text: the ”· · ·”’s include the contributions from higher Fock components and higher mass con- stituents. In following we will discuss separately each component presented in Eq.(2). pn Component: Kinematically one expects pionic degrees of freedom to become relevant √ − at deuteron internal momenta exceeding sthr 4m2N ≈ 370 MeV/c. However the empirical 2 evidence suggests that the pn component is dominant for deuteron internal momenta up to 650 MeV/c (see e.g. discussion in Ref.[20]). This can be understood based on the following facts: (i) the proportionality of the NπN vertex to the pion momentum, (ii) the form factor − of N → πN transition being hard ∼ exp λt with λ ≥ 3 GeV 2 [21] and (iii) the processes in which the high energy probe couples to the exchanged pion in the deuteron is signiﬁcantly suppressed at high momentum transfer. These facts indicate that the dominant inelastic component can be the ∆∆ rather than the NπN component which will extend the pn dominance for up to p ∼ 800 MeV/c. Despite the possibility for such a dominance, the pn component of the deuteron wave function currently is reliably understood only for up to 400 MeV/c. Many factors contribute to this uncertainty such as the insensitivity of the small distance phenomena to the accuracy of the NN phase shifts, relativistic eﬀects and the treatment of the pion threshold in the NN channel. ∆∆ Component: Due to the large cross section of the πN → ∆ transition and the above discussed suppression of the NπN transition one expects the largest non-nucleonic component in the deuteron to be the ∆∆ component. The current experimental constraints on the overall contribution of the ∆∆ component is ≤ 1%. ∗ ∗ NN Component: In principle the quantum numbers of the deuteron allow the NN component which will correspond to the radial excitation of one of the nucleons in the deuteron. Such an excitation will require an energy of ∼ 600 MeV/c corresponding to internal momenta similar to the ∆∆ component (∼ 800 MeV/c). However, empirically one ∗ expects a smaller NN → NN amplitude which may result in a contribution smaller than the contribution from the ∆∆ component. Currently there is no experimental evidence or ∗ constraint on the possible mixture of the NN component. Hiden Color Component: One of the unique predictions of QCD is the existence of the hidden color component in the deuteron wave function. In fact the color decomposition of a 6q system predicts almost 80% of the wave function strength to be due to the hidden color component[6, 7]. However one expects such a dominance only to occur at very large excitation energies of the NN system when the sum of the possible two-baryonic states in the deuteron is replaced by the six-quark representation. Since such large excitation energies are relevant to the nuclear core it rises an interesting possibility of the NN repulsive core being the result of the suppressed overlap between a hidden color dominated conﬁgurations and the NN component. NNπ Component: Finally, the most dominant three-particle Fock component of the deuteron is the NNπ component which one may expect to appear at excitation energies close to the pion threshold (corresponding to an internal momentum of ∼ 370 MeV/c) and to be sensitive to the external probe at low momentum transfer. There is plenty of evidence of this component from low and intermediate energy reactions which probed meson exchange currents that start to dominate at the missing momentum range of ∼ 350 MeV/c consistent with the above estimate of the pion threshold (see for example Refs.[2, 3]). 4 ##LTLine## ##LTLine##

text: the

text: ·”’s

text: include

text: the

text: contributions

text: from

text: higher

text: Fock

text: components

text: and

text: higher

text: mass

text: con-

text: stituents.

text: In

text: following

text: we

text: will

text: discuss

text: separately

text: each

text: component

text: presented

text: in

text: pn

text: Component:

text: Kinematically

text: one

text: expects

text: pionic

text: degrees

text: of

text: freedom

text: to

text: become

text: relevant

text: at

text: deuteron

text: internal

text: momenta

text: exceeding

text: sthr

text: MeV/c.

text: However

text: the

text: empirical

text: evidence

text: suggests

text: that

text: the

text: pn

text: component

text: is

text: dominant

text: for

text: deuteron

text: internal

text: momenta

text: up

text: to

text: MeV/c

text: (see

text: e.g.

text: discussion

text: in

text: This

text: can

text: be

text: understood

text: based

text: on

text: the

text: following

text: facts:

text: (i)

text: the

text: proportionality

text: of

text: the

text: NπN

text: vertex

text: to

text: the

text: pion

text: momentum,

text: (ii)

text: the

text: form

text: factor

text: of

text: N

text: πN

text: transition

text: being

text: hard

text: exp

text: λt

text: with

text: λ

text: GeV

text: and

text: (iii)

text: the

text: processes

text: in

text: which

text: the

text: high

text: energy

text: probe

text: couples

text: to

text: the

text: exchanged

text: pion

text: in

text: the

text: deuteron

text: is

text: signiﬁcantly

text: suppressed

text: at

text: high

text: momentum

text: transfer.

text: These

text: facts

text: indicate

text: that

text: the

text: dominant

text: inelastic

text: component

text: can

text: be

text: the

text: rather

text: than

text: the

text: NπN

text: component

text: which

text: will

text: extend

text: the

text: pn

text: dominance

text: for

text: up

text: to

text: p

text: MeV/c.

text: Despite

text: the

text: possibility

text: for

text: such

text: a

text: dominance,

text: the

text: pn

text: component

text: of

text: the

text: deuteron

text: wave

text: function

text: currently

text: is

text: reliably

text: understood

text: only

text: for

text: up

text: to

text: MeV/c.

text: Many

text: factors

text: contribute

text: to

text: this

text: uncertainty

text: such

text: as

text: the

text: insensitivity

text: of

text: the

text: small

text: distance

text: phenomena

text: to

text: the

text: accuracy

text: of

text: the

text: NN

text: phase

text: shifts,

text: relativistic

text: eﬀects

text: and

text: the

text: treatment

text: of

text: the

text: pion

text: threshold

text: in

text: the

text: NN

text: channel.

text: Component:

text: Due

text: to

text: the

text: large

text: cross

text: section

text: of

text: the

text: πN

text: transition

text: and

text: the

text: above

text: discussed

text: suppression

text: of

text: the

text: NπN

text: transition

text: one

text: expects

text: the

text: largest

text: non-nucleonic

text: component

text: in

text: the

text: deuteron

text: to

text: be

text: the

text: component.

text: The

text: current

text: experimental

text: constraints

text: on

text: the

text: overall

text: contribution

text: of

text: the

text: component

text: is

text: NN

text: Component:

text: In

text: principle

text: the

text: quantum

text: numbers

text: of

text: the

text: deuteron

text: allow

text: the

text: NN

text: component

text: which

text: will

text: correspond

text: to

text: the

text: radial

text: excitation

text: of

text: one

text: of

text: the

text: nucleons

text: in

text: the

text: deuteron.

text: Such

text: an

text: excitation

text: will

text: require

text: an

text: energy

text: of

text: MeV/c

text: corresponding

text: to

text: internal

text: momenta

text: similar

text: to

text: the

text: component

text: MeV/c).

text: However,

text: empirically

text: one

text: expects

text: a

text: smaller

text: NN

text: NN

text: amplitude

text: which

text: may

text: result

text: in

text: a

text: contribution

text: smaller

text: than

text: the

text: contribution

text: from

text: the

text: component.

text: Currently

text: there

text: is

text: no

text: experimental

text: evidence

text: or

text: constraint

text: on

text: the

text: possible

text: mixture

text: of

text: the

text: NN

text: component.

text: Hiden

text: Color

text: Component:

text: One

text: of

text: the

text: unique

text: predictions

text: of

text: QCD

text: is

text: the

text: existence

text: of

text: the

text: hidden

text: color

text: component

text: in

text: the

text: deuteron

text: wave

text: function.

text: In

text: fact

text: the

text: color

text: decomposition

text: of

text: a

text: system

text: predicts

text: almost

text: of

text: the

text: wave

text: function

text: strength

text: to

text: be

text: due

text: to

text: the

text: hidden

text: color

text: However

text: one

text: expects

text: such

text: a

text: dominance

text: only

text: to

text: occur

text: at

text: very

text: large

text: excitation

text: energies

text: of

text: the

text: NN

text: system

text: when

text: the

text: sum

text: of

text: the

text: possible

text: two-baryonic

text: states

text: in

text: the

text: deuteron

text: is

text: replaced

text: by

text: the

text: six-quark

text: representation.

text: Since

text: such

text: large

text: excitation

text: energies

text: are

text: relevant

text: to

text: the

text: nuclear

text: core

text: it

text: rises

text: an

text: interesting

text: possibility

text: of

text: the

text: NN

text: repulsive

text: core

text: being

text: the

text: result

text: of

text: the

text: suppressed

text: overlap

text: between

text: a

text: hidden

text: color

text: dominated

text: conﬁgurations

text: and

text: the

text: NN

text: component.

text: NNπ

text: Component:

text: Finally,

text: the

text: most

text: dominant

text: three-particle

text: Fock

text: component

text: of

text: the

text: deuteron

text: is

text: the

text: NNπ

text: component

text: which

text: one

text: may

text: expect

text: to

text: appear

text: at

text: excitation

text: energies

text: close

text: to

text: the

text: pion

text: threshold

text: (corresponding

text: to

text: an

text: internal

text: momentum

text: of

text: MeV/c)

text: and

text: to

text: be

text: sensitive

text: to

text: the

text: external

text: probe

text: at

text: low

text: momentum

text: transfer.

text: There

text: is

text: plenty

text: of

text: evidence

text: of

text: this

text: component

text: from

text: low

text: and

text: intermediate

text: energy

text: reactions

text: which

text: probed

text: meson

text: exchange

text: currents

text: that

text: start

text: to

text: dominate

text: at

text: the

text: missing

text: momentum

text: range

text: of

text: MeV/c

text: consistent

text: with

text: the

text: above

text: estimate

text: of

text: the

text: pion

text: threshold

text: (see

text: for

text: example

text: ##LTLine##

text: ##LTLine##


text: (e) Intermediate State Resonance Production: In Fig.1(e), the electromagnetic in- teraction excites the nucleon into a resonance state which then rescatters with the spectator nucleon into the ﬁnal proton and neutron. (f) Meson Exchange Contributions: In Fig.1(f)(g), the electromagnetic interaction takes places with the mesons which are exchanged between initial nucleons in the deuteron. (g) Non-Nucleonic Contributions: Final three terms contributing to the reaction (Fig.1(h) (i) and (j)) are sensitive to the non-nucleonic component of the deuteron wave function. Here the ﬁrst two represent the baryonic and the last, the hidden-color component contributions which was discussed in the Sec.. IV. CONCEPT OF PROBING THE HIGH MOMENTUM COMPONENT OF THE DEUTERON To probe the high momentum NN component of the deuteron, in the diagrammatic presentation of the scattering process of Eq.(1), is to isolate the contribution from the direct PWIA process1 (Fig.1(a)) at such momentum transfer q and ﬁnal proton momentum p p that the calculated missing momentum, according to Eq.(3), p > 300 MeV/c. m Such an isolation however requires a suppression or reliable accounting for all the remain- ing contributions (diagrams Fig.1(b)-(j)) discussed in the previous section. Our assertion is that this can be achieved if we consider the reaction (1) in high energy kinematics in which the transferred momentum q ≥ few GeV/c and the virtuality of the probe, Q2 > 1 GeV2, with an additional condition that the ﬁnal nucleons (proton in our case) carries almost all the momentum of the virtual photon while the recoiling nucleon (neutron) is signiﬁcantly less energetic, i.e.: p ∼ q ∼ few GeV/c and |p | = |(cid:126)q −p(cid:126) | ∼ few hunderd MeV/c and Q2 > 1 GeV 2. (4) p n p The eﬀects of the above conditions are diﬀerent for diﬀerent diagrams of Fig.1(b)-(j), which we can categorize as kinematical, dynamical and the mixture of both. The suppression of the spectator PWIA diagram, Fig.1(b), is purely kinematical, since in this case the amplitude of the scattering will be deﬁned by the deuteron wave function, ∼ ψ (p ) with the initial momentum of p ∼ few GeV/c as compared to the PWIA term d p p which is proportional to ∼ ψ (p ) with p ∼ few hundred MeV/c. d n n The diagrams containing meson exchange currents (Fig.1(f),(g)) will be suppressed dy- namically, since in the limit of Q2 (cid:29) m2 ∼ 1GeV2 they are suppressed compared to meson the PWIA term by an extra factor of Q6[22, 23]. Another dynamical suppression occurs in the scattering followed by the charge-interchange rescattering (Fig.1(d). In this case the − suppression is due to an extra s 1/2 factor as well as a much stronger t dependence in the pn → np amplitude as compared to the pn → pn amplitude that enters in the direct FSI 1 Hereafter refereed as PWIA process. 6 ##LTLine##

text: (e)

label: list

text: Intermediate

text: State

text: Resonance

text: Production:

text: In

text: the

text: electromagnetic

text: in-

text: teraction

text: excites

text: the

text: nucleon

text: into

text: a

text: resonance

text: state

text: which

text: then

text: rescatters

text: with

text: the

text: spectator

text: nucleon

text: into

text: the

text: ﬁnal

text: proton

text: and

text: neutron.

text: (f)

text: Meson

text: Exchange

text: Contributions:

text: In

text: the

text: electromagnetic

text: interaction

text: takes

text: places

text: with

text: the

text: mesons

text: which

text: are

text: exchanged

text: between

text: initial

text: nucleons

text: in

text: the

text: deuteron.

text: (g)

text: Non-Nucleonic

text: Contributions:

text: Final

text: three

text: terms

text: contributing

text: to

text: the

text: reaction

text: (i)

text: and

text: (j))

text: are

text: sensitive

text: to

text: the

text: non-nucleonic

text: component

text: of

text: the

text: deuteron

text: wave

text: function.

text: Here

text: the

text: ﬁrst

text: two

text: represent

text: the

text: baryonic

text: and

text: the

text: last,

text: the

text: hidden-color

text: component

text: contributions

text: which

text: was

text: discussed

text: in

text: the

text: Sec..

text: IV.

label: section

text: CONCEPT

text: OF

text: PROBING

text: THE

text: HIGH

text: MOMENTUM

text: COMPONENT

text: OF

text: THE

text: DEUTERON

text: To

text: probe

text: the

text: high

text: momentum

text: NN

text: component

text: of

text: the

text: deuteron,

text: in

text: the

text: diagrammatic

text: presentation

text: of

text: the

text: scattering

text: process

text: of

text: is

text: to

text: isolate

text: the

text: contribution

text: from

text: the

text: direct

text: PWIA

text: at

text: such

text: momentum

text: transfer

text: q

text: and

text: ﬁnal

text: proton

text: momentum

text: p

text: p

text: that

text: the

text: calculated

text: missing

text: momentum,

text: according

text: to

text: p

text: MeV/c.

text: m

text: Such

text: an

text: isolation

text: however

text: requires

text: a

text: suppression

text: or

text: reliable

text: accounting

text: for

text: all

text: the

text: remain-

text: ing

text: contributions

text: (diagrams

text: discussed

text: in

text: the

text: previous

text: section.

text: Our

text: assertion

text: is

text: that

text: this

text: can

text: be

text: achieved

text: if

text: we

text: consider

text: the

text: reaction

text: in

text: high

text: energy

text: kinematics

text: in

text: which

text: the

text: transferred

text: momentum

text: q

text: few

text: GeV/c

text: and

text: the

text: virtuality

text: of

text: the

text: probe,

text: with

text: an

text: additional

text: condition

text: that

text: the

text: ﬁnal

text: nucleons

text: (proton

text: in

text: our

text: case)

text: carries

text: almost

text: all

text: the

text: momentum

text: of

text: the

text: virtual

text: photon

text: while

text: the

text: recoiling

text: nucleon

text: (neutron)

text: is

text: signiﬁcantly

text: less

text: energetic,

text: i.e.:

text: p

text: q

text: few

text: GeV/c

text: and

text: |p

text: few

text: hunderd

text: MeV/c

text: and

text: GeV

text: p

text: n

text: p

text: The

text: eﬀects

text: of

text: the

text: above

text: conditions

text: are

text: diﬀerent

text: for

text: diﬀerent

text: diagrams

text: of

text: which

text: we

text: can

text: categorize

text: as

text: kinematical,

text: dynamical

text: and

text: the

text: mixture

text: of

text: both.

text: The

text: suppression

text: of

text: the

text: spectator

text: PWIA

text: diagram,

text: is

text: purely

text: kinematical,

text: since

text: in

text: this

text: case

text: the

text: amplitude

text: of

text: the

text: scattering

text: will

text: be

text: deﬁned

text: by

text: the

text: deuteron

text: wave

text: function,

text: ψ

text: (p

text: with

text: the

text: initial

text: momentum

text: of

text: p

text: few

text: GeV/c

text: as

text: compared

text: to

text: the

text: PWIA

text: term

text: d

text: p

text: p

text: which

text: is

text: proportional

text: to

text: ψ

text: (p

text: with

text: p

text: few

text: hundred

text: MeV/c.

text: d

text: n

text: n

text: The

text: diagrams

text: containing

text: meson

text: exchange

text: currents

text: will

text: be

text: suppressed

text: dy-

text: namically,

text: since

text: in

text: the

text: limit

text: of

text: they

text: are

text: suppressed

text: compared

text: to

text: meson

text: the

text: PWIA

text: term

text: by

text: an

text: extra

text: factor

text: of

text: Another

text: dynamical

text: suppression

text: occurs

text: in

text: the

text: scattering

text: followed

text: by

text: the

text: charge-interchange

text: rescattering

text: In

text: this

text: case

text: the

text: suppression

text: is

text: due

text: to

text: an

text: extra

text: s

text: factor

text: as

text: well

text: as

text: a

text: much

text: stronger

text: t

text: dependence

text: in

text: the

text: pn

text: np

text: amplitude

text: as

text: compared

text: to

text: the

text: pn

text: pn

text: amplitude

text: that

text: enters

text: in

text: the

text: direct

text: FSI

text: Hereafter

text: refereed

text: as

text: PWIA

text: process.

text: ##LTLine##


text: 1 Communication Theoretic Data Analytics Kwang-Cheng Chen, Shao-Lun Huang, Lizhong Zheng, H. Vincent Poor Abstract— Widespread use of the Internet and social from data/observations. Statistical methodologies deal networks invokes the generation of big data, which is with uncertain or nondeterministic reasoning and thus proving to be useful in a number of applications. To deal models, and are the focus of this paper. Machine learning with explosively growing amounts of data, data analytics and artiﬁcial intelligence are useful to analyze data, has emerged as a critical technology related to computing, signal processing, and information networking. In this pa- e.g. [2][3][7], typically via regression and/or classiﬁ- per, a formalism is considered in which data is modeled as a cation. With advances in supervised and unsupervised generalized social network and communication theory and learning, inferring the structure of knowledge, such as information theory are thereby extended to data analytics. inferring Bayesian network structure from data, is one of First, the creation of an equalizer to optimize information the most useful information technologies [8]. In recent transfer between two data variables is considered, and ﬁnancial data is used to demonstrate the advantages. Then, decades, considerable research effort has been devoted an information coupling approach based on information to various aspects of data mining and data analysis, geometry is applied for dimensionality reduction, with a but effective data analytics are still needed to address pattern recognition example to illustrate the effectiveness. the explosively growing amount of data resulting from These initial trials suggest the potential of communication Internet, mobile, and social networks. theoretic data analytics for a wide range of applications. A core technological direction in data analytics lies Index Terms—big data, social networks, data analysis, in processing high-dimensional data to obtain low- communication theory, information theory, information coupling, equalization, information fusion, data mining, dimensional information via computational reduction knowledge discovery, information centric processing algorithms, namely by nonlinear approaches [9][10], compressive sensing [11], or tensor geometric analy- sis [12]. In spite of remarkable achievements, with the I. INTRODUCTION exponential growth in data volume, it is very desirable to With the booming of Internet and mobile commu- develop more effective approaches to deal with existing nications, (big) data analytics has emerged as a crit- challenges including effective algorithms of scalable ical technology, adopting techniques such as machine computation, complexity and data size, outlier detec- learning, graphical models, etc. to mine desirable infor- tion and prediction, etc. Furthermore, modern wireless mation for a wide array of information communication communication systems and networks to support mobile technology (ICT) applications [1][2][3][4][5][6]. Data Internet and Internet of Things (IoT) applications require mining or knowledge discovery in databases (KDD) effective transport of information, while appropriate data has been used as a synonym for analytics on computer analytics enable communication spectral efﬁciency via generated data. To achieve the purpose of data analytics, proper context-aware computation [13]. The technologi- there are several major steps: (i) based on the selection cal challenge for data analytics due to very large numbers of data set(s), pre-processing of the data for effective of devices and data volume remains on the list of the or easy computation, (ii) processing of data or data most necessary avenues of inquiry. At this time, state-of- mining, likely adopting techniques from statistical infer- the-art data analytics primarily deal with data processing ence and artiﬁcial intelligence, and (iii) post-processing through computational models and techniques, such as to appropriately interpret results of data analytics as deep learning [14]. There lack efforts to examine the knowledge. Knowledge discovery aims at either veriﬁ- mechanism of data generation [15] and subsequent rela- cation of user hypotheses or prediction of future patterns tionships among data, which motivates the investigation of data analytics by leveraging communication theory This research was supported in part by the US Air Force under the and information theory in this paper. grant AOARD-14-4053, the US NSF grant CCF-1216476, the U.S. Army Research Ofﬁce under MURI Grant W911NF-11-1-0036, and As indicated in [16] and other works, relationships the U. S. National Science Foundation under Grant CCF-1420575. among data can be viewed as a type of generalized social K.C. Chen and S.-L. Huang are with the Graduate Institute of Com- network. The data variables can be treated as nodes in munication Engineering, National Taiwan University, ckc@ntu.edu.tw S.-L. Huang and Lizhong Zheng are with the Department of Elec- a network and their corresponding relationships can be trical Engineering and Computer Science, Massachusetts Institute of considered to be links governed by random variables (or Technology, shaolin@mit.edu and lizhong@mit.edu random processes by considering a time index). Such H.V. Poor is with the Department of Electrical Engineering, Prince- ton University, poor@princeton.edu scenarios are particularly relevant for today’s interactive

label: paragraph

text: Communication

label: title

text: Theoretic

text: Data

text: Analytics

text: Kwang-Cheng

label: author

text: Chen,

text: Shao-Lun

text: Huang,

text: Lizhong

text: Zheng,

text: H.

text: Vincent

text: Poor

text: Abstract—

text: Widespread

label: abstract

text: use

text: of

text: the

text: Internet

text: and

text: social

text: from

text: data/observations.

text: Statistical

text: methodologies

text: deal

text: networks

text: invokes

text: the

text: generation

text: of

text: big

text: data,

text: which

text: is

text: with

text: uncertain

text: or

text: nondeterministic

text: reasoning

text: and

text: thus

text: proving

text: to

text: be

text: useful

text: in

text: a

text: number

text: of

text: applications.

text: To

text: deal

text: models,

text: and

text: are

text: the

text: focus

text: of

text: this

text: paper.

text: Machine

text: learning

text: with

text: explosively

text: growing

text: amounts

text: of

text: data,

text: data

text: analytics

text: and

text: artiﬁcial

text: intelligence

text: are

text: useful

text: to

text: analyze

text: data,

text: has

text: emerged

text: as

text: a

text: critical

text: technology

text: related

text: to

text: computing,

text: signal

text: processing,

text: and

text: information

text: networking.

text: In

text: this

text: pa-

text: e.g.

text: typically

text: via

text: regression

text: and/or

text: classiﬁ-

text: per,

text: a

text: formalism

text: is

text: considered

text: in

text: which

text: data

text: is

text: modeled

text: as

text: a

text: cation.

text: With

text: advances

text: in

text: supervised

text: and

text: unsupervised

text: generalized

text: social

text: network

text: and

text: communication

text: theory

text: and

text: learning,

text: inferring

text: the

text: structure

text: of

text: knowledge,

text: such

text: as

text: information

text: theory

text: are

text: thereby

text: extended

text: to

text: data

text: analytics.

text: inferring

text: Bayesian

text: network

text: structure

text: from

text: data,

text: is

text: one

text: of

text: First,

text: the

text: creation

text: of

text: an

text: equalizer

text: to

text: optimize

text: information

text: the

text: most

text: useful

text: information

text: technologies

text: In

text: recent

text: transfer

text: between

text: two

text: data

text: variables

text: is

text: considered,

text: and

text: ﬁnancial

text: data

text: is

text: used

text: to

text: demonstrate

text: the

text: advantages.

text: Then,

text: decades,

text: considerable

text: research

text: effort

text: has

text: been

text: devoted

text: an

text: information

text: coupling

text: approach

text: based

text: on

text: information

text: to

text: various

text: aspects

text: of

text: data

text: mining

text: and

text: data

text: analysis,

text: geometry

text: is

text: applied

text: for

text: dimensionality

text: reduction,

text: with

text: a

text: but

text: effective

text: data

text: analytics

text: are

text: still

text: needed

text: to

text: address

text: pattern

text: recognition

text: example

text: to

text: illustrate

text: the

text: effectiveness.

text: the

text: explosively

text: growing

text: amount

text: of

text: data

text: resulting

text: from

text: These

text: initial

text: trials

text: suggest

text: the

text: potential

text: of

text: communication

text: Internet,

text: mobile,

text: and

text: social

text: networks.

text: theoretic

text: data

text: analytics

text: for

text: a

text: wide

text: range

text: of

text: applications.

text: A

text: core

text: technological

text: direction

text: in

text: data

text: analytics

text: lies

text: Index

text: Terms—big

text: data,

text: social

text: networks,

text: data

text: analysis,

text: in

text: processing

text: high-dimensional

text: data

text: to

text: obtain

text: low-

text: communication

text: theory,

text: information

text: theory,

text: information

text: coupling,

text: equalization,

text: information

text: fusion,

text: data

text: mining,

text: dimensional

text: information

text: via

text: computational

text: reduction

text: knowledge

text: discovery,

text: information

text: centric

text: processing

text: algorithms,

text: namely

text: by

text: nonlinear

text: approaches

text: compressive

text: sensing

text: or

text: tensor

text: geometric

text: analy-

text: sis

text: In

text: spite

text: of

text: remarkable

text: achievements,

text: with

text: the

text: I.

text: INTRODUCTION

text: exponential

text: growth

text: in

text: data

text: volume,

text: it

text: is

text: very

text: desirable

text: to

text: With

text: the

text: booming

text: of

text: Internet

text: and

text: mobile

text: commu-

text: develop

text: more

text: effective

text: approaches

text: to

text: deal

text: with

text: existing

text: nications,

text: (big)

text: data

text: analytics

text: has

text: emerged

text: as

text: a

text: crit-

text: challenges

text: including

text: effective

text: algorithms

text: of

text: scalable

text: ical

text: technology,

text: adopting

text: techniques

text: such

text: as

text: machine

text: computation,

text: complexity

text: and

text: data

text: size,

text: outlier

text: detec-

text: learning,

text: graphical

text: models,

text: etc.

text: to

text: mine

text: desirable

text: infor-

text: tion

text: and

text: prediction,

text: etc.

text: Furthermore,

text: modern

text: wireless

text: mation

text: for

text: a

text: wide

text: array

text: of

text: information

text: communication

text: communication

text: systems

text: and

text: networks

text: to

text: support

text: mobile

text: technology

text: (ICT)

text: applications

text: Data

text: Internet

text: and

text: Internet

text: of

text: Things

text: (IoT)

text: applications

text: require

text: mining

text: or

text: knowledge

text: discovery

text: in

text: databases

text: (KDD)

text: effective

text: transport

text: of

text: information,

text: while

text: appropriate

text: data

text: has

text: been

text: used

text: as

text: a

text: synonym

text: for

text: analytics

text: on

text: computer

text: analytics

text: enable

text: communication

text: spectral

text: efﬁciency

text: via

text: generated

text: data.

text: To

text: achieve

text: the

text: purpose

text: of

text: data

text: analytics,

text: proper

text: context-aware

text: computation

text: The

text: technologi-

text: there

text: are

text: several

text: major

text: steps:

text: (i)

text: based

text: on

text: the

text: selection

text: cal

text: challenge

text: for

text: data

text: analytics

text: due

text: to

text: very

text: large

text: numbers

text: of

text: data

text: set(s),

text: pre-processing

text: of

text: the

text: data

text: for

text: effective

text: of

text: devices

text: and

text: data

text: volume

text: remains

text: on

text: the

text: list

text: of

text: the

text: or

text: easy

text: computation,

text: (ii)

text: processing

text: of

text: data

text: or

text: data

text: most

text: necessary

text: avenues

text: of

text: inquiry.

text: At

text: this

text: time,

text: state-of-

text: mining,

text: likely

text: adopting

text: techniques

text: from

text: statistical

text: infer-

text: the-art

text: data

text: analytics

text: primarily

text: deal

text: with

text: data

text: processing

text: ence

text: and

text: artiﬁcial

text: intelligence,

text: and

text: (iii)

text: post-processing

text: through

text: computational

text: models

text: and

text: techniques,

text: such

text: as

text: to

text: appropriately

text: interpret

text: results

text: of

text: data

text: analytics

text: as

text: deep

text: learning

text: There

text: lack

text: efforts

text: to

text: examine

text: the

text: knowledge.

text: Knowledge

text: discovery

text: aims

text: at

text: either

text: veriﬁ-

text: mechanism

text: of

text: data

text: generation

text: and

text: subsequent

text: rela-

text: cation

text: of

text: user

text: hypotheses

text: or

text: prediction

text: of

text: future

text: patterns

text: tionships

text: among

text: data,

text: which

text: motivates

text: the

text: investigation

text: of

text: data

text: analytics

text: by

text: leveraging

text: communication

text: theory

text: This

text: research

text: was

text: supported

text: in

text: part

text: by

text: the

text: US

text: Air

text: Force

text: under

text: the

text: and

text: information

text: theory

text: in

text: this

text: paper.

text: grant

text: the

text: US

text: NSF

text: grant

text: the

text: U.S.

text: Army

text: Research

text: Ofﬁce

text: under

text: MURI

text: Grant

text: and

text: As

text: indicated

text: in

text: and

text: other

text: works,

text: relationships

text: the

text: U.

text: S.

text: National

text: Science

text: Foundation

text: under

text: Grant

text: among

text: data

text: can

text: be

text: viewed

text: as

text: a

text: type

text: of

text: generalized

text: social

text: K.C.

text: Chen

text: and

text: S.-L.

text: Huang

text: are

text: with

text: the

text: Graduate

text: Institute

text: of

text: Com-

text: network.

text: The

text: data

text: variables

text: can

text: be

text: treated

text: as

text: nodes

text: in

text: munication

text: Engineering,

text: National

text: Taiwan

text: University,

text: ckc@ntu.edu.tw

text: S.-L.

text: Huang

text: and

text: Lizhong

text: Zheng

text: are

text: with

text: the

text: Department

text: of

text: Elec-

text: a

text: network

text: and

text: their

text: corresponding

text: relationships

text: can

text: be

text: trical

text: Engineering

text: and

text: Computer

text: Science,

text: Massachusetts

text: Institute

text: of

text: considered

text: to

text: be

text: links

text: governed

text: by

text: random

text: variables

text: (or

text: Technology,

text: shaolin@mit.edu

text: and

text: lizhong@mit.edu

text: random

text: processes

text: by

text: considering

text: a

text: time

text: index).

text: Such

text: H.V.

text: Poor

text: is

text: with

text: the

text: Department

text: of

text: Electrical

text: Engineering,

text: Prince-

text: ton

text: University,

text: poor@princeton.edu

text: scenarios

text: are

text: particularly

text: relevant

text: for

text: today’s

text: interactive


text: 4 series, and thus allow better judgement or utilization of the causal relationship. This is one core issue in knowl- edge discovery of data analytics. In big data analytics, this knowledge problem has a very large number of variables. As in Figure 1, we focus on the problem of identifying the causal relationships between the set · · · of variables X , X , , X and the set of variables · · · 1 2 M Y , Y , , Y , speciﬁed by appropriate weights. This is 1 2 N identical to the following multiple access communication Fig. 4. Equalizer structure to extract the causal relationship between problem: two variables, from X to Y . A common and most fundamental scenario in (big) (X , X , . . . , X )[h ] × = (Y , Y , . . . , Y )T (1) 1 2 M ij M N 1 2 N data analytics is to ﬁnd the relationship between two variables, X and Y , while each variable represents a where [h ] = H is analogous to the channel ma- set or a series of data. Let us borrow the concept ij trix. This knowledge discovery problem in data an- of Bayesian networks or graphical models in machine alytics is thus equivalent to a blind channel estima- learning. We may subsequently infer the realization y of tion/identiﬁcation problem in multiple access commu- Y based on the observation of the realizatio→n x of X. nication. Since a feedback channel may not exist in Suppose there exists a causal relationship X Y , with general, this is a blind problem. However, for online uncertainty (or embedded in an unknown mechanism processing, equivalent feedback might not be impossible, allowing information transfer). As noted above, this which we leave as an open issue. We start from some causal relationship can be represented by the conditional | simple cases to illustrate this idea. or transition probability f(y x), which can be considered • Information diversity: A variable X may inﬂuence as a communication channel. Direct computation and · · · a number of variables, say Y , Y , , Y , which is inference therefore proceed based on this structure. If 1 2 N a form of information diversity. To identify a causal we intend to infer the knowledge structure such as this relationship between X and Y , this is precisely a simple causal relationship, machine learning on (big) n multi-channel estimation problem such as arises in data becomes a principal problem [2][5][6][7]. wireless communication. Since feedback in causal A. Equalizer to Optimize Information Transfer data relationships is generally impossible, such a Again, our view of this simple causal relationship class of problems falls under the category of blind considers information transfer from X to Y as a com- multi-channel estimation/identiﬁcation [24][25]. munication channel. In this context, if we want to deter- • Information fusion: Another class to consider mine the causal relationship of data due to information is the causal relationship from many data vari- transfer, according to communication theory, we should ables to inﬂuence a single data variable, say · · · establish an equalizer for the channel to better receive X , X , , X to Y . This corresponds to multi- 1 2 M such information. The equalizer in data analytics is most input-single-output channel estimation or identiﬁca- appropriately of the form of an adaptive equalizer, with tion, which is a rather overlooked subject. However, earlier observed data used for the purpose of training to another similar problem, source separation has been obtain the weighting coefﬁcients. The most challenging well studied. task in machine learning is knowledge discovery, i.e., identifying the causal relationship among variables. The B. Applications to Inference on Financial Time Series communication theoretic approach supplies a new angle from which to examine this task, and its information A useful way to demonstrate our analytical method- theoretic insight will be investigated in Section IV via ology is to consider ﬁnancial time series data, which information coupling. has been well studied in the literature. The purpose Well known in communication theory, an equalizer is to demonstrate the prediction of stock prices from serves as a part of an optimal receiver to detect a other factors. In this example, we are trying to predict transmitted signal from a noisy channel distorted by the stock price of Taiwan Semiconductor Manufacturing nonlinearities and other impairments. In order to infer Corp. (TSMC), which is the world’s largest contract from another data variable or time series, we may take chip fabrication company. To demonstrate information advantage of the same concept to process the data. transfer and thus communication theoretic data analytics, Proper implementation of an equalizer could enhance we consider two factors, which appear to be somewhat the causal relationship between data variables and time indirectly related to stock prices in Taiwan but are ##LTFigure##

text: series,

text: and

text: thus

text: allow

text: better

text: judgement

text: or

text: utilization

text: of

text: the

text: causal

text: relationship.

text: This

text: is

text: one

text: core

text: issue

text: in

text: knowl-

text: edge

text: discovery

text: of

text: data

text: analytics.

text: In

text: big

text: data

text: analytics,

text: this

text: knowledge

text: problem

text: has

text: a

text: very

text: large

text: number

text: of

text: variables.

text: As

text: in

text: Figure

text: we

text: focus

text: on

text: the

text: problem

text: of

text: identifying

text: the

text: causal

text: relationships

text: between

text: the

text: set

text: of

text: variables

text: X

text: X

text: X

text: and

text: the

text: set

text: of

text: variables

text: M

text: Y

text: Y

text: Y

text: speciﬁed

text: by

text: appropriate

text: weights.

text: This

text: is

text: N

text: identical

text: to

text: the

text: following

text: multiple

text: access

text: communication

text: Fig.

text: Equalizer

text: structure

text: to

text: extract

text: the

text: causal

text: relationship

text: between

text: problem:

text: two

text: variables,

text: from

text: X

text: to

text: Y

text: A

text: common

text: and

text: most

text: fundamental

text: scenario

text: in

text: (big)

text: (X

text: X

text: X

text: )[h

text: (Y

text: Y

text: Y

text: )T

text: M

text: ij

text: M

text: N

text: N

text: data

text: analytics

text: is

text: to

text: ﬁnd

text: the

text: relationship

text: between

text: two

text: variables,

text: X

text: and

text: Y

text: while

text: each

text: variable

text: represents

text: a

text: where

text: [h

text: H

text: is

text: analogous

text: to

text: the

text: channel

text: ma-

text: set

text: or

text: a

text: series

text: of

text: data.

text: Let

text: us

text: borrow

text: the

text: concept

text: ij

text: trix.

text: This

text: knowledge

text: discovery

text: problem

text: in

text: data

text: an-

text: of

text: Bayesian

text: networks

text: or

text: graphical

text: models

text: in

text: machine

text: alytics

text: is

text: thus

text: equivalent

text: to

text: a

text: blind

text: channel

text: estima-

text: learning.

text: We

text: may

text: subsequently

text: infer

text: the

text: realization

text: y

text: of

text: tion/identiﬁcation

text: problem

text: in

text: multiple

text: access

text: commu-

text: Y

text: based

text: on

text: the

text: observation

text: of

text: the

text: realizatio→n

text: x

text: of

text: X.

text: nication.

text: Since

text: a

text: feedback

text: channel

text: may

text: not

text: exist

text: in

text: Suppose

text: there

text: exists

text: a

text: causal

text: relationship

text: X

text: Y

text: with

text: general,

text: this

text: is

text: a

text: blind

text: problem.

text: However,

text: for

text: online

text: uncertainty

text: (or

text: embedded

text: in

text: an

text: unknown

text: mechanism

text: processing,

text: equivalent

text: feedback

text: might

text: not

text: be

text: impossible,

text: allowing

text: information

text: transfer).

text: As

text: noted

text: above,

text: this

text: which

text: we

text: leave

text: as

text: an

text: open

text: issue.

text: We

text: start

text: from

text: some

text: causal

text: relationship

text: can

text: be

text: represented

text: by

text: the

text: conditional

text: simple

text: cases

text: to

text: illustrate

text: this

text: idea.

text: or

text: transition

text: probability

text: f(y

text: x),

text: which

text: can

text: be

text: considered

text: Information

text: diversity:

text: A

text: variable

text: X

text: may

text: inﬂuence

text: as

text: a

text: communication

text: channel.

text: Direct

text: computation

text: and

text: a

text: number

text: of

text: variables,

text: say

text: Y

text: Y

text: Y

text: which

text: is

text: inference

text: therefore

text: proceed

text: based

text: on

text: this

text: structure.

text: If

text: N

text: a

text: form

text: of

text: information

text: diversity.

text: To

text: identify

text: a

text: causal

text: we

text: intend

text: to

text: infer

text: the

text: knowledge

text: structure

text: such

text: as

text: this

text: relationship

text: between

text: X

text: and

text: Y

text: this

text: is

text: precisely

text: a

text: simple

text: causal

text: relationship,

text: machine

text: learning

text: on

text: (big)

text: n

text: multi-channel

text: estimation

text: problem

text: such

text: as

text: arises

text: in

text: data

text: becomes

text: a

text: principal

text: problem

text: wireless

text: communication.

text: Since

text: feedback

text: in

text: causal

text: A.

text: Equalizer

text: to

text: Optimize

text: Information

text: Transfer

text: data

text: relationships

text: is

text: generally

text: impossible,

text: such

text: a

text: Again,

text: our

text: view

text: of

text: this

text: simple

text: causal

text: relationship

text: class

text: of

text: problems

text: falls

text: under

text: the

text: category

text: of

text: blind

text: considers

text: information

text: transfer

text: from

text: X

text: to

text: Y

text: as

text: a

text: com-

text: multi-channel

text: estimation/identiﬁcation

text: munication

text: channel.

text: In

text: this

text: context,

text: if

text: we

text: want

text: to

text: deter-

text: Information

text: fusion:

text: Another

text: class

text: to

text: consider

text: mine

text: the

text: causal

text: relationship

text: of

text: data

text: due

text: to

text: information

text: is

text: the

text: causal

text: relationship

text: from

text: many

text: data

text: vari-

text: transfer,

text: according

text: to

text: communication

text: theory,

text: we

text: should

text: ables

text: to

text: inﬂuence

text: a

text: single

text: data

text: variable,

text: say

text: establish

text: an

text: equalizer

text: for

text: the

text: channel

text: to

text: better

text: receive

text: X

text: X

text: X

text: to

text: Y

text: This

text: corresponds

text: to

text: multi-

text: M

text: such

text: information.

text: The

text: equalizer

text: in

text: data

text: analytics

text: is

text: most

text: input-single-output

text: channel

text: estimation

text: or

text: identiﬁca-

text: appropriately

text: of

text: the

text: form

text: of

text: an

text: adaptive

text: equalizer,

text: with

text: tion,

text: which

text: is

text: a

text: rather

text: overlooked

text: subject.

text: However,

text: earlier

text: observed

text: data

text: used

text: for

text: the

text: purpose

text: of

text: training

text: to

text: another

text: similar

text: problem,

text: source

text: separation

text: has

text: been

text: obtain

text: the

text: weighting

text: coefﬁcients.

text: The

text: most

text: challenging

text: well

text: studied.

text: task

text: in

text: machine

text: learning

text: is

text: knowledge

text: discovery,

text: i.e.,

text: identifying

text: the

text: causal

text: relationship

text: among

text: variables.

text: The

text: B.

text: Applications

text: to

text: Inference

text: on

text: Financial

text: Time

text: Series

text: communication

text: theoretic

text: approach

text: supplies

text: a

text: new

text: angle

text: from

text: which

text: to

text: examine

text: this

text: task,

text: and

text: its

text: information

text: A

text: useful

text: way

text: to

text: demonstrate

text: our

text: analytical

text: method-

text: theoretic

text: insight

text: will

text: be

text: investigated

text: in

text: Section

text: IV

text: via

text: ology

text: is

text: to

text: consider

text: ﬁnancial

text: time

text: series

text: data,

text: which

text: information

text: coupling.

text: has

text: been

text: well

text: studied

text: in

text: the

text: literature.

text: The

text: purpose

text: Well

text: known

text: in

text: communication

text: theory,

text: an

text: equalizer

text: is

text: to

text: demonstrate

text: the

text: prediction

text: of

text: stock

text: prices

text: from

text: serves

text: as

text: a

text: part

text: of

text: an

text: optimal

text: receiver

text: to

text: detect

text: a

text: other

text: factors.

text: In

text: this

text: example,

text: we

text: are

text: trying

text: to

text: predict

text: transmitted

text: signal

text: from

text: a

text: noisy

text: channel

text: distorted

text: by

text: the

text: stock

text: price

text: of

text: Taiwan

text: Semiconductor

text: Manufacturing

text: nonlinearities

text: and

text: other

text: impairments.

text: In

text: order

text: to

text: infer

text: Corp.

text: (TSMC),

text: which

text: is

text: the

text: world’s

text: largest

text: contract

text: from

text: another

text: data

text: variable

text: or

text: time

text: series,

text: we

text: may

text: take

text: chip

text: fabrication

text: company.

text: To

text: demonstrate

text: information

text: advantage

text: of

text: the

text: same

text: concept

text: to

text: process

text: the

text: data.

text: transfer

text: and

text: thus

text: communication

text: theoretic

text: data

text: analytics,

text: Proper

text: implementation

text: of

text: an

text: equalizer

text: could

text: enhance

text: we

text: consider

text: two

text: factors,

text: which

text: appear

text: to

text: be

text: somewhat

text: the

text: causal

text: relationship

text: between

text: data

text: variables

text: and

text: time

text: indirectly

text: related

text: to

text: stock

text: prices

text: in

text: Taiwan

text: but

text: are

text: ##LTFigure##


text: Quasi-Periodic Schro¨ dinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology Yiqian Wang Jiangong You Abstract One knows that the set of quasi-periodic Schro¨dinger cocycles with positive Lya- punov exponent is open and dense in the analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be approximated by ones with C ≤ ≤ ∞ zero Lyapunov exponent in the space of l (1 l ) smooth quasi-periodic cocy- cles. As a consequence, the set of quasi-periodic Schro¨dinger cocycles with positive C Lyapunov exponent is not l open. Keywords. Lyapunov exponent; Smooth quasi-periodic cocycles; Schro¨dinger oper- ators. 1 Introduction and Results C → Let X be a r compact manifold, T : X X be ergodic with a normalized invari- R ant measure µ and A(x) be a SL(2, )-valued function on X. The dynamical system: → × R R (x, w) (T (x), A(x)w) in X 2 is called a SL(2, ) cocycle (or cocycle for sim- plicity) over the base dynamics (X, T ). We will simply denoted it as (T, A). If the base T R \Z (cid:18) → (cid:19) system is a rotation on torus, i.e., X = m = m m, T = T : x x + ω with ω rational independent ω, we call (T , A) a quasi-periodic cocycle, which is simply denoted ω − v(x) 1 by (ω, A). If furthermore A(x) = S (x) is of the form S (x) = with v v 1 0 Y. Wang: Department of Mathematics, Nanjing University, Nanjing 210093, China; e-mail: yiqianw@nju.edu.cn J. You: Department of Mathematics, Nanjing University, Nanjing 210093, China; e-mail: jyou@nju.edu.cn Mathematics Subject Classiﬁcation (2010): Primary 37; Secondary 37D25 1 ##LTLine##

text: Quasi-Periodic

text: Schro¨

text: dinger

text: Cocycles

text: with

text: Positive

text: Lyapunov

text: Exponent

text: are

text: not

text: Open

text: in

text: the

text: Smooth

text: Topology

text: Yiqian

text: Wang

text: Jiangong

text: You

text: Abstract

text: One

text: knows

text: that

text: the

text: set

text: of

text: quasi-periodic

text: Schro¨dinger

text: cocycles

text: with

text: positive

text: Lya-

text: punov

text: exponent

text: is

text: open

text: and

text: dense

text: in

text: the

text: analytic

text: topology.

text: In

text: this

text: paper,

text: we

text: construct

text: cocycles

text: with

text: positive

text: Lyapunov

text: exponent

text: which

text: can

text: be

text: approximated

text: by

text: ones

text: with

text: C

text: zero

text: Lyapunov

text: exponent

text: in

text: the

text: space

text: of

text: l

text: l

text: smooth

text: quasi-periodic

text: cocy-

text: cles.

text: As

text: a

text: consequence,

text: the

text: set

text: of

text: quasi-periodic

text: Schro¨dinger

text: cocycles

text: with

text: positive

text: C

text: Lyapunov

text: exponent

text: is

text: not

text: l

text: open.

text: Keywords.

text: Lyapunov

text: exponent;

text: Smooth

text: quasi-periodic

text: cocycles;

text: Schro¨dinger

text: oper-

text: ators.

text: Introduction

text: and

text: Results

text: C

text: Let

text: X

text: be

text: a

text: r

text: compact

text: manifold,

text: T

text: X

text: X

text: be

text: ergodic

text: with

text: a

text: normalized

text: invari-

text: R

text: ant

text: measure

text: µ

text: and

text: A(x)

text: be

text: a

text: )-valued

text: function

text: on

text: X.

text: The

text: dynamical

text: system:

text: R

text: R

text: (x,

text: w)

text: (T

text: (x),

text: A(x)w)

text: in

text: X

text: is

text: called

text: a

text: cocycle

text: (or

text: cocycle

text: for

text: sim-

text: plicity)

text: over

text: the

text: base

text: dynamics

text: (X,

text: T

text: We

text: will

text: simply

text: denoted

text: it

text: as

text: (T,

text: A).

text: If

text: the

text: base

text: T

text: R

text: \Z

text: system

text: is

text: a

text: rotation

text: on

text: torus,

text: i.e.,

text: X

text: m

text: m

text: m,

text: T

text: T

text: x

text: x

text: ω

text: with

text: ω

text: rational

text: independent

text: ω,

text: we

text: call

text: (T

text: A)

text: a

text: quasi-periodic

text: cocycle,

text: which

text: is

text: simply

text: denoted

text: ω

text: v(x)

text: by

text: (ω,

text: A).

text: If

text: furthermore

text: A(x)

text: S

text: (x)

text: is

text: of

text: the

text: form

text: S

text: (x)

text: with

text: v

text: v

text: Y.

text: Wang:

text: Department

text: of

text: Mathematics,

text: Nanjing

text: University,

text: Nanjing

text: China;

text: e-mail:

text: yiqianw@nju.edu.cn

text: J.

text: You:

text: Department

text: of

text: Mathematics,

text: Nanjing

text: University,

text: Nanjing

text: China;

text: e-mail:

text: jyou@nju.edu.cn

text: Mathematics

text: Subject

text: Classiﬁcation

text: Primary

text: Secondary

text: ##LTLine##


text: 22 Yiqian Wang, Jiangong You (cid:12) (cid:12) On the other hand, we have(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − (cid:12) dk 1( 1 ) (cid:46) | |− − ≥ 1−+f2 f k 1 if k 1. dfk 1 | | (cid:38) (cid:107) (cid:107) · | | Thus from f A 2 cot θ we obtain |∂iφ| ≤ | − π |− · 1 ≤ (cid:107) (cid:107)− | − π |− − C(i) θ i | | C(i) A 2 θ i 1. ∂θi 2 f 2 (cid:80) Next we estimate |∂∂(cid:107)Aiφ(cid:107)i | ≤ (cid:80)l1+···+lk=i |dk−d1f(k1−+11f2 )| · |∂(cid:107)∂Al1(cid:107)fl1 | · · · |∂(cid:107)∂Alk(cid:107)flk | lj ≥ 1, 1 ≤ j ≤ k (A.26) ≤ | |− − · | | · · · | | ··· f k 1 (cid:107)∂l1(cid:107)f (cid:107)∂lk(cid:107)f . l1+ +lk=i ∂ A l1 ∂ A lk | | ∼ | | · | | | − |− (cid:28) (cid:107) (cid:107) It is easy to see that f a cot θ with the condition θ π 1 A 2. We also 2 have | ∂lsf | ≤ | || ∂lsa | | || ∂lsb | (cid:107) (cid:107) cot θ (cid:107) (cid:107) + tan θ (cid:107) (cid:107) . ∂ A ls ∂ A ls ∂ A ls (cid:12) (cid:18) (cid:19)(cid:12) By a direct comput(cid:12)ation, we obtain (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107)− (cid:107) (cid:107)− | ∂lsa | ∂ls A 2 B 2 A 2 B 2 ≤ · | | · (cid:107) (cid:107)− ∂(cid:107)A(cid:107)ls = ∂(cid:107)A(cid:107)ls (cid:107)B(cid:107)2 − (cid:107)B(cid:107)−2 C(ls) a A ls (cid:12) (cid:16) (cid:17)(cid:12) and (cid:12) (cid:12) (cid:12) (cid:12) | | (cid:107) (cid:107) (cid:107) (cid:107)− −(cid:107) (cid:107)− (cid:107) (cid:107) ≤ · (cid:107) (cid:107)− (cid:107) (cid:107) · (cid:107) (cid:107)− ∂(cid:107)∂Als(cid:107)bls = ∂(cid:107)∂Als(cid:107)ls A 2 (cid:107)BB(cid:107)22−(cid:107)BA(cid:107)−22 B 2 C(ls) ( A 2 + (cid:107)BA(cid:107)24 ) A ls. (cid:110) (cid:111) Thus we have | | ≤ · | − |(cid:107) (cid:107) − | − |− · (cid:107) (cid:107)− · (cid:107) (cid:107)− (cid:107) (cid:107) (cid:107)∂ls(cid:107)f C(l ) θ π A 2 ls + θ π 1 A ls ( A 2 + (cid:107)A(cid:107)2 ) ∂ A ls s 2 2 B 4 (A.27) ≤ · | | · (cid:107) (cid:107)− C(l ) f A ls. s | | (cid:38) (cid:107) (cid:107) · | | With the fact that f A 2 cot θ , it follows that | |− · | | ≤ | |− · (cid:107) (cid:107)− f 2 (cid:107)∂ls(cid:107)f C(l ) f 1 A ls ∂ A ls s (A.28) ≤ · (cid:107) (cid:107)− − · | − |− C(l ) A 2 ls θ π 1. s 2 Combining (A.26), (A.27) with (A.28), we obtain (A.21). Similarly, we have (A.22) (cid:116)(cid:117) and (A.23). The estimates for ψ can be proved similarly. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Yiqian

text: Wang,

text: Jiangong

text: You

text: On

text: the

text: other

text: hand,

text: we

text: dk

text: f

text: k

text: if

text: k

text: dfk

text: Thus

text: from

text: f

text: A

text: cot

text: θ

text: we

text: obtain

text: |∂iφ|

text: π

text: π

text: C(i)

text: θ

text: i

text: C(i)

text: A

text: θ

text: i

text: ∂θi

text: f

text: Next

text: we

text: estimate

text: lj

text: j

text: k

text: f

text: k

text: +lk=i

text: A

text: A

text: lk

text: It

text: is

text: easy

text: to

text: see

text: that

text: f

text: a

text: cot

text: θ

text: with

text: the

text: condition

text: θ

text: π

text: A

text: We

text: also

text: have

text: ∂lsf

text: ∂lsa

text: ∂lsb

text: cot

text: θ

text: tan

text: θ

text: A

text: ls

text: A

text: ls

text: A

text: ls

text: By

text: a

text: direct

text: we

text: obtain

text: ∂lsa

text: ∂ls

text: A

text: B

text: A

text: B

text: C(ls)

text: a

text: A

text: ls

text: and

text: A

text: B

text: C(ls)

text: A

text: A

text: ls.

text: Thus

text: we

text: have

text: C(l

text: θ

text: π

text: A

text: ls

text: θ

text: π

text: A

text: ls

text: A

text: A

text: ls

text: s

text: B

text: C(l

text: f

text: A

text: ls.

text: s

text: With

text: the

text: fact

text: that

text: f

text: A

text: cot

text: θ

text: it

text: follows

text: that

text: f

text: C(l

text: f

text: A

text: ls

text: A

text: ls

text: s

text: C(l

text: A

text: ls

text: θ

text: π

text: s

text: Combining

text: with

text: we

text: obtain

text: Similarly,

text: we

text: have

text: and

text: The

text: estimates

text: for

text: ψ

text: can

text: be

text: proved

text: similarly.

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Positive Lyapunov Exponent are not Open 27 [18] J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. 152 (2000), 835-879. Z [19] J. Bourgain, M. Goldstein and W. Schlag, Anderson localization for Schr¨odinger operators on with potentials given by skew-shift, Commun. Math. Physics 220(3)(2001), 583-621. [20] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, Journal of Statistical Physics 108(5-6)(2002), 1203-1218. Z [21] J. Bourgain and W. Schlag, Anderson localization for Schro¨dinger operators on with strongly mixing potentials, Commun. Math. Phys. 215(2000), 143-175. [22] P. Duarte and S.Klein, Continuity of the Lyapunov exponents for quasiperiodic cocycles, Comm. Math. Phys. 332(3)(2014), 1113-1166. [23] L.H. Eliasson, Discrete one-dimensional quasi-periodic Schro¨dinger operators with pure point spec- trum, Acta Math. 179(2) (1997), 153-196. [24] J. Fro¨hlich, T. Spencer and P. Wittwer, Localization for a class of one-dimensional quasi-periodic Schro¨dinger operators, Comm. Math. Phys. 132(1990), 5-25. [25] A. Furman, On the multiplicative ergodic theorem for the uniquely ergodic systems, Ann. Inst. Henri Poincare´ 33(1997), 797-815. [26] H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428. [27] H. Furstenberg and Y. Kifer, Random matrix products and measures in projective spaces, Israel J. Math 10(1983), 12-32. [28] M. Goldstein and W. Schlag, H¨older continuity of the integrated density of states for quasi-periodic Schr¨odinger equations and averages of shifts of subharmonic functions, Annals of Math. 154(2001), 155-203. [29] H. Hennion, Loi des grands nombres et perturbations pour des produits r´eductibles de matrices al´eatoires ind´ependantes, Z. Wahrsch. Verw. Gebiete 67(1984), 265-278. [30] M. Herman, Une me´thode pour minorer les exposants de Lyapounov et quelques exemples montrant le caracte`re local d’un the´ore`me d’Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58(1983), 453-502. [31] K. Ishii, Localization of eigenstates and transport phenomena in one-dimensional disordered sys- tems, Suppl. Prog. Theoret. Phys. 53 (1973), 77-138. [32] S. Jitomirskaya, D. Koslover and M. Schulteis, Continuity of the Lyapunov exponent for general analytic quasiperiodic cocycles, Ergodic theory and dynamical systems 29(2009), 1881-1905. [33] S. Jitomirskaya and C. Marx, Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities, Journal of Fixed Point Theory and Applications 10(2011), 129-146. [34] S. Jitomirskaya and C. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model, Comm. Math. Phys. 316(1)(2012), 237-267. [35] S. Jitomirskaya and C. Marx, Analytic quasi-periodic Schr¨odinger operators and rational frequency approximants, Geom. Funct. Anal. 22(5)(2012), 1407-1443. [36] S. Jitomirskaya and R. Mavi, Continuity of the measure of the spectrum for quasiperiodic Schro¨dinger operators with rough potentials, Commun. Math. Phys. 325(2) (2014), 585-601. [37] S. Jitomirskaya and R. Mavi, Dynamical bounds for quasiperiodic Schro¨dinger operators with rough potentials, arXiv:1412.0309(2014). [38] S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one- dimensional Schro¨dinger operators, Stochastic Analysis(Katata/Kyoto, 1982), (North-Holland Math. Library 32, North-Holland, Amsterdam)(1984), 225-247.

text: Positive

text: Lyapunov

text: Exponent

text: are

text: not

text: Open

label: reference

text: J.

text: Bourgain

text: and

text: M.

text: Goldstein,

text: On

text: nonperturbative

text: localization

text: with

text: quasi-periodic

text: potential,

text: Ann.

text: of

text: Math.

text: Z

text: J.

text: Bourgain,

text: M.

text: Goldstein

text: and

text: W.

text: Schlag,

text: Anderson

text: localization

text: for

text: Schr¨odinger

text: operators

text: on

text: with

text: potentials

text: given

text: by

text: skew-shift,

text: Commun.

text: Math.

text: Physics

text: J.

text: Bourgain

text: and

text: S.

text: Jitomirskaya,

text: Continuity

text: of

text: the

text: Lyapunov

text: exponent

text: for

text: quasiperiodic

text: operators

text: with

text: analytic

text: potential,

text: Journal

text: of

text: Statistical

text: Physics

text: Z

text: J.

text: Bourgain

text: and

text: W.

text: Schlag,

text: Anderson

text: localization

text: for

text: Schro¨dinger

text: operators

text: on

text: with

text: strongly

text: mixing

text: potentials,

text: Commun.

text: Math.

text: Phys.

text: P.

text: Duarte

text: and

text: S.Klein,

text: Continuity

text: of

text: the

text: Lyapunov

text: exponents

text: for

text: quasiperiodic

text: cocycles,

text: Comm.

text: Math.

text: Phys.

text: L.H.

text: Eliasson,

text: Discrete

text: one-dimensional

text: quasi-periodic

text: Schro¨dinger

text: operators

text: with

text: pure

text: point

text: spec-

text: trum,

text: Acta

text: Math.

text: J.

text: Fro¨hlich,

text: T.

text: Spencer

text: and

text: P.

text: Wittwer,

text: Localization

text: for

text: a

text: class

text: of

text: one-dimensional

text: quasi-periodic

text: Schro¨dinger

text: operators,

text: Comm.

text: Math.

text: Phys.

text: A.

text: Furman,

text: On

text: the

text: multiplicative

text: ergodic

text: theorem

text: for

text: the

text: uniquely

text: ergodic

text: systems,

text: Ann.

text: Inst.

text: Henri

text: Poincare´

text: H.

text: Furstenberg,

text: Noncommuting

text: random

text: products,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: H.

text: Furstenberg

text: and

text: Y.

text: Kifer,

text: Random

text: matrix

text: products

text: and

text: measures

text: in

text: projective

text: spaces,

text: Israel

text: J.

text: Math

text: M.

text: Goldstein

text: and

text: W.

text: Schlag,

text: H¨older

text: continuity

text: of

text: the

text: integrated

text: density

text: of

text: states

text: for

text: quasi-periodic

text: Schr¨odinger

text: equations

text: and

text: averages

text: of

text: shifts

text: of

text: subharmonic

text: functions,

text: Annals

text: of

text: Math.

text: H.

text: Hennion,

text: Loi

text: des

text: grands

text: nombres

text: et

text: perturbations

text: pour

text: des

text: produits

text: r´eductibles

text: de

text: matrices

text: al´eatoires

text: ind´ependantes,

text: Z.

text: Wahrsch.

text: Verw.

text: Gebiete

text: M.

text: Herman,

text: Une

text: me´thode

text: pour

text: minorer

text: les

text: exposants

text: de

text: Lyapounov

text: et

text: quelques

text: exemples

text: montrant

text: le

text: caracte`re

text: local

text: d’un

text: the´ore`me

text: d’Arnold

text: et

text: de

text: Moser

text: sur

text: le

text: tore

text: de

text: dimension

text: Comment.

text: Math.

text: Helv.

text: K.

text: Ishii,

text: Localization

text: of

text: eigenstates

text: and

text: transport

text: phenomena

text: in

text: one-dimensional

text: disordered

text: sys-

text: tems,

text: Suppl.

text: Prog.

text: Theoret.

text: Phys.

text: S.

text: Jitomirskaya,

text: D.

text: Koslover

text: and

text: M.

text: Schulteis,

text: Continuity

text: of

text: the

text: Lyapunov

text: exponent

text: for

text: general

text: analytic

text: quasiperiodic

text: cocycles,

text: Ergodic

text: theory

text: and

text: dynamical

text: systems

text: S.

text: Jitomirskaya

text: and

text: C.

text: Marx,

text: Continuity

text: of

text: the

text: Lyapunov

text: exponent

text: for

text: analytic

text: quasi-periodic

text: cocycles

text: with

text: singularities,

text: Journal

text: of

text: Fixed

text: Point

text: Theory

text: and

text: Applications

text: S.

text: Jitomirskaya

text: and

text: C.

text: Marx,

text: Analytic

text: quasi-perodic

text: cocycles

text: with

text: singularities

text: and

text: the

text: Lyapunov

text: exponent

text: of

text: extended

text: Harper’s

text: model,

text: Comm.

text: Math.

text: Phys.

text: S.

text: Jitomirskaya

text: and

text: C.

text: Marx,

text: Analytic

text: quasi-periodic

text: Schr¨odinger

text: operators

text: and

text: rational

text: frequency

text: approximants,

text: Geom.

text: Funct.

text: Anal.

text: S.

text: Jitomirskaya

text: and

text: R.

text: Mavi,

text: Continuity

text: of

text: the

text: measure

text: of

text: the

text: spectrum

text: for

text: quasiperiodic

text: Schro¨dinger

text: operators

text: with

text: rough

text: potentials,

text: Commun.

text: Math.

text: Phys.

text: S.

text: Jitomirskaya

text: and

text: R.

text: Mavi,

text: Dynamical

text: bounds

text: for

text: quasiperiodic

text: Schro¨dinger

text: operators

text: with

text: rough

text: potentials,

text: S.

text: Kotani,

text: Ljapunov

text: indices

text: determine

text: absolutely

text: continuous

text: spectra

text: of

text: stationary

text: random

text: one-

text: dimensional

text: Schro¨dinger

text: operators,

text: Stochastic

text: Analysis(Katata/Kyoto,

text: (North-Holland

text: Math.

text: Library

text: North-Holland,


text: 1 Enhanced Mixtures of Part Model for Human Pose Estimation Wenjuan Gong, Yongzhen Huang, Jordi Gonza`lez, and Liang Wang Abstract— Mixture of parts model has been successfully we are able to verify body part localization from more feature applied to 2D human pose estimation problem either as explicitly responses. Compared with the greedy solution in the original trained body part model or as latent variables for the whole MoP method, we are able to solve double-counting problem human body model. Mixture of parts model usually utilize tree with a global optimization. structure for representing relations between body parts. Tree structures facilitate training and referencing of the model but With detected 2D body part locations from the proposed could not deal with double counting problems, which hinder method, we are able to predict 3D poses by feeding 2D body its applications in 3D pose estimation. While most of work part detections to a 2D to 3D pose estimator. For 2D to targeted to solve these problems tend to modify the tree models 3D pose estimator, we choose Gaussian process regression, or the optimization target. We incorporate other cues from which has been proved to be effective in modeling non-linear input features. For example, in surveillance environments, human silhouettes can be extracted relative easily although not ﬂawlessly. regression problems. We further validate the whole pipeline In this condition, we can combine extracted human blobs with on a public available dataset for pose estimation: HumanEva histogram of gradient feature, which is commonly used in mixture dataset. We visualized two types of results: enhanced 2D body of parts model for training body part templates. The method can part detections and 3D poses estimated from enhanced 2D be easily extend to other candidate features under our generalized body part detections. Figure 1 shows main steps as a pipeline framework. We show 2D body part detection results on a public available dataset: HumanEva dataset. Furthermore, a 2D to 3D for the whole algorithm. The pipeline includes several major pose estimator is trained with Gaussian process regression model steps: feature extractions (MoP and background subtraction, in and 2D body part detections from the proposed method is fed to our case), global optimization, and 2D to 3D pose estimation. the estimator, thus 3D poses are predictable given new 2D body From the input, the original mixtures of part model is trained part detections. We also show results of 3D pose estimation on and applied to detect body part positions. Meanwhile, human HumanEva dataset. blobs extracted with background subtractions are used as Index Terms—Pose estimation, double counting problem, mix- another cue in our method. Then in the third step, this two ture of parts Model cues are combined with the proposed algorithm. The generalized framework in our algorithm are able to I. INTRODUCTION incorporate multiple features other than human blobs. We Pose estimation from still images has wide applications extract multiple cues from input images and combine them in image and video indexing, video surveillance and human under the proposed framework. With augmented inputs from computer interaction. For example, online solutions of this multiple features, we are able improve 2D body part localiza- problem can be applied for single frame initialization in tion and solve double counting problem. As in MoP method, tracking human poses. Yet pose estimation from still images, a human body is modeled by a tree structure where kinematic that is, 2D body part localization is a difﬁcult problem, due constraints between connecting parts are kept. First, feature to the fact that human body is highly ﬂexible resulting human models of different features are trained separately and the poses with high degrees of freedom even in 2D images. optimization target is modiﬁed to reach a global optimization A state-of-art and currently widely used solution for 2D target by incorporating multiple feature cues. The inference body part detection is the mixture-of-parts (MoP) method [1], of the optimal pose in a test image is also carried out with in which a human body is modeled as a tree structure and multiple cues. The bottom up message passing procedure com- body parts are encoded as nodes in the tree. Maximum bines multiple feature cues so as to reduce false positive body responses from detection are passed from the leaf nodes to the part detections and the top down back tracing procedure are root. One problem with this solution is the double-counting optimized globally so as to tackle double counting problem. problem, that is, one detected body part is counted twice The contributions of this papers are as followings: by for both sides of the human body. In this paper, we tackle combing multiple cues, we boost 2D body part localizations the double-counting problem in MoP model with multiple under a general framework; enhanced 2D body part detectors feature inputs. Additional input features are incorporated so are validated on a public available dataset: HumanEva dataset; 3D pose estimation is shown as an examplar application of W. Gong is with the Department of Computer and Communication Engi- detected 2D poses and this application is also validated on neering, China University of Petroleum, Qingdao, China, e-mail: wenjuan- gong@upc.edu.cn. HumanEva dataset. The rest of the paper is organized as Y. Huang and W. Liang are with Institute of Automation, Chinese Academy following: in section , we introduce related works on 2D of Science. pose estimation and related works on solving double counting J. Gonza`lez is with Computer Vision Center, Autonomous University of Barcelona. problems; in section , we introduce details of the proposed

text: Enhanced

text: Mixtures

text: of

text: Part

text: Model

text: for

text: Human

text: Pose

text: Estimation

text: Wenjuan

text: Gong,

text: Yongzhen

text: Huang,

text: Jordi

text: Gonza`lez,

text: and

text: Liang

text: Wang

text: Abstract—

text: Mixture

text: of

text: parts

text: model

text: has

text: been

text: successfully

text: we

text: are

text: able

text: to

text: verify

text: body

text: part

text: localization

text: from

text: more

text: feature

text: applied

text: to

text: human

text: pose

text: estimation

text: problem

text: either

text: as

text: explicitly

text: responses.

text: Compared

text: with

text: the

text: greedy

text: solution

text: in

text: the

text: original

text: trained

text: body

text: part

text: model

text: or

text: as

text: latent

text: variables

text: for

text: the

text: whole

text: MoP

text: method,

text: we

text: are

text: able

text: to

text: solve

text: double-counting

text: problem

text: human

text: body

text: model.

text: Mixture

text: of

text: parts

text: model

text: usually

text: utilize

text: tree

text: with

text: a

text: global

text: optimization.

text: structure

text: for

text: representing

text: relations

text: between

text: body

text: parts.

text: Tree

text: structures

text: facilitate

text: training

text: and

text: referencing

text: of

text: the

text: model

text: but

text: With

text: detected

text: body

text: part

text: locations

text: from

text: the

text: proposed

text: could

text: not

text: deal

text: with

text: double

text: counting

text: problems,

text: which

text: hinder

text: method,

text: we

text: are

text: able

text: to

text: predict

text: poses

text: by

text: feeding

text: body

text: its

text: applications

text: in

text: pose

text: estimation.

text: While

text: most

text: of

text: work

text: part

text: detections

text: to

text: a

text: to

text: pose

text: estimator.

text: For

text: to

text: targeted

text: to

text: solve

text: these

text: problems

text: tend

text: to

text: modify

text: the

text: tree

text: models

text: pose

text: estimator,

text: we

text: choose

text: Gaussian

text: process

text: regression,

text: or

text: the

text: optimization

text: target.

text: We

text: incorporate

text: other

text: cues

text: from

text: which

text: has

text: been

text: proved

text: to

text: be

text: effective

text: in

text: modeling

text: non-linear

text: input

text: features.

text: For

text: example,

text: in

text: surveillance

text: environments,

text: human

text: silhouettes

text: can

text: be

text: extracted

text: relative

text: easily

text: although

text: not

text: ﬂawlessly.

text: regression

text: problems.

text: We

text: further

text: validate

text: the

text: whole

text: pipeline

text: In

text: this

text: condition,

text: we

text: can

text: combine

text: extracted

text: human

text: blobs

text: with

text: on

text: a

text: public

text: available

text: dataset

text: for

text: pose

text: estimation:

text: HumanEva

text: histogram

text: of

text: gradient

text: feature,

text: which

text: is

text: commonly

text: used

text: in

text: mixture

text: dataset.

text: We

text: visualized

text: two

text: types

text: of

text: results:

text: enhanced

text: body

text: of

text: parts

text: model

text: for

text: training

text: body

text: part

text: templates.

text: The

text: method

text: can

text: part

text: detections

text: and

text: poses

text: estimated

text: from

text: enhanced

text: be

text: easily

text: extend

text: to

text: other

text: candidate

text: features

text: under

text: our

text: generalized

text: body

text: part

text: detections.

text: Figure

text: shows

text: main

text: steps

text: as

text: a

text: pipeline

text: framework.

text: We

text: show

text: body

text: part

text: detection

text: results

text: on

text: a

text: public

text: available

text: dataset:

text: HumanEva

text: dataset.

text: Furthermore,

text: a

text: to

text: for

text: the

text: whole

text: algorithm.

text: The

text: pipeline

text: includes

text: several

text: major

text: pose

text: estimator

text: is

text: trained

text: with

text: Gaussian

text: process

text: regression

text: model

text: steps:

text: feature

text: extractions

text: (MoP

text: and

text: background

text: subtraction,

text: in

text: and

text: body

text: part

text: detections

text: from

text: the

text: proposed

text: method

text: is

text: fed

text: to

text: our

text: case),

text: global

text: optimization,

text: and

text: to

text: pose

text: estimation.

text: the

text: estimator,

text: thus

text: poses

text: are

text: predictable

text: given

text: new

text: body

text: From

text: the

text: input,

text: the

text: original

text: mixtures

text: of

text: part

text: model

text: is

text: trained

text: part

text: detections.

text: We

text: also

text: show

text: results

text: of

text: pose

text: estimation

text: on

text: and

text: applied

text: to

text: detect

text: body

text: part

text: positions.

text: Meanwhile,

text: human

text: HumanEva

text: dataset.

text: blobs

text: extracted

text: with

text: background

text: subtractions

text: are

text: used

text: as

text: Index

text: Terms—Pose

text: estimation,

text: double

text: counting

text: problem,

text: mix-

text: another

text: cue

text: in

text: our

text: method.

text: Then

text: in

text: the

text: third

text: step,

text: this

text: two

text: ture

text: of

text: parts

text: Model

text: cues

text: are

text: combined

text: with

text: the

text: proposed

text: algorithm.

text: The

text: generalized

text: framework

text: in

text: our

text: algorithm

text: are

text: able

text: to

text: I.

text: INTRODUCTION

text: incorporate

text: multiple

text: features

text: other

text: than

text: human

text: blobs.

text: We

text: Pose

text: estimation

text: from

text: still

text: images

text: has

text: wide

text: applications

text: extract

text: multiple

text: cues

text: from

text: input

text: images

text: and

text: combine

text: them

text: in

text: image

text: and

text: video

text: indexing,

text: video

text: surveillance

text: and

text: human

text: under

text: the

text: proposed

text: framework.

text: With

text: augmented

text: inputs

text: from

text: computer

text: interaction.

text: For

text: example,

text: online

text: solutions

text: of

text: this

text: multiple

text: features,

text: we

text: are

text: able

text: improve

text: body

text: part

text: localiza-

text: problem

text: can

text: be

text: applied

text: for

text: single

text: frame

text: initialization

text: in

text: tion

text: and

text: solve

text: double

text: counting

text: problem.

text: As

text: in

text: MoP

text: method,

text: tracking

text: human

text: poses.

text: Yet

text: pose

text: estimation

text: from

text: still

text: images,

text: a

text: human

text: body

text: is

text: modeled

text: by

text: a

text: tree

text: structure

text: where

text: kinematic

text: that

text: is,

text: body

text: part

text: localization

text: is

text: a

text: difﬁcult

text: problem,

text: due

text: constraints

text: between

text: connecting

text: parts

text: are

text: kept.

text: First,

text: feature

text: to

text: the

text: fact

text: that

text: human

text: body

text: is

text: highly

text: ﬂexible

text: resulting

text: human

text: models

text: of

text: different

text: features

text: are

text: trained

text: separately

text: and

text: the

text: poses

text: with

text: high

text: degrees

text: of

text: freedom

text: even

text: in

text: images.

text: optimization

text: target

text: is

text: modiﬁed

text: to

text: reach

text: a

text: global

text: optimization

text: A

text: state-of-art

text: and

text: currently

text: widely

text: used

text: solution

text: for

text: target

text: by

text: incorporating

text: multiple

text: feature

text: cues.

text: The

text: inference

text: body

text: part

text: detection

text: is

text: the

text: mixture-of-parts

text: (MoP)

text: method

text: of

text: the

text: optimal

text: pose

text: in

text: a

text: test

text: image

text: is

text: also

text: carried

text: out

text: with

text: in

text: which

text: a

text: human

text: body

text: is

text: modeled

text: as

text: a

text: tree

text: structure

text: and

text: multiple

text: cues.

text: The

text: bottom

text: up

text: message

text: passing

text: procedure

text: com-

text: body

text: parts

text: are

text: encoded

text: as

text: nodes

text: in

text: the

text: tree.

text: Maximum

text: bines

text: multiple

text: feature

text: cues

text: so

text: as

text: to

text: reduce

text: false

text: positive

text: body

text: responses

text: from

text: detection

text: are

text: passed

text: from

text: the

text: leaf

text: nodes

text: to

text: the

text: part

text: detections

text: and

text: the

text: top

text: down

text: back

text: tracing

text: procedure

text: are

text: root.

text: One

text: problem

text: with

text: this

text: solution

text: is

text: the

text: double-counting

text: optimized

text: globally

text: so

text: as

text: to

text: tackle

text: double

text: counting

text: problem.

text: problem,

text: that

text: is,

text: one

text: detected

text: body

text: part

text: is

text: counted

text: twice

text: The

text: contributions

text: of

text: this

text: papers

text: are

text: as

text: followings:

text: by

text: for

text: both

text: sides

text: of

text: the

text: human

text: body.

text: In

text: this

text: paper,

text: we

text: tackle

text: combing

text: multiple

text: cues,

text: we

text: boost

text: body

text: part

text: localizations

text: the

text: double-counting

text: problem

text: in

text: MoP

text: model

text: with

text: multiple

text: under

text: a

text: general

text: framework;

text: enhanced

text: body

text: part

text: detectors

text: feature

text: inputs.

text: Additional

text: input

text: features

text: are

text: incorporated

text: so

text: are

text: validated

text: on

text: a

text: public

text: available

text: dataset:

text: HumanEva

text: dataset;

text: pose

text: estimation

text: is

text: shown

text: as

text: an

text: examplar

text: application

text: of

text: W.

text: Gong

text: is

text: with

text: the

text: Department

text: of

text: Computer

text: and

text: Communication

text: Engi-

text: detected

text: poses

text: and

text: this

text: application

text: is

text: also

text: validated

text: on

text: neering,

text: China

text: University

text: of

text: Petroleum,

text: Qingdao,

text: China,

text: e-mail:

text: wenjuan-

text: gong@upc.edu.cn.

text: HumanEva

text: dataset.

text: The

text: rest

text: of

text: the

text: paper

text: is

text: organized

text: as

text: Y.

text: Huang

text: and

text: W.

text: Liang

text: are

text: with

text: Institute

text: of

text: Automation,

text: Chinese

text: Academy

text: following:

text: in

text: section

text: we

text: introduce

text: related

text: works

text: on

text: of

text: Science.

text: pose

text: estimation

text: and

text: related

text: works

text: on

text: solving

text: double

text: counting

text: J.

text: Gonza`lez

text: is

text: with

text: Computer

text: Vision

text: Center,

text: Autonomous

text: University

text: of

text: Barcelona.

text: problems;

text: in

text: section

text: we

text: introduce

text: details

text: of

text: the

text: proposed