text: Ejecta Velocities in Tycho’s SNR 5 Table 3 Best-ﬁt Parameters of the Double Gaussian Model from the Central Regionsa Using Chandra − − Widthb E E 2δE v⊥c v c (km s 1) v (km s 1) red blue − exp exp Lines (eV) (keV) (keV) (eV) (km s 1) (this work) (Hayato et al. 2010) ± ± ± ± Si-Heα 12.6 1.821 0.003 1.880+−0.002 59 4 4780 320 5010 340 4730+−30 0.003 ±20 S-Heα 15 2.393+−0.006 2.478+−0.004 87+−7 5240+−430 5490+−450 4660 50 ±0.009 ±0.006 ±11 ±680 ±710 ± Fe-Kα 41 6.36 0.02 6.53 0.02 170 30 4000 700 4200 800 4000 300 Using only high surface brightness pixels ± ± ± ± Si-Heα 12.6 1.822 0.003 1.884+−0.002 62 4 5020 320 5260 340 — ± 0.003 S-Heα 15 2.389 0.007 2.478+−0.004 89+−8 5490+−490 5750+−510 — 0.005 9 550 580 Using only low surface brightness pixels Si-Heα 12.6 1.821+−0.004 1.880+−0.003 59+−4 4780+−320 5010+−340 — 0.005 0.004 5 410 430 S-Heα 15 2.395+−0.008 2.479+−0.007 84+−11 5170+−680 5410+−710 — 0.009 0.008 12 740 770 for S-Heα). In all cases the derived energy diﬀerences (cid:6)#(cid:5)! between the red- and blueshifted components for both Si and S were in agreement. The ﬁnal uncertainty on the energy diﬀerence includes the statistical uncertainty (cid:6)# plus the range from the diﬀerent spectral models. Re- sults of the Chandra ﬁts for the three species are given (cid:5)!(cid:4)"(cid:6)# in Table 3 (for now we only consider the top three en- tries that correspond to the integrated spectrum from the central region). For the Si+S band the best ﬁt yields (cid:9)% χ2 = 146.75 for 92 degrees of freedom and for the Fe band χ2 = 137.32 for 127 degrees of freedom. Figure 3 (cid:5)! shows that the double Gaussian model is a good ﬁt to (cid:1)!(cid:9)% the spectra data. For the spectral ﬁts, line widths were ﬁxed at the val- (cid:7)$ (cid:7)$(cid:4)"(cid:9)% (cid:8)& (cid:10)'(cid:15)((cid:12))(cid:16)*(cid:13)+(cid:17),(cid:1)-(cid:2).(cid:14)/(cid:12))(cid:11)0(cid:3)1 ues determined from the minima in each radial proﬁle. The radial velocity (v⊥) was calculated from the average !(cid:5)"(cid:4)!(cid:5)$(cid:7) energy of the red- and blue-shifted components and δE. To convert v⊥ to the shell expansion speed we needed to (cid:48) !(cid:5)"(cid:4)!(cid:5)#(cid:6) correct for the projection factor over the central 1.41 of the remnant. For Si-Heα and S-Heα, we assumed a spherically symmetric shell extending over 190(cid:48)(cid:48)–220(cid:48)(cid:48) '(cid:9)%(cid:19)#(cid:6)!(cid:5)!(cid:1)&(cid:8) and for Fe-Kα, we assumed the shell covered the radial (cid:48)(cid:48) (cid:48)(cid:48) range 180 –200 . We calculated the projection factor $(cid:7)%(cid:19)#(cid:6)!(cid:5)!(cid:1)&(cid:8) using the method in section A.1 of Hayato et al. (2010). '(cid:9) We determined projection factors of 0.955 for Si-Heα and S-Heα and 0.948 for Fe-Kα. For Suzaku, an additional !(cid:5) correction was necessary to account for the ﬂux spread- !(cid:1)'(cid:9) ing due to the broad Suzaku PSF. This can be ignored for the Chandra analysis. '(cid:9)"(cid:4)'(cid:9) ((cid:10) ((cid:10)"(cid:4)'(cid:9) The shell expansion velocities from Chandra are in the )(cid:11)*(cid:16)+(cid:13),(cid:17)-(cid:14).(cid:18)/(cid:1)0(cid:2)1(cid:15)+(cid:13)2(cid:12)3(cid:3) ∼ − range of 4200–5500 km s 1; all are consistent with the Suzaku results. The large uncertainty on the Fe shell ex- Figure 3. Double Gaussian model ﬁt to the Chandra spectrum pansion from Chandra alone does not allow us to exclude from the center of Tycho’s SNR (Sky1+Sky2 regions) in the Si+S that it is moving at the same speed as the other species. band (top∼panel) and the Fe-Kα band (bottom panel). The weak However, by combining all of the Si and S measurements bump at 5.6 keV in the bottom panel is Cr Kα line emission. The continuum emission is ﬁtted with a power-law model. (both Chandra and Suzaku, i.e., using a weighted combi- nation of the values in the last two columns of Table 3) we arrive at an expansion velocity for the Si+S shell of − ple, there is a correlation between expansion speed and 4724+−27 km s 1 that is signiﬁcantly greater (>4σ) than 19 emission-measure (say, because higher intensity spots, the (similarly combined) Fe-shell expansion velocity of ± − such as compact blobs of ejecta, tend to expand more 4025 280 km s 1. quickly) then we will get a biased velocity diﬀerence. The results quoted above correspond to an emission- However, thanks to Chandra’s high angular resolution we measure-weighted mean velocity diﬀerence for the ap- can assess this eﬀect. We divided all pixels within the proaching and receding hemispheres. Although this is (cid:48) central 1.41 region by surface brightness into either high a practical approach from an observational perspective, ∼ × − or low values (splitting at the mean level of 6 10 4 it may not produce an unbiased result. If, for exam- − − − ph s 1 cm 2 arcmin 2) and extracted a separate spec- 11−−VVeekk 11−−ss ssttnnuuooCC χχ 11−−VVeekk 11−−ss ssttnnuuooCC χχ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ejecta

text: Velocities

text: in

text: Tycho’s

text: SNR

text: Table

text: Best-ﬁt

text: Parameters

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text: the

text: Central

text: Regionsa

text: Using

text: Chandra

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text: E

text: E

text: v⊥c

text: v

text: c

text: (km

text: s

text: v

text: (km

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text: red

text: blue

text: exp

text: exp

text: Lines

text: (eV)

text: (keV)

text: (keV)

text: (eV)

text: (km

text: s

text: (this

text: work)

text: (Hayato

text: et

text: al.

text: Si-Heα

text: S-Heα

text: Fe-Kα

text: Using

text: only

text: high

text: surface

text: brightness

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text: Si-Heα

text: S-Heα

text: Using

text: only

text: low

text: surface

text: brightness

text: pixels

text: Si-Heα

text: S-Heα

text: for

text: S-Heα).

text: In

text: all

text: cases

text: the

text: derived

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text: diﬀerences

text: between

text: the

text: red-

text: and

text: blueshifted

text: components

text: for

text: both

text: Si

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text: The

text: ﬁnal

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text: the

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text: For

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text: best

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text: χ2

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text: Figure

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text: a

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text: ﬁt

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text: spectra

text: data.

text: For

text: the

text: spectral

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text: were

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text: determined

text: from

text: the

text: minima

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text: each

text: radial

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text: The

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text: the

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text: red-

text: and

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text: δE.

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text: v⊥

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text: needed

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text: the

text: remnant.

text: For

text: Si-Heα

text: and

text: S-Heα,

text: we

text: assumed

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text: spherically

text: symmetric

text: shell

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text: over

text: and

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text: Fe-Kα,

text: we

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text: covered

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text: calculated

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text: For

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text: This

text: can

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text: the

text: Chandra

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text: The

text: shell

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text: with

text: the

text: Suzaku

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text: The

text: large

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text: Figure

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text: ACFI-T16-14 x Right-Handed Neutrinos and T-Violating, P-Conserving Interactions Basem Kamal El-Menouﬁ,1 Michael J. Ramsey-Musolf,1,2 and Chien-Yeah Seng1 1Amherst Center for Fundamental Interactions, Department of Physics, University of Massachusetts Amherst, MA 01003, USA 2Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125 USA We show that experimental probes of the P-conserving, T-violating triple correlation in polarized neutron or nuclear β-decay provide a unique probe of possible T-violation at the TeV scale in the presence of right-handed neutrinos. In contrast to other possible sources of semileptonic T- violation involving only left-handed neutrinos, those involving right-handed neutrinos are relatively unconstrained by present limits on the permanent electric dipole moments of the electron, neutral → atoms, and the neutron. On the other hand, LHC results for pp e+ missing transverse energy imply that an order of magnitude of improvement in D-coeﬃcient sensitivity would be needed for discovery. Finally, we discuss the interplay with the scale of neutrino mass and naturalness considerations. 1. INTRODUCTION gument may occur [14, 15], the question remains as to the prospective impact of, and motivation for, improved probes of ﬂavor-conserving PCTV observables. Recently, The search for time-reversal violation (TV) has long the authors of Ref. [16] addressed this question in the been a subject of considerable experimental and theo- EFT context, studying the contribution of the “left-right retical interest. It is partially motivated by the need for four fermion” (LR4F) operator to the D-coeﬃcient of CP-violation beyond that encoded in the Standard Model the neutron β-decay (deﬁned below). They ﬁnd that the (SM) Cabibbo-Kobayashi-Maskawa (CKM) matrix to ex- neutron EDM sets an indirect bound on the D-coeﬃcient plain the cosmic baryon asymmetry[1]. Assuming CPT that is three orders of magnitude more stringent than its is a good symmetry of nature, searches for TV provide direct experimental bound. a probe of this possible CP-violation. Experimentally, In this paper, we observe that there exists a set the parity (P)- and T-violating (PVTV) sector is be- of dimension-six four-fermion operators involving right- ing probed with great sensitivity through electric dipole moment (EDM) searches, with the three most stringent handed neutrinos that (a) contribute to the D-coeﬃcient and (b) are relatively unconstrained by EDM limits. Be- limits having been obtained for the 199Hg atom[2], the cause the SM charge changing weak interaction involves electron (extracted from the ThO molecule)[3], and the purely left-handed leptons, this contribution to neutron neutron[4, 5]. hand, the P-conserving and T-violating decay does not interfere linearly with the SM contribu- (PCTV) sector ( equivalent to C- and CP-violation as- tion, resulting in a quadratic, rather than linear, de- suming CPT) has received considerably less attention. pendence on the operator Wilson coeﬃcients. Nonethe- Experimental eﬀorts in the sector include measurement of anomalous η-decay channels such as η → 2π0γ, 3π0γ, less, present limits on D probe the TeV mass scale. We also show that while a subset of these operators gener- 3γ [6] and the D-coeﬃcient in the β-decay of polarized ate hadronic EDMs, their eﬀects are suppressed by loop neutrons [7] and 19Ne [8]. These processes are sensi- ∼ tive probes of “new physics” because the Standard Model factors as well as Λ /v where Λ 1 GeV is the chiral χ χ (SM) contributions are usually small [9, 10]. There is a symmetry breaking scale and v = 246 GeV is the Higgs vacuum expectation value (VEV). The resulting neutron SM ﬁnal-state interaction that could mimic a non-zero D- coeﬃcient in β-decay at order 10−5 for neutron [11] and EDM sensitivity to Λ is also at the TeV scale and does 10−4 for 19Ne [8] but the application of heavy baryon ef- not preclude a non-zero result in a next generation D- coeﬃcient probe. fective ﬁeld theory allows a precise computation of this contribution (up to 1% accuracy in the case of neutron Interestingly, indirect constraints from T-conserving [12]). observables may be more severe. These observables in- clude Large Hadron Collider (LHC) results for the pro- Theoretically, the eﬀect of PCTV physics due to be- → yond Standard Model (BSM) interactions can be studied cess pp e+X +MET (missing transverse energy) and neutrino mass. The latter constraints also rely on nat- in a model-independent way using eﬀective ﬁeld theory uralness considerations, a somewhat subjective criteria. (EFT). In this approach, one has integrated out the BSM The former imply that an order of magnitude improve- heavy degrees of freedom (DOF). In this context, it was observed in Ref. [13] that any EDM limits imply severe ment in D-coeﬃcient sensitivity would be required in or- der to discover evidence for PCTV right-handed neutrino bounds on PCTV observables since a PCTV interaction interactions. in the presence of P-violating SM radiative corrections will induce an EDM. While special exceptions to this ar- Our analysis leading to these conclusions is organized

label: paragraph

text: x

text: Right-Handed

label: title

text: Neutrinos

text: and

text: T-Violating,

text: P-Conserving

text: Interactions

text: Basem

label: author

text: Kamal

text: Michael

text: J.

text: and

text: Chien-Yeah

text: Center

text: for

text: Fundamental

text: Interactions,

text: Department

text: of

text: Physics,

text: University

text: of

text: Massachusetts

text: Amherst,

text: MA

text: USA

text: Radiation

text: Laboratory,

text: California

text: Institute

text: of

text: Technology,

text: Pasadena,

text: CA

text: USA

text: We

label: abstract

text: show

text: that

text: experimental

text: probes

text: of

text: the

text: P-conserving,

text: T-violating

text: triple

text: correlation

text: in

text: polarized

text: neutron

text: or

text: nuclear

text: β-decay

text: provide

text: a

text: unique

text: probe

text: of

text: possible

text: T-violation

text: at

text: the

text: TeV

text: scale

text: in

text: the

text: presence

text: of

text: right-handed

text: neutrinos.

text: In

text: contrast

text: to

text: other

text: possible

text: sources

text: of

text: semileptonic

text: T-

text: violation

text: involving

text: only

text: left-handed

text: neutrinos,

text: those

text: involving

text: right-handed

text: neutrinos

text: are

text: relatively

text: unconstrained

text: by

text: present

text: limits

text: on

text: the

text: permanent

text: electric

text: dipole

text: moments

text: of

text: the

text: electron,

text: neutral

text: atoms,

text: and

text: the

text: neutron.

text: On

text: the

text: other

text: hand,

text: LHC

text: results

text: for

text: pp

text: e+

text: missing

text: transverse

text: energy

text: imply

text: that

text: an

text: order

text: of

text: magnitude

text: of

text: improvement

text: in

text: D-coeﬃcient

text: sensitivity

text: would

text: be

text: needed

text: for

text: discovery.

text: Finally,

text: we

text: discuss

text: the

text: interplay

text: with

text: the

text: scale

text: of

text: neutrino

text: mass

text: and

text: naturalness

text: considerations.

text: INTRODUCTION

text: gument

text: may

text: occur

text: the

text: question

text: remains

text: as

text: to

text: the

text: prospective

text: impact

text: of,

text: and

text: motivation

text: for,

text: improved

text: probes

text: of

text: ﬂavor-conserving

text: PCTV

text: observables.

text: Recently,

text: The

text: search

text: for

text: time-reversal

text: violation

text: (TV)

text: has

text: long

text: the

text: authors

text: of

text: Ref.

text: addressed

text: this

text: question

text: in

text: the

text: been

text: a

text: subject

text: of

text: considerable

text: experimental

text: and

text: theo-

text: EFT

text: context,

text: studying

text: the

text: contribution

text: of

text: the

text: “left-right

text: retical

text: interest.

text: It

text: is

text: partially

text: motivated

text: by

text: the

text: need

text: for

text: four

text: fermion”

text: operator

text: to

text: the

text: D-coeﬃcient

text: of

text: CP-violation

text: beyond

text: that

text: encoded

text: in

text: the

text: Standard

text: Model

text: the

text: neutron

text: β-decay

text: (deﬁned

text: below).

text: They

text: ﬁnd

text: that

text: the

text: (SM)

text: Cabibbo-Kobayashi-Maskawa

text: (CKM)

text: matrix

text: to

text: ex-

text: neutron

text: EDM

text: sets

text: an

text: indirect

text: bound

text: on

text: the

text: D-coeﬃcient

text: plain

text: the

text: cosmic

text: baryon

text: Assuming

text: CPT

text: that

text: is

text: three

text: orders

text: of

text: magnitude

text: more

text: stringent

text: than

text: its

text: is

text: a

text: good

text: symmetry

text: of

text: nature,

text: searches

text: for

text: TV

text: provide

text: direct

text: experimental

text: bound.

text: a

text: probe

text: of

text: this

text: possible

text: CP-violation.

text: Experimentally,

text: In

text: this

text: paper,

text: we

text: observe

text: that

text: there

text: exists

text: a

text: set

text: the

text: parity

text: (P)-

text: and

text: T-violating

text: (PVTV)

text: sector

text: is

text: be-

text: of

text: dimension-six

text: four-fermion

text: operators

text: involving

text: right-

text: ing

text: probed

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text: Trait Data distribution Trait Data distribution Gender Male (50.8%), Female (49.2%) Gender 0.9601 (APA) Ethnicity White (79.5%), Other (20.5%) Ethnicity 0.913 (APA) Hair Color Dark (60%), Bright (40.0%) Hair Color 0.719 (SA) Makeup Wears (39.4%), Does not wear (60.6%) Makeup 0.697 (SA) Age Young (67.8%), Elder (32.2%) Age 0.563 (MA) Emotions Joy (64.2%), Other (35.8%) Emotions 0.688 (SA) Attractive Yes (65.9%), No (34.1%) Attractive 0.29 (FA) Humorous Yes (55.6%), No (44.4%) Humorous 0.171 (SLA) Chubby Yes (57.3%), No (42.7%) Chubby 0.153 (SLA) Table 1: Attributes / Traits in FAD: Personal traits in FAD Table 2: Inter-rater agreement (Fleiss’ Kappa) measured along with the corresponding classes are listed. For each trait, for each of the traits in FAD. A value of 1 indicates per- the distribution of images across the corresponding classes is fect agreement, while a value of 0 indicates no agreement. given. As is evident, some traits have more skewness across APA stands for Almost perfect agreement; MA for Moderate their classes as compared to others. For all our experiments, agreement; SA for Substantial agreement; FA for Fair agree- our training and test sets contain a similar distribution. ment; and SLA for Slight agreement. (cid:883) (cid:884) (cid:885) (cid:886) (cid:887) (cid:888) (cid:889) (cid:890) ticipants examine several images and provide labels for each iﬁfethmaonalsrlvoWakiniugt,nheggewgehiofeneoffeafamrfaeccfelrhaotlaedrctidttehmhh(easeaootuatgrfrrcaeaehtblhillewtaeMplbesataaerTbhratluetilaiiseltrvcfssdkeiaplfc3iepoatsnoarnttieorrtm1tsiduil0cyegioisnohpns.faiTnnnMoaoctubneTrla-reureiemrp)1dka.iauyngmnmedtihsega.ennhtTttaolnonanfoabocte$tcl6aesotx,uifeobonrnyrtt Input Image Conv + Pool Conv + Pool Conv Conv Conv + Pool Fully-Connected Fully-Connected Fully-Connected SoftmaxLoss Layer (other variant possible) To further ensure the quality of the labels, we included some very simple questions designed to identify partici- Figure 1: Block Illustration of AlexNet [Krizhevsky et al., pants who could be randomly clicking answers or not pay- 2012]: a deep CNN architecture. The deep convolutional ing enough attention to the task.2 We removed the responses neural net has eight layers (denoted as L1, . . . , L8) after the of participants who failed to correctly answer these ques- input. The last fully connected layer is conventionally fol- tions. Also, we excluded the responses of participants who lowed by a softmax loss layer, but can also be replaced by the disagreed with their peers on over a third of the labels for likes of Sigmoid Cross Entropy Loss Layer [Jia et al., 2014]. the objective traits (e.g. participants who did not agree with We use this CNN architecture for all our experiments. their peers on the gender or ethnicity labels for a third of their images). Our goal is to use the annotations of the images in FAD Table 1. Due to the various advantages offered by deep nets, to train an automated system to infer personal traits from fa- we begin by applying one of the most famous CNN architec- cial images. However, some traits are clearly more difﬁcult tures [Krizhevsky et al., 2012] for our prediction task, widely than others. When people ﬁnd it easy to infer a certain prop- known as AlexNet. The block-level architecture of AlexNet erty from an image, we expect a high degree of agreement is shown in Figure 1. between the raters. In contrast, when inferring a target vari- Brief Overview of AlexNet: The fully-connected layers able is difﬁcult, we expect our annotators to often disagree have 4096 neurons each. Max-pooling is done to facilitate regarding the correct label for an image. local translation-invariance and dimension reduction. For the In cases where a trait exhibits a low degree of inter-rater fully connected layers, a drop-out [Srivastava et al., 2014] agreement in the dataset, even an excellent learning method probability of 0.5 is used to avoid overﬁtting. The ﬁnal fully would ﬁnd it difﬁcult to achieve a high degree of accuracy in connected layer takes the outputs of L7 as its input, produces the task. Table 2 presents the inter-rater agreement, as mea- outputs equal to the number of classes through a fully con- sured by Fleiss’ Kappa [Fleiss et al., 2013], for each of the nected architecture, then passes these outputs through a soft- traits, evaluated on our dataset. max function, and ﬁnally applies the negative log likelihood 3 Prediction Algorithm loss. With softmax loss layer, each input image is expected to have only one label. When the softmax loss layer is re- Our goal is to predict the various traits of a person from placed by a sigmoid cross-entropy loss layer, the outputs of their facial images. We consider FAD for all our experiments, L8 are applied to a sigmoid function to produce predicted and use the traits and the corresponding classes as listed in probabilities, using which a cross-entropy loss is computed. 2For example, we asked simple mathematical questions for Here each input can have multiple label probabilities. 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text: Trait

label: table

text: Data

text: distribution

text: Trait

text: Data

text: distribution

text: Gender

text: Male

text: Female

text: Gender

text: (APA)

text: Ethnicity

text: White

text: Other

text: Ethnicity

text: (APA)

text: Hair

text: Color

text: Dark

text: Bright

text: Hair

text: Color

text: (SA)

text: Makeup

text: Wears

text: Does

text: not

text: wear

text: Makeup

text: (SA)

text: Age

text: Young

text: Elder

text: Age

text: (MA)

text: Emotions

text: Joy

text: Other

text: Emotions

text: (SA)

text: Attractive

text: Yes

text: No

text: Attractive

text: (FA)

text: Humorous

text: Yes

text: No

text: Humorous

text: (SLA)

text: Chubby

text: Yes

text: No

text: Chubby

text: (SLA)

text: Table

text: Attributes

label: caption

text: Traits

text: in

text: FAD:

text: Personal

text: traits

text: in

text: FAD

text: Table

text: Inter-rater

text: agreement

text: (Fleiss’

text: Kappa)

text: measured

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text: agreement,

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text: traits

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text: for

text: Almost

text: perfect

text: agreement;

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text: classes

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text: compared

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text: others.

text: For

text: all

text: our

text: experiments,

text: agreement;

text: SA

text: for

text: Substantial

text: agreement;

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text: Fair

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text: our

text: training

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text: test

text: sets

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text: similar

text: distribution.

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text: SLA

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text: Slight

text: agreement.

text: ticipants

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text: images

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text: provide

text: labels

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text: each

text: Input

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text: Conv

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text: Fully-Connected

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text: Limits of Embedded Graphs and Universality Conjectures for the Network Flow Benjamin Schweinhart June 2017 Abstract We deﬁne notions of local topological convergence and local geometric convergence for R embedded graphs in n, and study their properties. The former is related to Benjamini- Schramm convergence, and the latter to weak convergence of probability measures with respect to a certain topology on the space of embedded graphs. These are used to state universality conjectures for the long-term behavior of the network ﬂow, or curvature ﬂow on embedded graphs. To provide evidence these conjectures, we develop and apply computational methods to test for local topological and local geometric convergence. 1 Introduction It has long been observed, in experiments and simulations, that scale-invariant statistics of grain boundary networks appear to be universal. [22, 15] That is, the statistics of their long-term scale-invariant properties are largely independent of the initial conditions. This is particularly interesting because the universal condition is both statistical and transient - a ﬁnite graph that approaches it will not stop ﬂowing, but will rather continue evolving until it has very few edges. This observation motivates the introduction of two notions of graph limit for embedded graphs, one topological and one geometric. We use these to formally state universality conjectures for the network ﬂow on graphs. We computationally test the local topological and local geometric convergence of simulations of the network ﬂow in two dimensions. R In Section we brieﬂy introduce the network ﬂow for embedded graphs in n, which is also known as curvature ﬂow on embedded graphs. The planar case is the simplest model of the physical phenomenon of grain growth in a polycrystal [23], and the evolution of graphs in higher dimensions is also of mathematical interest. Most of the concepts deﬁned here can be extended to general k- R dimensional regular cell complexes in n, and a similar universality phenomenon has been observed R for the most physically important case of 2-dimensional cell complexes in 3. The eventual goal of the program outlined here would be to prove the existence of universal statistics in that case, and to study their properties. However, we will focus on embedded graphs in this paper for the purpose of clarity. We deﬁne two notions of of convergence for embedded graphs - local topological, or Benjamini- Schramm, convergence in Section and local geometric convergence in Section . The former implies convergence of all local topological statistics, and is connected to the method of swatches introduced 1

text: Limits

text: of

text: Embedded

text: Graphs

text: and

text: Universality

text: Conjectures

text: for

text: the

text: Network

text: Flow

text: Benjamin

text: Schweinhart

text: June

label: date

text: Abstract

text: We

text: deﬁne

text: notions

text: of

text: local

text: topological

text: convergence

text: and

text: local

text: geometric

text: convergence

text: for

text: R

text: embedded

text: graphs

text: in

text: n,

text: and

text: study

text: their

text: properties.

text: The

text: former

text: is

text: related

text: to

text: Benjamini-

text: Schramm

text: convergence,

text: and

text: the

text: latter

text: to

text: weak

text: convergence

text: of

text: probability

text: measures

text: with

text: respect

text: to

text: a

text: certain

text: topology

text: on

text: the

text: space

text: of

text: embedded

text: graphs.

text: These

text: are

text: used

text: to

text: state

text: universality

text: conjectures

text: for

text: the

text: long-term

text: behavior

text: of

text: the

text: network

text: ﬂow,

text: or

text: curvature

text: ﬂow

text: on

text: embedded

text: graphs.

text: To

text: provide

text: evidence

text: these

text: conjectures,

text: we

text: develop

text: and

text: apply

text: computational

text: methods

text: to

text: test

text: for

text: local

text: topological

text: and

text: local

text: geometric

text: convergence.

text: Introduction

text: It

text: has

text: long

text: been

text: observed,

text: in

text: experiments

text: and

text: simulations,

text: that

text: scale-invariant

text: statistics

text: of

text: grain

text: boundary

text: networks

text: appear

text: to

text: be

text: universal.

text: That

text: is,

text: the

text: statistics

text: of

text: their

text: long-term

text: scale-invariant

text: properties

text: are

text: largely

text: independent

text: of

text: the

text: initial

text: conditions.

text: This

text: is

text: particularly

text: interesting

text: because

text: the

text: universal

text: condition

text: is

text: both

text: statistical

text: and

text: transient

text: a

text: ﬁnite

text: graph

text: that

text: approaches

text: it

text: will

text: not

text: stop

text: ﬂowing,

text: but

text: will

text: rather

text: continue

text: evolving

text: until

text: it

text: has

text: very

text: few

text: edges.

text: This

text: observation

text: motivates

text: the

text: introduction

text: of

text: two

text: notions

text: of

text: graph

text: limit

text: for

text: embedded

text: graphs,

text: one

text: topological

text: and

text: one

text: geometric.

text: We

text: use

text: these

text: to

text: formally

text: state

text: universality

text: conjectures

text: for

text: the

text: network

text: ﬂow

text: on

text: graphs.

text: We

text: computationally

text: test

text: the

text: local

text: topological

text: and

text: local

text: geometric

text: convergence

text: of

text: simulations

text: of

text: the

text: network

text: ﬂow

text: in

text: two

text: dimensions.

text: R

text: In

text: Section

text: we

text: brieﬂy

text: introduce

text: the

text: network

text: ﬂow

text: for

text: embedded

text: graphs

text: in

text: n,

text: which

text: is

text: also

text: known

text: as

text: curvature

text: ﬂow

text: on

text: embedded

text: graphs.

text: The

text: planar

text: case

text: is

text: the

text: simplest

text: model

text: of

text: the

text: physical

text: phenomenon

text: of

text: grain

text: growth

text: in

text: a

text: polycrystal

text: and

text: the

text: evolution

text: of

text: graphs

text: in

text: higher

text: dimensions

text: is

text: also

text: of

text: mathematical

text: interest.

text: Most

text: of

text: the

text: concepts

text: deﬁned

text: here

text: can

text: be

text: extended

text: to

text: general

text: k-

text: R

text: dimensional

text: regular

text: cell

text: complexes

text: in

text: n,

text: and

text: a

text: similar

text: universality

text: phenomenon

text: has

text: been

text: observed

text: R

text: for

text: the

text: most

text: physically

text: important

text: case

text: of

text: cell

text: complexes

text: in

text: The

text: eventual

text: goal

text: of

text: the

text: program

text: outlined

text: here

text: would

text: be

text: to

text: prove

text: the

text: existence

text: of

text: universal

text: statistics

text: in

text: that

text: case,

text: and

text: to

text: study

text: their

text: properties.

text: However,

text: we

text: will

text: focus

text: on

text: embedded

text: graphs

text: in

text: this

text: paper

text: for

text: the

text: purpose

text: of

text: clarity.

text: We

text: deﬁne

text: two

text: notions

text: of

text: of

text: convergence

text: for

text: embedded

text: graphs

text: local

text: topological,

text: or

text: Benjamini-

text: Schramm,

text: convergence

text: in

text: Section

text: and

text: local

text: geometric

text: convergence

text: in

text: Section

text: The

text: former

text: implies

text: convergence

text: of

text: all

text: local

text: topological

text: statistics,

text: and

text: is

text: connected

text: to

text: the

text: method

text: of

text: swatches

text: introduced


text: (a) (b) Figure 1: Small regions of the conjectured universal state in (a) 2 and (b) 3 dimensions. They were produced by simulations discussed in Section R The second hypothesis implies that two embedded graphs in n are homeomorphic if and only if R they are combinatorially equivalent. The intersection of an embedded graph in n with any open or R compact subset of n is also an embedded graph, though new edges may be created in the process (as well as vertices at the boundary, if the set is compact) .In general, it is not always the case that U ∩ G = U¯ ∩ G. The sets of embedded graphs in Rn and the open n−ball of radius r are denoted G G n and n, respectively. We will deﬁne a topology for these sets in Section . r 1.2 Summary of Concepts Here, we include a table summarizing the notions of graph limits deﬁned in this paper, and a list of deﬁnitions of the terms appearing therein. Convergence Local Topological Local Geometric R Space of Abstract Graphs Embedded Graphs in n with Bounded Degree with Smooth Geometric Cloth Local Probability Distribution Probability Distribution S G Distributions on on n r,k r Universal Probability Distribution Probability Distribution • G Distribution on G on n Symmetry Involution Invariance Translation Invariance Addnl. Symmetry Topological Homogeneity Homogeneity (Ergodicity of Translations) Convergence Convergence of discrete Weak Convergence of Prob. Distributions Unif. Separating Sequence Controls Local Topological Stats Local Geometric Stats 1. Local topological convergence - this is the same notion as Benjamin-Schramm convergence. We 3 ##LTFigure## ##LTFigure## ##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## ##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## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: Figure

text: Small

text: regions

text: of

text: the

text: conjectured

text: universal

text: state

text: in

text: (a)

text: and

text: (b)

text: dimensions.

text: They

text: were

text: produced

text: by

text: simulations

text: discussed

text: in

text: Section

text: R

text: The

text: second

text: hypothesis

text: implies

text: that

text: two

text: embedded

text: graphs

text: in

text: n

text: are

text: homeomorphic

text: if

text: and

text: only

text: if

text: R

text: they

text: are

text: combinatorially

text: equivalent.

text: The

text: intersection

text: of

text: an

text: embedded

text: graph

text: in

text: n

text: with

text: any

text: open

text: or

text: R

text: compact

text: subset

text: of

text: n

text: is

text: also

text: an

text: embedded

text: graph,

text: though

text: new

text: edges

text: may

text: be

text: created

text: in

text: the

text: process

text: (as

text: well

text: as

text: vertices

text: at

text: the

text: boundary,

text: if

text: the

text: set

text: is

text: compact)

text: .In

text: general,

text: it

text: is

text: not

text: always

text: the

text: case

text: that

text: U

text: G

text: U¯

text: G.

text: The

text: sets

text: of

text: embedded

text: graphs

text: in

text: Rn

text: and

text: the

text: open

text: n−ball

text: of

text: radius

text: r

text: are

text: denoted

text: G

text: G

text: n

text: and

text: n,

text: respectively.

text: We

text: will

text: deﬁne

text: a

text: topology

text: for

text: these

text: sets

text: in

text: Section

text: r

text: Summary

text: of

text: Concepts

text: Here,

text: we

text: include

text: a

text: table

text: summarizing

text: the

text: notions

text: of

text: graph

text: limits

text: deﬁned

text: in

text: this

text: paper,

text: and

text: a

text: list

text: of

text: deﬁnitions

text: of

text: the

text: terms

text: appearing

text: therein.

text: Convergence

text: Local

text: Topological

text: Local

text: Geometric

text: R

text: Space

text: of

text: Abstract

text: Graphs

text: Embedded

text: Graphs

text: in

text: n

text: with

text: Bounded

text: Degree

text: with

text: Smooth

text: Geometric

text: Cloth

text: Local

text: Probability

text: Distribution

text: Probability

text: Distribution

text: S

text: G

text: Distributions

text: on

text: on

text: n

text: r,k

text: r

text: Universal

text: Probability

text: Distribution

text: Probability

text: Distribution

text: G

text: Distribution

text: on

text: G

text: on

text: n

text: Symmetry

text: Involution

text: Invariance

text: Translation

text: Invariance

text: Addnl.

text: Symmetry

text: Topological

text: Homogeneity

text: Homogeneity

text: (Ergodicity

text: of

text: Translations)

text: Convergence

text: Convergence

text: of

text: discrete

text: Weak

text: Convergence

text: of

text: Prob.

text: Distributions

text: Unif.

text: Separating

text: Sequence

text: Controls

text: Local

text: Topological

text: Stats

text: Local

text: Geometric

text: Stats

text: Local

text: topological

text: convergence

text: this

text: is

text: the

text: same

text: notion

text: as

text: Benjamin-Schramm

text: convergence.

text: We

text: ##LTFigure##

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text: Figure 2: A planar graph evolving by curvature ﬂow. → G Deﬁnition 2.1. A continuous function f : (T , T ) n is a curvature ﬂow on regular graphs ∈ 1 2 if f(t) is a regular graph for all t (T , T ), and there there exists a collection of parametrizations { → R1 } 2 of the edges of f(t), γ : (T , T ) n , satisfying i 1 2 ∂γ (t) ∀ ∈ i = κn t (T , T ) ∂t 1 2 where κ is the curvature and n is the unit normal vector with orientation given by the parametriza- tion. The restriction to regular graphs is quite severe, but the deﬁnition of the the network ﬂow can be extended to more general graphs as follows: → G Deﬁnition 2.2. A continuous function f : [T , T ] n is a curvature ﬂow on graphs 1 2 ⊂ R and a network ﬂow if for any bounded region B n, there is a ﬁnite collection of times { } ∩ T = s , . . . , s = T such that f (t) := f(t) B deﬁnes a curvature ﬂow on regular graphs 1 1 m 2 B on the interval (s , s ) and i i+1 lim f(t) = f(s ) → i t si ≤ ≤ { } for each 1 i n, in the varifold topology. The s are called singular times. i [12] discusses hypotheses under which a network ﬂow exists starting from a non-regular initial condition in two dimensions.Note that the behavior of the ﬂow after a singular time is not necessarily uniquely determined. 5 ##LTFigure## ##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## ##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## ##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## ##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: Figure

text: A

text: planar

text: graph

text: evolving

text: by

text: curvature

text: ﬂow.

text: G

text: Deﬁnition

text: A

text: continuous

text: function

text: f

text: (T

text: T

text: n

text: is

text: a

text: curvature

text: ﬂow

text: on

text: regular

text: graphs

text: if

text: f(t)

text: is

text: a

text: regular

text: graph

text: for

text: all

text: t

text: (T

text: T

text: and

text: there

text: there

text: exists

text: a

text: collection

text: of

text: parametrizations

text: of

text: the

text: edges

text: of

text: f(t),

text: γ

text: (T

text: T

text: n

text: satisfying

text: i

text: ∂γ

text: (t)

text: i

text: κn

text: t

text: (T

text: T

text: ∂t

text: where

text: κ

text: is

text: the

text: curvature

text: and

text: n

text: is

text: the

text: unit

text: normal

text: vector

text: with

text: orientation

text: given

text: by

text: the

text: parametriza-

text: tion.

text: The

text: restriction

text: to

text: regular

text: graphs

text: is

text: quite

text: severe,

text: but

text: the

text: deﬁnition

text: of

text: the

text: the

text: network

text: ﬂow

text: can

text: be

text: extended

text: to

text: more

text: general

text: graphs

text: as

text: follows:

text: G

text: Deﬁnition

text: A

text: continuous

text: function

text: f

text: [T

text: T

text: n

text: is

text: a

text: curvature

text: ﬂow

text: on

text: graphs

text: R

text: and

text: a

text: network

text: ﬂow

text: if

text: for

text: any

text: bounded

text: region

text: B

text: n,

text: there

text: is

text: a

text: ﬁnite

text: collection

text: of

text: times

text: T

text: s

text: s

text: T

text: such

text: that

text: f

text: (t)

text: f(t)

text: B

text: deﬁnes

text: a

text: curvature

text: ﬂow

text: on

text: regular

text: graphs

text: m

text: B

text: on

text: the

text: interval

text: (s

text: s

text: and

text: i

text: lim

text: f(t)

text: f(s

text: i

text: t

text: si

text: for

text: each

text: i

text: n,

text: in

text: the

text: varifold

text: topology.

text: The

text: s

text: are

text: called

text: singular

text: times.

text: i

text: discusses

text: hypotheses

text: under

text: which

text: a

text: network

text: ﬂow

text: exists

text: starting

text: from

text: a

text: non-regular

text: initial

text: condition

text: in

text: two

text: dimensions.Note

text: that

text: the

text: behavior

text: of

text: the

text: ﬂow

text: after

text: a

text: singular

text: time

text: is

text: not

text: necessarily

text: uniquely

text: determined.

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text: Figure 1. Distributions of ε at LHC for MCGlauber and MCKLN (with two diﬀerent 2 smoothing scales) produce initial conditions that are at one end very circular and at the other end elliptical. mckln Λ(cid:61)0.3 PbPb 2.76TeV 0.15 mckln Λ(cid:61)1.0 glauber (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224) 0.10 (cid:230)(cid:224) (cid:242) (cid:242) (cid:242) (cid:230)(cid:224) (cid:242) (cid:230)(cid:224) (cid:242) (cid:242) 0.05 (cid:230)(cid:224) (cid:242) (cid:230)(cid:230)(cid:224)(cid:224)(cid:242)(cid:242) (cid:230)(cid:224)(cid:242)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224) 0.00 0 10 20 30 40 50 60 centrality (cid:37) { } { } Figure 2. v 2 and v 2 for MCGlauber and MCKLN (with two diﬀerent smoothing scales) 2 3 at LHC. even making predictions for the highest LHC energies [18]. While there has been some progress made in combining the knowledge of hydrodynamical backgrounds [19, 20] and ﬂuctuating initial conditions [21] with energy loss models, it was not until [22] that full event-by-event viscous hydrodynamical backgrounds (v-USPhydro [23, 24]) were combined with an energy loss model (BBMG [5, 25, 26]). The eﬀects of event-by-event ﬂuctuations as well as the correlation between soft and hard ﬂow harmonics in [22] provided the key step needed in order to solve the ⊗ longstanding R v puzzle. AA 2 2. Event-by-event viscous hydrodynamics+energy loss In [22] there were two crucial diﬀerences between all previous calculations of v (p > 10GeV ): 2 T (i) Full event-by-event viscous hydrodynamical backgrounds were fed into the energy loss model to obtain vhard. 2 (ii) vexp(p ) from Eq. (1) was calculated correlating the soft integrated vsoft and hard vhard(p ) 2 T 2 2 T to simulate the two particle (one soft, one hard) correlations used to measure elliptical ﬂow experimentally. The soft vsoft was taken using the best ﬁtting parameters in the soft sector, see [22] for more 2 details. The vhard (the 2nd Fourier coeﬃcient of R (p , φ)) is computed using the BBMG 2 AA T code [5, 25, 26] in the pQCD-like scenario. In order to make comparisons to experimental data 2 v, 2 v 3 2 (cid:60) (cid:56) (cid:60) (cid:56) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Figure 1: captioncolorDistributed asynchronous dual free stochastic dual coordinate ascent for parameter server framework. In iteration t, the server receives gradient message v from worker k, and sends the k up-to-date wt back to the worker k. Global variables in other workers are stale. For example worker 1 and K − − store stale global variables wt 2 and wt 5 respectively. Distributed machine learning methods are required to optimize problem (1) when the dataset is distributed over multiple machines. In [Jaggi et al., 2014], the authors proposed communication-efﬁcient distributed dual coordinate ascent (CoCoA) for primal-dual distributed optimization. In each iteration, the CoCoA framework allows workers to optimize subproblems independently at ﬁrst. After that, it calls “Reduce" operation to collect local solution from all workers, and updates global variable and broadcasts the up-to- date global variable to workers in the end. It uses stochastic dual coordinate ascent (SDCA) as the local solver which is one of the most successful methods proposed for solving the problem (1) [Hsieh et al., 2008, Shalev-Shwartz and Zhang, 2013a]. In [Shalev-Shwartz and Zhang, 2013b], the authors proved that SDCA has linear convergence if the convex function φ (w) is smooth, which is much faster than stochastic gradient i descent (SGD). [Yang, 2013, Takácˇ et al., 2015] also proposed distributed SDCA and analyzed the tradeoff between computation and communication. [Ma et al., 2015, Ma et al., 2017] accelerated the CoCoA by allowing for more aggressive updates, and proved that CoCoA has linear primal-dual convergence for the smooth convex problem and sublinear convergence for the non-smooth convex problem. However, there are two issues for these primal-dual distributed methods. Firstly, all of them use SDCA as the local solver. SDCA is not applicable when the dual problem is unknown, e.g. φ is non-convex. Therefore, the applications of i these primal-dual distributed methods are limited. Secondly, all of these methods assume that the workers have similar computing speed, which is not true in practice. Straggler problem is an unavoidable practical issue in the distributed optimization. Thus, the computing time of CoCoA and distributed SDCA is dependent on the slowest worker. Even if there is only one bad worker, they will work far slower than expectation. In [Shalev-Shwartz, 2015, Shalev-Shwartz, 2016], the authors proposed dual free stochastic dual coor- dinate ascent (dfSDCA). It was proved to admit similar convergence rate to SDCA while it did not rely on duality at all. However there is no distributed optimization method using dfSDCA, and its convergence analysis is still unknown yet. In this paper, we solve the above two challenging issues in previous primal-dual distributed optimization methods by proposing Distributed Dual Free Stochastic Dual Coordinate Ascent (Dis-dfSDCA). We use dfSDCA as the local solver such that Dis-dfSDCA can be applied to the non-convex problem easily. We alleviate the effect of straggler problem by allowing asynchronous communication between server and workers. As shown in Figure 1, the server does not wait and workers may store the stale global variable in the local. We also analyze the convergence rate of our method and prove that it admits linear convergence rate even if the individual losses (φ ) are non-convex, as long as the sum of losses f is convex. Finally, i we conduct simulation on the distributed system with straggler problem. Experimental results verify our 2 ##LTFigure## ##LTFigure##

text: Figure

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text: are

text: required

text: to

text: optimize

text: problem

text: when

text: the

text: dataset

text: is

text: distributed

text: over

text: multiple

text: machines.

text: In

text: [Jaggi

text: et

text: al.,

text: the

text: authors

text: proposed

text: communication-efﬁcient

text: distributed

text: dual

text: coordinate

text: ascent

text: (CoCoA)

text: for

text: primal-dual

text: distributed

text: optimization.

text: In

text: each

text: iteration,

text: the

text: CoCoA

text: framework

text: allows

text: workers

text: to

text: optimize

text: subproblems

text: independently

text: at

text: ﬁrst.

text: After

text: that,

text: it

text: calls

text: “Reduce"

text: operation

text: to

text: collect

text: local

text: solution

text: from

text: all

text: workers,

text: and

text: updates

text: global

text: variable

text: and

text: broadcasts

text: the

text: up-to-

text: date

text: global

text: variable

text: to

text: workers

text: in

text: the

text: end.

text: It

text: uses

text: stochastic

text: dual

text: coordinate

text: ascent

text: (SDCA)

text: as

text: the

text: local

text: solver

text: which

text: is

text: one

text: of

text: the

text: most

text: successful

text: methods

text: proposed

text: for

text: solving

text: the

text: problem

text: [Hsieh

text: et

text: al.,

text: Shalev-Shwartz

text: and

text: Zhang,

text: In

text: [Shalev-Shwartz

text: and

text: Zhang,

text: the

text: authors

text: proved

text: that

text: SDCA

text: has

text: linear

text: convergence

text: if

text: the

text: convex

text: function

text: φ

text: (w)

text: is

text: smooth,

text: which

text: is

text: much

text: faster

text: than

text: stochastic

text: gradient

text: i

text: descent

text: (SGD).

text: [Yang,

text: Takácˇ

text: et

text: al.,

text: also

text: proposed

text: distributed

text: SDCA

text: and

text: analyzed

text: the

text: tradeoff

text: between

text: computation

text: and

text: communication.

text: [Ma

text: et

text: al.,

text: Ma

text: et

text: al.,

text: accelerated

text: the

text: CoCoA

text: by

text: allowing

text: for

text: more

text: aggressive

text: updates,

text: and

text: proved

text: that

text: CoCoA

text: has

text: linear

text: primal-dual

text: convergence

text: for

text: the

text: smooth

text: convex

text: problem

text: and

text: sublinear

text: convergence

text: for

text: the

text: non-smooth

text: convex

text: problem.

text: However,

text: there

text: are

text: two

text: issues

text: for

text: these

text: primal-dual

text: distributed

text: methods.

text: Firstly,

text: all

text: of

text: them

text: use

text: SDCA

text: as

text: the

text: local

text: solver.

text: SDCA

text: is

text: not

text: applicable

text: when

text: the

text: dual

text: problem

text: is

text: unknown,

text: e.g.

text: φ

text: is

text: non-convex.

text: Therefore,

text: the

text: applications

text: of

text: i

text: these

text: primal-dual

text: distributed

text: methods

text: are

text: limited.

text: Secondly,

text: all

text: of

text: these

text: methods

text: assume

text: that

text: the

text: workers

text: have

text: similar

text: computing

text: speed,

text: which

text: is

text: not

text: true

text: in

text: practice.

text: Straggler

text: problem

text: is

text: an

text: unavoidable

text: practical

text: issue

text: in

text: the

text: distributed

text: optimization.

text: Thus,

text: the

text: computing

text: time

text: of

text: CoCoA

text: and

text: distributed

text: SDCA

text: is

text: dependent

text: on

text: the

text: slowest

text: worker.

text: Even

text: if

text: there

text: is

text: only

text: one

text: bad

text: worker,

text: they

text: will

text: work

text: far

text: slower

text: than

text: expectation.

text: In

text: [Shalev-Shwartz,

text: Shalev-Shwartz,

text: the

text: authors

text: proposed

text: dual

text: free

text: stochastic

text: dual

text: coor-

text: dinate

text: ascent

text: (dfSDCA).

text: It

text: was

text: proved

text: to

text: admit

text: similar

text: convergence

text: rate

text: to

text: SDCA

text: while

text: it

text: did

text: not

text: rely

text: on

text: duality

text: at

text: all.

text: However

text: there

text: is

text: no

text: distributed

text: optimization

text: method

text: using

text: dfSDCA,

text: and

text: its

text: convergence

text: analysis

text: is

text: still

text: unknown

text: yet.

text: In

text: this

text: paper,

text: we

text: solve

text: the

text: above

text: two

text: challenging

text: issues

text: in

text: previous

text: primal-dual

text: distributed

text: optimization

text: methods

text: by

text: proposing

text: Distributed

text: Dual

text: Free

text: Stochastic

text: Dual

text: Coordinate

text: Ascent

text: (Dis-dfSDCA).

text: We

text: use

text: dfSDCA

text: as

text: the

text: local

text: solver

text: such

text: that

text: Dis-dfSDCA

text: can

text: be

text: applied

text: to

text: the

text: non-convex

text: problem

text: easily.

text: We

text: alleviate

text: the

text: effect

text: of

text: straggler

text: problem

text: by

text: allowing

text: asynchronous

text: communication

text: between

text: server

text: and

text: workers.

text: As

text: shown

text: in

text: Figure

text: the

text: server

text: does

text: not

text: wait

text: and

text: workers

text: may

text: store

text: the

text: stale

text: global

text: variable

text: in

text: the

text: local.

text: We

text: also

text: analyze

text: the

text: convergence

text: rate

text: of

text: our

text: method

text: and

text: prove

text: that

text: it

text: admits

text: linear

text: convergence

text: rate

text: even

text: if

text: the

text: individual

text: losses

text: (φ

text: are

text: non-convex,

text: as

text: long

text: as

text: the

text: sum

text: of

text: losses

text: f

text: is

text: convex.

text: Finally,

text: i

text: we

text: conduct

text: simulation

text: on

text: the

text: distributed

text: system

text: with

text: straggler

text: problem.

text: Experimental

text: results

text: verify

text: our

text: ##LTFigure##

text: ##LTFigure##


text: (cid:10) (cid:11) − ∗ ∇ ≥ where the ﬁrst inequality follows from the strong convexity of P such that wt w , P (wt) − ∗ (cid:107) − ∗(cid:107) − ∗ ≥ (cid:107) − ∗(cid:107) P (wt) P (w ) + λ wt w 2 and P (wt) P (w ) λ wt w 2. Deﬁning γ = λ and substituting 2 2 2 (cid:3) above two inequalities into (43) and (45) respectively, we complete the proof. Proof to Theorem 3 Proof 6 We deﬁne C = c A + c B and set c = 1 , c = 1. Inputting Lemma 2 in the equation, t+1 a t+1 b t+1 a 4L2 b we have: equationcolor E [C ] = c A + c B t+1 a t+1 b t+1 ≤ c ((cid:0)1 − ηλH)E(cid:107)αt − α∗(cid:1)(cid:107)2 + (c +(cid:18)2c ηλHL2 − 2c ηH(λ − γ ))E(cid:107)wt(cid:19)− w(cid:88)∗(cid:107)2 a i i b a b 2 − − − E(cid:107) (cid:107) Hτ(L + λ)2η3 t 1 E(cid:107) (cid:107) + c η2 c ηλ(1 β) vt 2 + 2c λHτL2η3 +(cid:18)c vj 2 (cid:19) (54) b a (cid:0) a (cid:1) b γ (cid:88) j=d(t) − − E − − E(cid:107) (cid:107) Hτ(L + λ)2η3 t 1 E(cid:107) (cid:107) = (1 ηλH) [C ] + c η2 c ηλ(1 β) vt 2 + 2c λHτL2η3 + c vj 2 t b a a b γ j=d(t) − where γ = λ. Adding the above inequality from t = 0 to t = T 1, we have that: 2 (cid:88)equationcolor (cid:88) (cid:0) (cid:1) (cid:88) − − − T 1 T 1 T 1 E ≤ − E − − E(cid:107) (cid:107) [Ct+1] (1(cid:18) ηλH) [Ct] + cbη2 caηλ((cid:19)1 (cid:88)β)(cid:88) vt 2 t=0 t=0 t=0 − − Hτ(L + λ)2η3 T 1 t 1 E(cid:107) (cid:107) + 2c λHτL2η3 + c (cid:18) vj 2 (cid:19) a (cid:88) b γ (cid:88) t=0 j=d(t) − − ≤ − T 1 E − − Hτ2(L + λ)2η3 T 1 E(cid:107) (cid:107) ((cid:88)1 ηλH) [Ct] + cbη2 caηλ(1 β) + 2caλHτ2L2η3 + cb vt 2 γ t=0 t=0 − T 1 ≤ E − E [C ] + (1 ηλH) [C ] (55) t 0 t=1 where the last inequality holds as long as: equationcolor ≤ λ2 η (56) 2HLτ2λ2 + 8HLτ2(L + λ)2 + 4λL2 + nλ3 such that 2c λHL2τ2η3 + c H(L+λ)2τ2η3 + c η2 − c ηλ(1 − β) ≤ 0. We complete the proof. (cid:3) a b γ b a sectioncolor 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: the

text: ﬁrst

text: inequality

text: follows

text: from

text: the

text: strong

text: convexity

text: of

text: P

text: such

text: that

text: wt

text: w

text: P

text: (wt)

text: P

text: (wt)

text: P

text: (w

text: λ

text: wt

text: w

text: and

text: P

text: (wt)

text: P

text: (w

text: λ

text: wt

text: w

text: Deﬁning

text: γ

text: λ

text: and

text: substituting

text: above

text: two

text: inequalities

text: into

text: and

text: respectively,

text: we

text: complete

text: the

text: proof.

text: Proof

text: to

text: Theorem

text: Proof

text: We

text: deﬁne

text: C

text: c

text: A

text: c

text: B

text: and

text: set

text: c

text: c

text: Inputting

text: Lemma

text: in

text: the

text: equation,

text: a

text: b

text: a

text: b

text: we

text: have:

text: equationcolor

text: E

text: [C

text: c

text: A

text: c

text: B

text: a

text: b

text: c

text: (c

text: ηH(λ

text: γ

text: a

text: i

text: i

text: b

text: a

text: b

text: Hτ(L

text: t

text: c

text: η2

text: c

text: ηλ(1

text: β)

text: vt

text: vj

text: b

text: a

text: a

text: b

text: γ

text: j=d(t)

text: E

text: Hτ(L

text: t

text: ηλH)

text: [C

text: c

text: η2

text: c

text: ηλ(1

text: β)

text: vt

text: c

text: vj

text: t

text: b

text: a

text: a

text: b

text: γ

text: j=d(t)

text: where

text: γ

text: λ.

text: Adding

text: the

text: above

text: inequality

text: from

text: t

text: to

text: t

text: T

text: we

text: have

text: that:

text: T

text: T

text: T

text: E

text: E

text: ηλH)

text: [Ct]

text: vt

text: Hτ(L

text: T

text: t

text: c

text: vj

text: a

text: b

text: γ

text: j=d(t)

text: T

text: E

text: T

text: ηλH)

text: [Ct]

text: β)

text: cb

text: vt

text: γ

text: T

text: E

text: E

text: [C

text: ηλH)

text: [C

text: t

text: where

text: the

text: last

text: inequality

text: holds

text: as

text: long

text: as:

text: equationcolor

text: λ2

text: η

text: such

text: that

text: c

text: c

text: η2

text: c

text: ηλ(1

text: β)

text: We

text: complete

text: the

text: proof.

text: a

text: b

text: γ

text: b

text: a

text: sectioncolor

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##