

text: et

text: al.,

text: for

text: an

text: input

text: graph

text: G

text: with

text: n

text: vertices

text: and

text: m

text: edges.

text: The

text: minimum

text: ﬁll-in

text: heuristic

text: tends

text: to

text: be

text: slower

text: to

text: compute

text: but

text: yields

text: slightly

text: better

text: treewidth

text: approximations

text: in

text: practice.

text: Moreover,

text: it

text: will

text: ﬁnd

text: a

text: perfect

text: elimination

text: ordering

text: (ı.e.,

text: adding

text: no

text: ﬁll-in

text: edges)

text: if

text: it

text: exists,

text: thus

text: recognizing

text: chordal

text: graphs

text: and

text: it

text: returns

text: the

text: optimal

text: treewidth

text: in

text: this

text: particular

text: case

text: ((this

text: can

text: be

text: easily

text: established

text: from

text: results

text: in

text: Bodlaender

text: et

text: al.,

text: Notice

text: that

text: there

text: exists

text: linear

text: time

text: O(n

text: m)

text: algorithms

text: to

text: detect

text: chordal

text: graphs

text: as

text: the

text: Maxi-

text: mum

text: Cardinality

text: Search

text: (MCS)

text: greedy

text: algorithm

text: (Tarjan

text: and

text: Yannakakis,

text: but

text: the

text: treewidth

text: approximation

text: they

text: return

text: is

text: usually

text: worse

text: than

text: the

text: previous

text: heuristic

text: approaches.

text: A

text: simple

text: way

text: to

text: improve

text: the

text: treewidth

text: bound

text: found

text: by

text: these

text: greedy

text: algorithms

text: is

text: to

text: break

text: ties

text: for

text: the

text: selected

text: criterion

text: using

text: a

text: second

text: criterion,

text: such

text: as

text: minimum

text: ﬁll-in

text: ﬁrst

text: and

text: then

text: maximum

text: degree,

text: or

text: to

text: break

text: ties

text: at

text: random

text: and

text: to

text: iterate

text: on

text: the

text: resulting

text: randomized

text: algorithms

text: as

text: done

text: in

text: Kask

text: et

text: al.

text: We

text: compared

text: the

text: mean

text: treewidth

text: bound

text: found

text: by

text: these

text: four

text: approaches

text: (minimum

text: degree,

text: minimum

text: ﬁll-in,

text: MCS

text: and

text: randomized

text: iterative

text: minimum

text: ﬁll-in)

text: on

text: a

text: set

text: of

text: ﬁve

text: CSP

text: and

text: MRF

text: benchmarks

text: used

text: as

text: combinatorial

text: optimization

text: problems

text: in

text: various

text: solver

text: competitions.

text: Par-

text: ityLearning

text: is

text: an

text: optimization

text: variant

text: of

text: the

text: minimal

text: disagreement

text: parity

text: CSP

text: problem

text: originally

text: contributed

text: to

text: the

text: DIMACS

text: benchmark

text: and

text: used

text: in

text: the

text: Minizinc

text: challenge

text: (Optimization

text: Re-

text: search

text: Group,

text: Linkage

text: is

text: a

text: genetic

text: linkage

text: analysis

text: benchmark

text: (Elidan

text: and

text: Globerson,

text: GeomSurf

text: and

text: SceneDecomp

text: are

text: respectively

text: geometric

text: surface

text: labeling

text: and

text: scene

text: de-

text: composition

text: problems

text: in

text: computer

text: vision

text: (Andres

text: et

text: al.,

text: The

text: number

text: of

text: instances

text: per

text: problem

text: as

text: well

text: as

text: their

text: mean

text: characteristics

text: are

text: given

text: in

text: Table

text: Results

text: are

text: reported

text: in

text: Figure

text: (Left).The

text: randomized

text: iterative

text: minimum

text: ﬁll-in

text: algorithm

text: used

text: a

text: maximum

text: of

text: iterations

text: or

text: seconds

text: (respectively

text: iterations

text: and

text: seconds

text: for

text: ParityLearning

text: and

text: Linkage),

text: compared

text: to

text: a

text: maximum

text: of

text: second

text: used

text: by

text: the

text: non-iterative

text: approaches.

text: The

text: minimum

text: ﬁll-

text: in

text: algorithm

text: (using

text: maximum

text: degree

text: for

text: ties

text: breaking)

text: performed

text: better

text: than

text: the

text: other

text: greedy

text: approaches,

text: being

text: slightly

text: improved

text: by

text: its

text: randomized

text: iterative

text: version.

text: On

text: the

text: same

text: benchmark,

text: we

text: also

text: compared

text: three

text: exact

text: methods

text: for

text: the

text: task

text: of

text: mode

text: evaluation

text: that

text: exploit

text: either

text: minimum

text: ﬁll-in

text: ordering

text: or

text: its

text: randomized

text: iterative

text: version:

text: variable

text: elimina-

text: tion

text: (ELIM),

text: BTD

text: (de

text: Givry

text: et

text: al.,

text: and

text: AND/OR

text: search

text: (Marinescu

text: and

text: Dechter,

text: Elim

text: and

text: BTD

text: exploit

text: the

text: minimum

text: ﬁll-in

text: ordering

text: while

text: AND/OR

text: search

text: used

text: its

text: randomized

text: iterative

text: version.

text: In

text: addition,

text: BTD

text: and

text: AND/OR

text: Search

text: exploit

text: a

text: tree

text: decomposition

text: during

text: a

text: Depth

text: First

text: Branch

text: and

text: Bound

text: method

text: in

text: order

text: to

text: get

text: a

text: good

text: trade-off

text: between

text: memory

text: space

text: and

text: search

text: effort.

text: As

text: variable

text: elimination,

text: they

text: have

text: a

text: worst-case

text: time

text: complexity

text: exponential

text: in

text: the

text: treewidth.

text: All

text: methods

text: were

text: allocated

text: a

text: maximum

text: of

text: hour

text: and

text: GB

text: of

text: RAM

text: on

text: an

text: AMD

text: Operon

text: at

text: GHz.

text: The

text: results,

text: as

text: reported

text: in

text: Figure

text: (Right)

text: shown

text: that

text: BTD

text: was

text: able

text: to

text: solve

text: more

text: problems

text: than

text: the

text: two

text: other

text: methods

text: for

text: a

text: ﬁxed

text: CPU

text: time.

text: However,

text: on

text: a

text: given


text: It has been established by Yedidia et al. (2005) that the ﬁxed points of LBP (when they exist, convergence is still not well understood, see Weiss (2000) and Mooij and Kappen (2007)) are || stationary points of the problem of minimizing the Bethe free energy, or equivalently KL(q p) Q with q in the class Bethe of distributions. Q Furthermore Yedidia et al. (2005) showed that for any class of distributions corresponding to a particular CVM method, it is possible to deﬁne a generalized BP algorithm whose ﬁxed points || Q are stationary points of the problem of minimizing KL(q p) in . The drawback of the LBP algorithm and its extensions (Yedidia et al., 2005) it that they are not associated with any theoretical bound on the error made on the marginals approximations. Nevertheless, this algorithm is increasingly used for inference in graphical models for its good behavior in practice (Murphy et al., 1999). It is implemented in several pieces of software for inference in graphical models like libDAI (Mooij, 2010) or OpenGM2 (Andres et al., 2012). 7 Illustration with coupled HMM We now illustrate how the the procedures we have described have been used for parameter estimation in a elaborated example: coupled HMM. Consider a set of I signals observed at times ∈ { } t 1, . . . T and denote Oi the variable corresponding to the observed signal i at time t. HMM t models assume that the distribution of each Oi is conditional to some hidden state Hi, where t t the series (Hi) is a Markov chain. Coupled HMM further assumes that the hidden states t t=1,...T display a correlation at each time (see Jordan, 2004; Wainwright and Jordan, 2008), resulting in the graphical model displayed in Figure 8. Such models have been considered in a series of domains such as bioinformatics (Choi et al., 2013), electroencephalogram analysis (Zhong and Ghosh, 2002) or speech recognition (Nock and Ostendorf, 2003). More complex versions are sometimes considered, assuming dependency between two times series at two consecutive time steps. For this model, the joint distribution of the hidden variables H = (Hi) and observed (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) t i,t (cid:33) variables O = (Oi) factorizes as t i,t (cid:89) (cid:89) (cid:89) (cid:89) (cid:89) ∝ × × p(h, o) ψM(hi− , hi) ψC(h ) ψE(hi, oi) , (8) t 1 t t t t i t t i t where h = (hi) stands the vector of all hidden variables at time t and where ψM encodes the t t i Markovian dependency of the hidden variables within a series, ψC encodes the coupling between the hidden variables of all series at a given time and ψE encodes the emission of the observed signal given the corresponding hidden state. A fairly comprehensive exploration of these models can be found in (Murphy, 2002). 7.1 Exact EM algorithm for coupled HMM Coupled HMM are examples of incomplete data models, as they involve variables (O, H) where only the variables O are observed. Maximum likelihood inference for such a model aims at ﬁnding the value of the parameter θ that maximizes the (log-)likelihood of the observed data o, that is to solve max log pθ(o). The most popular algorithm to achieve this task is the EM algorithm, θ 27

text: It

text: has

text: been

text: established

text: by

text: Yedidia

text: et

text: al.

text: that

text: the

text: ﬁxed

text: points

text: of

text: LBP

text: (when

text: they

text: exist,

text: convergence

text: is

text: still

text: not

text: well

text: understood,

text: see

text: Weiss

text: and

text: Mooij

text: and

text: Kappen

text: are

text: stationary

text: points

text: of

text: the

text: problem

text: of

text: minimizing

text: the

text: Bethe

text: free

text: energy,

text: or

text: equivalently

text: KL(q

text: p)

text: Q

text: with

text: q

text: in

text: the

text: class

text: Bethe

text: of

text: distributions.

text: Q

text: Furthermore

text: Yedidia

text: et

text: al.

text: showed

text: that

text: for

text: any

text: class

text: of

text: distributions

text: corresponding

text: to

text: a

text: particular

text: CVM

text: method,

text: it

text: is

text: possible

text: to

text: deﬁne

text: a

text: generalized

text: BP

text: algorithm

text: whose

text: ﬁxed

text: points

text: Q

text: are

text: stationary

text: points

text: of

text: the

text: problem

text: of

text: minimizing

text: KL(q

text: p)

text: in

text: The

text: drawback

text: of

text: the

text: LBP

text: algorithm

text: and

text: its

text: extensions

text: (Yedidia

text: et

text: al.,

text: it

text: that

text: they

text: are

text: not

text: associated

text: with

text: any

text: theoretical

text: bound

text: on

text: the

text: error

text: made

text: on

text: the

text: marginals

text: approximations.

text: Nevertheless,

text: this

text: algorithm

text: is

text: increasingly

text: used

text: for

text: inference

text: in

text: graphical

text: models

text: for

text: its

text: good

text: behavior

text: in

text: practice

text: (Murphy

text: et

text: al.,

text: It

text: is

text: implemented

text: in

text: several

text: pieces

text: of

text: software

text: for

text: inference

text: in

text: graphical

text: models

text: like

text: libDAI

text: (Mooij,

text: or

text: (Andres

text: et

text: al.,

text: Illustration

text: with

text: coupled

text: HMM

text: We

text: now

text: illustrate

text: how

text: the

text: the

text: procedures

text: we

text: have

text: described

text: have

text: been

text: used

text: for

text: parameter

text: estimation

text: in

text: a

text: elaborated

text: example:

text: coupled

text: HMM.

text: Consider

text: a

text: set

text: of

text: I

text: signals

text: observed

text: at

text: times

text: t

text: T

text: and

text: denote

text: Oi

text: the

text: variable

text: corresponding

text: to

text: the

text: observed

text: signal

text: i

text: at

text: time

text: t.

text: HMM

text: t

text: models

text: assume

text: that

text: the

text: distribution

text: of

text: each

text: Oi

text: is

text: conditional

text: to

text: some

text: hidden

text: state

text: Hi,

text: where

text: t

text: t

text: the

text: series

text: (Hi)

text: is

text: a

text: Markov

text: chain.

text: Coupled

text: HMM

text: further

text: assumes

text: that

text: the

text: hidden

text: states

text: t

text: display

text: a

text: correlation

text: at

text: each

text: time

text: (see

text: Jordan,

text: Wainwright

text: and

text: Jordan,

text: resulting

text: in

text: the

text: graphical

text: model

text: displayed

text: in

text: Figure

text: Such

text: models

text: have

text: been

text: considered

text: in

text: a

text: series

text: of

text: domains

text: such

text: as

text: bioinformatics

text: (Choi

text: et

text: al.,

text: electroencephalogram

text: analysis

text: (Zhong

text: and

text: Ghosh,

text: or

text: speech

text: recognition

text: (Nock

text: and

text: Ostendorf,

text: More

text: complex

text: versions

text: are

text: sometimes

text: considered,

text: assuming

text: dependency

text: between

text: two

text: times

text: series

text: at

text: two

text: consecutive

text: time

text: steps.

text: For

text: this

text: model,

text: the

text: joint

text: distribution

text: of

text: the

text: hidden

text: variables

text: H

text: (Hi)

text: and

text: observed

text: t

text: i,t

text: variables

text: O

text: (Oi)

text: factorizes

text: as

text: t

text: i,t

text: p(h,

text: o)

text: ψM(hi−

text: hi)

text: ψC(h

text: ψE(hi,

text: oi)

text: t

text: t

text: t

text: t

text: t

text: i

text: t

text: t

text: i

text: t

text: where

text: h

text: (hi)

text: stands

text: the

text: vector

text: of

text: all

text: hidden

text: variables

text: at

text: time

text: t

text: and

text: where

text: ψM

text: encodes

text: the

text: t

text: t

text: i

text: Markovian

text: dependency

text: of

text: the

text: hidden

text: variables

text: within

text: a

text: series,

text: ψC

text: encodes

text: the

text: coupling

text: between

text: the

text: hidden

text: variables

text: of

text: all

text: series

text: at

text: a

text: given

text: time

text: and

text: ψE

text: encodes

text: the

text: emission

text: of

text: the

text: observed

text: signal

text: given

text: the

text: corresponding

text: hidden

text: state.

text: A

text: fairly

text: comprehensive

text: exploration

text: of

text: these

text: models

text: can

text: be

text: found

text: in

text: (Murphy,

text: Exact

text: EM

text: algorithm

text: for

text: coupled

text: HMM

text: Coupled

text: HMM

text: are

text: examples

text: of

text: incomplete

text: data

text: models,

text: as

text: they

text: involve

text: variables

text: (O,

text: H)

text: where

text: only

text: the

text: variables

text: O

text: are

text: observed.

text: Maximum

text: likelihood

text: inference

text: for

text: such

text: a

text: model

text: aims

text: at

text: ﬁnding

text: the

text: value

text: of

text: the

text: parameter

text: θ

text: that

text: maximizes

text: the

text: (log-)likelihood

text: of

text: the

text: observed

text: data

text: o,

text: that

text: is

text: to

text: solve

text: max

text: log

text: pθ(o).

text: The

text: most

text: popular

text: algorithm

text: to

text: achieve

text: this

text: task

text: is

text: the

text: EM

text: algorithm,

text: θ


text: HI− HI HI t 1 t t+1 OI− OI OI t 1 t t+1 Hi− Hi Hi t 1 t t+1 Oi− Oi Oi t 1 t t+1 H1− H1 H1 t 1 t t+1 O1− O1 O1 t 1 t t+1 Figure 8: Graphical representation of p(h, o) for a coupled HMM (Dempster et al., 1977), that can be rephrased in the following way. Observe that | − | | log pθ(o) = E(log pθ(o, H) o) E(log pθ(H o) o) = max F (θ, q), q where − − || | F (θ, q) = E (log pθ(o, H)) E (log q(H)) = log pθ(o) KL(q(H) pθ(H o)), q q and q stands for any distribution on the hidden variables H, E stands for the expectation under the true distribution p and E under the arbitrary distribution q. The EM algorithm consists in q alternatively maximizing F (θ, q) with respect to q (E-step) and to θ (M-step). The solution of the | E-step is q(h) = p(h o) since the Kullback-Leibler divergence is minimal (and null) in this case. | (cid:32) (cid:33) (cid:32) (cid:33) Exact computation of p(h o)(cid:89)can be performed by obs(cid:89)erv(cid:89)ing that (8) can be rewritten as ∝ (cid:48) × p(h, o) ψM (h − , h ) ψE(hi, oi) , t 1 t t t t i t (cid:48) where ΨM encodes both the Markovian dependency and the coupling of the hidden variables within a given time step. This writing is equivalent to merging all hidden variables of a given time step and corresponds to the graphical model given in Figure 9. Denoting K the number of possible values for each hidden variables, we end up with a regular hidden Markov model with KI | possible hidden states. Both p(h o) (and its mode) can then be computed in a exact manner with either the forward-backward recursion or the Viterbi algorithm, which have the same complexity: O(T K2I). The exact calculation can therefore be achieved provided that KI remains small enough, but becomes intractable when the number of signals I exceeds few tens. 7.2 Approximate E step for the EM algorithm A ﬁrst approach to derive an approximate E step it to seek for a variational approximation | Q of p(h o) assuming that q(h) is restricted to a family of tractable distributions, as described in Section . This approach results in the maximization of a lower bound of the original log-likelihood. 28 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HI−

text: HI

text: HI

text: t

text: t

text: OI−

text: OI

text: OI

text: t

text: t

text: Hi−

text: Hi

text: Hi

text: t

text: t

text: Oi−

text: Oi

text: Oi

text: t

text: t

text: t

text: t

text: t

text: t

text: Figure

text: Graphical

label: caption

text: representation

text: of

text: p(h,

text: o)

text: for

text: a

text: coupled

text: HMM

text: (Dempster

text: et

text: al.,

text: that

text: can

text: be

text: rephrased

text: in

text: the

text: following

text: way.

text: Observe

text: that

text: log

text: pθ(o)

text: E(log

text: pθ(o,

text: H)

text: o)

text: E(log

text: pθ(H

text: o)

text: o)

text: max

text: F

text: (θ,

text: q),

text: q

text: where

text: F

text: (θ,

text: q)

text: E

text: (log

text: pθ(o,

text: H))

text: E

text: (log

text: q(H))

text: log

text: pθ(o)

text: KL(q(H)

text: pθ(H

text: o)),

text: q

text: q

text: and

text: q

text: stands

text: for

text: any

text: distribution

text: on

text: the

text: hidden

text: variables

text: H,

text: E

text: stands

text: for

text: the

text: expectation

text: under

text: the

text: true

text: distribution

text: p

text: and

text: E

text: under

text: the

text: arbitrary

text: distribution

text: q.

text: The

text: EM

text: algorithm

text: consists

text: in

text: q

text: alternatively

text: maximizing

text: F

text: (θ,

text: q)

text: with

text: respect

text: to

text: q

text: (E-step)

text: and

text: to

text: θ

text: (M-step).

text: The

text: solution

text: of

text: the

text: E-step

text: is

text: q(h)

text: p(h

text: o)

text: since

text: the

text: Kullback-Leibler

text: divergence

text: is

text: minimal

text: (and

text: null)

text: in

text: this

text: case.

text: Exact

text: computation

text: of

text: p(h

text: be

text: performed

text: by

text: that

text: can

text: be

text: rewritten

text: as

text: p(h,

text: o)

text: ψM

text: (h

text: h

text: ψE(hi,

text: oi)

text: t

text: t

text: t

text: t

text: t

text: i

text: t

text: where

text: ΨM

text: encodes

text: both

text: the

text: Markovian

text: dependency

text: and

text: the

text: coupling

text: of

text: the

text: hidden

text: variables

text: within

text: a

text: given

text: time

text: step.

text: This

text: writing

text: is

text: equivalent

text: to

text: merging

text: all

text: hidden

text: variables

text: of

text: a

text: given

text: time

text: step

text: and

text: corresponds

text: to

text: the

text: graphical

text: model

text: given

text: in

text: Figure

text: Denoting

text: K

text: the

text: number

text: of

text: possible

text: values

text: for

text: each

text: hidden

text: variables,

text: we

text: end

text: up

text: with

text: a

text: regular

text: hidden

text: Markov

text: model

text: with

text: KI

text: possible

text: hidden

text: states.

text: Both

text: p(h

text: o)

text: (and

text: its

text: mode)

text: can

text: then

text: be

text: computed

text: in

text: a

text: exact

text: manner

text: with

text: either

text: the

text: forward-backward

text: recursion

text: or

text: the

text: Viterbi

text: algorithm,

text: which

text: have

text: the

text: same

text: complexity:

text: O(T

text: The

text: exact

text: calculation

text: can

text: therefore

text: be

text: achieved

text: provided

text: that

text: KI

text: remains

text: small

text: enough,

text: but

text: becomes

text: intractable

text: when

text: the

text: number

text: of

text: signals

text: I

text: exceeds

text: few

text: tens.

text: Approximate

text: E

text: step

text: for

text: the

text: EM

text: algorithm

text: A

text: ﬁrst

text: approach

text: to

text: derive

text: an

text: approximate

text: E

text: step

text: it

text: to

text: seek

text: for

text: a

text: variational

text: approximation

text: Q

text: of

text: p(h

text: o)

text: assuming

text: that

text: q(h)

text: is

text: restricted

text: to

text: a

text: family

text: of

text: tractable

text: distributions,

text: as

text: described

text: in

text: Section

text: This

text: approach

text: results

text: in

text: the

text: maximization

text: of

text: a

text: lower

text: bound

text: of

text: the

text: original

text: log-likelihood.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: been applied in practice with different approximate distributions, each of them corresponding to a different underlying treewidth (Section ). 2 Graphical Models 2.1 Models deﬁnition (cid:81) Consider a a stochastic system deﬁned by a set of random variables X = (X , . . . , X ). Each 1 ∈n variable X takes values in Λ . Then, a realization of X is a set x = (x , . . . , x ), with x Λ . The i i 1 n i i set of all possible realizations is called the state space, and is denoted Λ = n Λ . If A is a subset { } i=1 i { ∈ } of V = 1, . . . , n , X , x and Λ are respectively the subset of random variables X , i A , {A A∈ } A i the set of realizations x , i A and the state space of X . If p is the joint probability distribution i A of X on Λ, we denote ∀ ∈ x Λ, p(x) = p(X = x). Note that we focus here on discrete variables (we will discuss inference in the case of contin- uous variables on examples in Section ). A joint distribution p on Λ is said to be a probabilistic B graphical model (Lauritzen, 1996; Bishop, 2006; Koller and Friedman, 2009) indexed on a set { } R of parts of V if there exists a set Ψ = ψ ∈B of maps from Λ to +, called potential functions, B B B B indexed by such that p can be expressed in the(cid:89)following factorized form: 1 p(x) = ψ (x ), (1) (cid:80) (cid:81) Z B B ∈B B where Z = ∈ ∈B ψ (x ) is the normalizing constant, also called partition function. (cid:83) ∈ xBΛ B B B | | The elements B are the scopes of the potential functions and B is the arity of the potential function ψ . The set of scopes of all the potential functions involving variable X is denoted B B{ } i = ∈B B . i B(cid:51) B i One desirable property of graphical models is that of Markov local independence: if p(X = x) can be expressed as (1) then a variable X is (stochastically) independent of all others in X con- i ditionally to the set of variables X ∪ \ . This set is called the Markov blanket of X , or its neighborhood. We will denote it N(.BT∈hBieBs)e iconditional independences can be representedigraph- i ically, by a graph with one vertex per variable in X. The question of encoding the independence properties associated with a given distribution into a graph structure is well described in (Koller and Friedman, 2009), and we will not discuss it here. We will consider the classical graph G = (V, E) associated to a decomposition of the form (1) where an edge is drawn between two vertices i and ∈ B j if there exists B such that i and j are in B. Such a representation of a graphical model is B {{ }} actually not as rich as the representation (1). For instance, if n = 3, the two cases = 1, 2, 3 B {{ } { } { }} and = 1, 2 , 2, 3 , 3, 1 are represented by the same graph G, namely a clique of size 3. The factor graph representation goes beyond this limit: this graphical representation is a bipartite graph with one vertex per potential function and one vertex per variable. edges are only between functions and variables. An edge is present between a function vertex (also called factor vertex) and a variable vertex if the variable is in the scope of the potential function. Figure 1 displays examples of the two graphical representations. 4 ##LTLine##

text: been

text: applied

text: in

text: practice

text: with

text: different

text: approximate

text: distributions,

text: each

text: of

text: them

text: corresponding

text: to

text: a

text: different

text: underlying

text: treewidth

text: (Section

label: section

text: Graphical

text: Models

text: Models

text: deﬁnition

text: Consider

text: a

text: a

text: stochastic

text: system

text: deﬁned

text: by

text: a

text: set

text: of

text: random

text: variables

text: X

text: (X

text: X

text: Each

text: ∈n

text: variable

text: X

text: takes

text: values

text: in

text: Λ

text: Then,

text: a

text: realization

text: of

text: X

text: is

text: a

text: set

text: x

text: (x

text: x

text: with

text: x

text: Λ

text: The

text: i

text: i

text: n

text: i

text: i

text: set

text: of

text: all

text: possible

text: realizations

text: is

text: called

text: the

text: state

text: space,

text: and

text: is

text: denoted

text: Λ

text: n

text: Λ

text: If

text: A

text: is

text: a

text: subset

text: i

text: of

text: V

text: n

text: X

text: x

text: and

text: Λ

text: are

text: respectively

text: the

text: subset

text: of

text: random

text: variables

text: X

text: i

text: A

text: {A

text: A∈

text: A

text: i

text: the

text: set

text: of

text: realizations

text: x

text: i

text: A

text: and

text: the

text: state

text: space

text: of

text: X

text: If

text: p

text: is

text: the

text: joint

text: probability

text: distribution

text: i

text: A

text: of

text: X

text: on

text: Λ,

text: we

text: denote

text: x

text: Λ,

text: p(x)

text: p(X

text: x).

text: Note

text: that

text: we

text: focus

text: here

text: on

text: discrete

text: variables

text: (we

text: will

text: discuss

text: inference

text: in

text: the

text: case

text: of

text: contin-

text: uous

text: variables

text: on

text: examples

text: in

text: Section

text: A

text: joint

text: distribution

text: p

text: on

text: Λ

text: is

text: said

text: to

text: be

text: a

text: probabilistic

text: B

text: graphical

text: model

text: (Lauritzen,

text: Bishop,

text: Koller

text: and

text: Friedman,

text: indexed

text: on

text: a

text: set

text: R

text: of

text: parts

text: of

text: V

text: if

text: there

text: exists

text: a

text: set

text: Ψ

text: ψ

text: ∈B

text: of

text: maps

text: from

text: Λ

text: to

text: called

text: potential

text: functions,

text: B

text: B

text: B

text: B

text: indexed

text: by

text: such

text: that

text: p

text: can

text: be

text: expressed

text: in

text: factorized

text: form:

text: p(x)

text: ψ

text: (x

text: Z

text: B

text: B

text: ∈B

text: B

text: where

text: Z

text: ∈B

text: ψ

text: (x

text: is

text: the

text: normalizing

text: constant,

text: also

text: called

text: partition

text: function.

text: xBΛ

text: B

text: B

text: B

text: The

text: elements

text: B

text: are

text: the

text: scopes

text: of

text: the

text: potential

text: functions

text: and

text: B

text: is

text: the

text: arity

text: of

text: the

text: potential

text: function

text: ψ

text: The

text: set

text: of

text: scopes

text: of

text: all

text: the

text: potential

text: functions

text: involving

text: variable

text: X

text: is

text: denoted

text: B

text: B{

text: i

text: ∈B

text: B

text: i

text: B

text: i

text: One

text: desirable

text: property

text: of

text: graphical

text: models

text: is

text: that

text: of

text: Markov

text: local

text: independence:

text: if

text: p(X

text: x)

text: can

text: be

text: expressed

text: as

text: then

text: a

text: variable

text: X

text: is

text: (stochastically)

text: independent

text: of

text: all

text: others

text: in

text: X

text: con-

text: i

text: ditionally

text: to

text: the

text: set

text: of

text: variables

text: X

text: This

text: set

text: is

text: called

text: the

text: Markov

text: blanket

text: of

text: X

text: or

text: its

text: neighborhood.

text: We

text: will

text: denote

text: it

text: N(.BT∈hBieBs)e

text: iconditional

text: independences

text: can

text: be

text: representedigraph-

text: i

text: ically,

text: by

text: a

text: graph

text: with

text: one

text: vertex

text: per

text: variable

text: in

text: X.

text: The

text: question

text: of

text: encoding

text: the

text: independence

text: properties

text: associated

text: with

text: a

text: given

text: distribution

text: into

text: a

text: graph

text: structure

text: is

text: well

text: described

text: in

text: (Koller

text: and

text: Friedman,

text: and

text: we

text: will

text: not

text: discuss

text: it

text: here.

text: We

text: will

text: consider

text: the

text: classical

text: graph

text: G

text: (V,

text: E)

text: associated

text: to

text: a

text: decomposition

text: of

text: the

text: form

text: where

text: an

text: edge

text: is

text: drawn

text: between

text: two

text: vertices

text: i

text: and

text: B

text: j

text: if

text: there

text: exists

text: B

text: such

text: that

text: i

text: and

text: j

text: are

text: in

text: B.

text: Such

text: a

text: representation

text: of

text: a

text: graphical

text: model

text: is

text: B

text: actually

text: not

text: as

text: rich

text: as

text: the

text: representation

text: For

text: instance,

text: if

text: n

text: the

text: two

text: cases

text: B

text: and

text: are

text: represented

text: by

text: the

text: same

text: graph

text: G,

text: namely

text: a

text: clique

text: of

text: size

text: The

text: factor

text: graph

text: representation

text: goes

text: beyond

text: this

text: limit:

text: this

text: graphical

text: representation

text: is

text: a

text: bipartite

text: graph

text: with

text: one

text: vertex

text: per

text: potential

text: function

text: and

text: one

text: vertex

text: per

text: variable.

text: edges

text: are

text: only

text: between

text: functions

text: and

text: variables.

text: An

text: edge

text: is

text: present

text: between

text: a

text: function

text: vertex

text: (also

text: called

text: factor

text: vertex)

text: and

text: a

text: variable

text: vertex

text: if

text: the

text: variable

text: is

text: in

text: the

text: scope

text: of

text: the

text: potential

text: function.

text: Figure

text: displays

text: examples

text: of

text: the

text: two

text: graphical

text: representations.

text: ##LTLine##


text: (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) The function p then satisﬁes: A ⊕ (cid:12) ⊕ (cid:12) (cid:12) (cid:12) p Z = p ψ = ψ A A ∈B B ∈B B xV B xA¯ B (cid:12) × ⊕ where combines functions using and eliminates variables using +. Marginal evaluation is also interesting in the case where some variables are observed. If the val- ⊂ ues of some variables x (O V ) have been observed, we can compute the marginal conditional O distribution by restricting the domains of variables X to the observed value. O The entropy H of a probabilistic graphical model p is deﬁned as − H(p) = E[ln(p(x))], (3) · where E[ ] denotes the mathematical expectation. In the case of a graphical model, by linearity (cid:88) (cid:88) of the expectation, the entropy is equal to − H(p) = ln(Z) p(x ) ln(ψ (x )). B B B ∈B ∈ B xB ΛB (cid:12) ⊕ This expression is an alternation of use of and operators (for p(x ) evaluation, for each B B and x ). B Optimization task An optimization task in a graphical model corresponds to the evaluation of the most probable state x∗ of the random vect(cid:89)or X, deﬁned as (cid:88) ∗ x = arg max p(x) = arg max ψ (x ) = arg max ln ψ (x ). (4) ∈ ∈ B B B B x Λ x Λ ∈B ∈B B B ⊕ (cid:12) ⊕ (cid:12) The maximum itself is ∈B ln ψ (x ) with set to max and to +. The computation ∗ xV B B B of the mode x does not require the computation of the normalizing constant Z, however computing ∗ the mode probability p(x ) does. Therefore counting and optimization tasks can be interpreted as two instantiations of⊕the s(cid:12)ame computational task expressed in terms of combination and elimination operators, namely ∈B ψ ⊂ (cid:12) ⊕ xA B B where A V . When the combination operator and the elimination operator are respectively × set to and +, this computational problem is known as a sum-product problem in the Artiﬁcial ⊕ (cid:12) Intelligence literature (Pearl, 1988),(Bishop, 2006, chapter 8). When is set to max and to the sum operator it is a max-sum problem (Bishop, 2006, chapter 8). We will see in Section that there exists an exact algorithm solving this general task that exploits the distributivity of the combination and elimination operators to perform operations in a smart order. From this generic algorithm, known as variable elimination (Bertele´ and Brioshi, 1972) or bucket elimination (Dechter, 1999), one can deduce exact algorithms to solve counting and ⊕ (cid:12) optimization tasks in a graphical model, by instantiating the operators and . 7

text: The

text: function

text: p

text: then

text: satisﬁes:

text: A

text: p

text: Z

text: p

text: ψ

text: ψ

text: A

text: A

text: ∈B

text: B

text: ∈B

text: B

text: xV

text: B

text: xA¯

text: B

text: where

text: combines

text: functions

text: using

text: and

text: eliminates

text: variables

text: using

text: Marginal

text: evaluation

text: is

text: also

text: interesting

text: in

text: the

text: case

text: where

text: some

text: variables

text: are

text: observed.

text: If

text: the

text: val-

text: ues

text: of

text: some

text: variables

text: x

text: (O

text: V

text: have

text: been

text: observed,

text: we

text: can

text: compute

text: the

text: marginal

text: conditional

text: O

text: distribution

text: by

text: restricting

text: the

text: domains

text: of

text: variables

text: X

text: to

text: the

text: observed

text: value.

text: O

text: The

text: entropy

text: H

text: of

text: a

text: probabilistic

text: graphical

text: model

text: p

text: is

text: deﬁned

text: as

text: H(p)

text: E[ln(p(x))],

text: where

text: E[

text: denotes

text: the

text: mathematical

text: expectation.

text: In

text: the

text: case

text: of

text: a

text: graphical

text: model,

text: by

text: linearity

text: of

text: the

text: expectation,

text: the

text: entropy

text: is

text: equal

text: to

text: H(p)

text: ln(Z)

text: p(x

text: ln(ψ

text: (x

text: B

text: B

text: B

text: ∈B

text: B

text: xB

text: ΛB

text: This

text: expression

text: is

text: an

text: alternation

text: of

text: use

text: of

text: and

text: operators

text: (for

text: p(x

text: evaluation,

text: for

text: each

text: B

text: B

text: and

text: x

text: B

text: Optimization

text: task

text: An

text: optimization

text: task

text: in

text: a

text: graphical

text: model

text: corresponds

text: to

text: the

text: evaluation

text: of

text: the

text: most

text: probable

text: state

text: x∗

text: of

text: the

text: random

text: X,

text: deﬁned

text: as

text: x

text: arg

text: max

text: p(x)

text: arg

text: max

text: ψ

text: (x

text: arg

text: max

text: ln

text: ψ

text: (x

text: B

text: B

text: B

text: B

text: x

text: Λ

text: x

text: Λ

text: ∈B

text: ∈B

text: B

text: B

text: The

text: maximum

text: itself

text: is

text: ∈B

text: ln

text: ψ

text: (x

text: with

text: set

text: to

text: max

text: and

text: to

text: The

text: computation

text: xV

text: B

text: B

text: B

text: of

text: the

text: mode

text: x

text: does

text: not

text: require

text: the

text: computation

text: of

text: the

text: normalizing

text: constant

text: Z,

text: however

text: computing

text: the

text: mode

text: probability

text: p(x

text: does.

text: Therefore

text: counting

text: and

text: optimization

text: tasks

text: can

text: be

text: interpreted

text: as

text: two

text: instantiations

text: of⊕the

text: computational

text: task

text: expressed

text: in

text: terms

text: of

text: combination

text: and

text: elimination

text: operators,

text: namely

text: ∈B

text: ψ

text: xA

text: B

text: B

text: where

text: A

text: V

text: When

text: the

text: combination

text: operator

text: and

text: the

text: elimination

text: operator

text: are

text: respectively

text: set

text: to

text: and

text: this

text: computational

text: problem

text: is

text: known

text: as

text: a

text: sum-product

text: problem

text: in

text: the

text: Artiﬁcial

text: Intelligence

text: literature

text: (Pearl,

text: chapter

text: When

text: is

text: set

text: to

text: max

text: and

text: to

text: the

text: sum

text: operator

text: it

text: is

text: a

text: max-sum

text: problem

text: (Bishop,

text: chapter

text: We

text: will

text: see

text: in

text: Section

text: that

text: there

text: exists

text: an

text: exact

text: algorithm

text: solving

text: this

text: general

text: task

text: that

text: exploits

text: the

text: distributivity

text: of

text: the

text: combination

text: and

text: elimination

text: operators

text: to

text: perform

text: operations

text: in

text: a

text: smart

text: order.

text: From

text: this

text: generic

text: algorithm,

text: known

text: as

text: variable

text: elimination

text: (Bertele´

text: and

text: Brioshi,

text: or

text: bucket

text: elimination

text: (Dechter,

text: one

text: can

text: deduce

text: exact

text: algorithms

text: to

text: solve

text: counting

text: and

text: optimization

text: tasks

text: in

text: a

text: graphical

text: model,

text: by

text: instantiating

text: the

text: operators

text: and