text: G G the other features characterize only p, but not p. We sum- exhibit a longer average path length than news whose inter- t marize these different features in Table 1, along with their actions happened mostly from the seed node. Finally, the formal deﬁnitions, and describe them in the following. diameter of a graph, deﬁned as the longer shortest path be- tween any pair of nodes of the graph, indicates the spreading High Level Properties. These features represent high- distance of posts. level properties of the complete propagation cascade repre- G (cid:104) (cid:105) sented by p V p, Ep . Evolution Properties. These features represent evolution Some features represent very general properties related to properties over time of the post p propagation, from its cre- the virality of the content. These general properties are the ation time t to a subsequent point t + δ. To compute these lifetime of the cascade, measured as the distance in minutes features, we construct the propagation graphs at different G G from the ﬁrst to the last captured interaction with the con- time steps , ..., . We then calculate the value of t+30 t+δ tent; the size of the cascade in terms of number of nodes three of our high-level features for each graph at each time (users who interacted with the content); the number of total step: (i) Friendships Ratio, (ii) Size, and (iii) Interactions interactions; and the time required for the cascade to reach Ratio. For each of these high-level feature, we obtain a time its 90% total interactions (referred to as 90% interactions series time). v , v , ..., v , 1 2 δ/30 Other features derived from high level properties attempt on which we compute a set of well-known statistical mea- to capture different possible types of interaction with the sures that represent the evolution over time of the se- content. Friendships ratio is deﬁned as the proportion of ries (Wiens, Horvitz, and Guttag 2012). Namely, these sta- edges whose type is “friendship” over the total number of tistical measures are the mean, standard deviation, linear edges and represents the number of times, in proportion, that weighted mean and quadratic weighted mean, average ab- the post potentially propagated among friends, rather than solute change, and maximum of the series. These measures directly from the seed node. Indeed, if no friends of users capture the evolution of the time series up to the current time interact with some content, its potential propagation graph and, especially, do not require us to know the whole time se- only contains edges with the seed page. Interactions ratio, ries, an important property for our Early Stage classiﬁcation instead, represents the average exposure to interactions from experiment. We decided to derive time-series only for the friends of users with the content. Since vertices are interact- three high-level features listed above. Indeed, we argue that ing users, and edges are potential interactions, higher values the other features either have no temporal properties (e.g., of this metric mean lower exposure (little interaction with lifetime, time to reach 90% interactions), evolve in similar or the content by one’s friends). These features are motivated predictable ways (e.g., diameter), or describe behaviors that by the observation that it is possible that the users’ fruition are already captured by the selected time series. Moreover, of different types of content is different, with some types of evolution properties of time series would be especially used content being interacted directly from the source, and other in Early Stage classiﬁcation. Social networks need to per- types of content relying more on word-of-mouth propaga- form feature extraction on this scenario at every time step: tion (Romero, Meeder, and Kleinberg 2011). features derived from topological properties are too compu- tationally expensive (e.g., diameter calculation runs in more Topological Properties. We also select as features some than quadratic complexity w.r.t. the number of vertices) to well-known properties of the topological structure of graphs. be continuously calculated for each new post. These properties are commonly used to learn information about graph structures, and have been applied in solving 5 Results problems such as link analysis and prediction (Al Hasan In this section we present the results of our experiments. et al. 2006), and especially in cascade and virality predic- In particular, we ﬁrst discuss evaluation metrics and baseline tion (Hong, Dan, and Davison 2011; Cheng et al. 2014). The (Section 5.1). We then report the results on the Early Stage average vertex degree feature represents the average number Classiﬁcation scenario (Section 5.2), and the results on the of possible propagation edges for the content at any given Final Stage Classiﬁcation scenario (Section 5.3). hop. Higher values of this metric indicate the presence of 5.1 Evaluation Metrics interacting users greatly exposed to the content, or able to inﬂuence many of their social friends. The global clustering As usual in binary classiﬁcation, the classiﬁcation base- coefﬁcient (Holland and Leinhardt 1971), a measure of the line is the performance of a random classiﬁer on the data: density of connections of graphs, is another indication of without any information regarding the propagation graph, it whether the possible propagation paths are generated from guesses either science or conspiracy with equal probability, interactions between friends, or directly with the seed page. as a coin ﬂip. The goal of our experiments is to show that Assortativity coefﬁcient, deﬁned as the degree correlation structural features do not help more sophisticated models in between pairs of linked nodes (Newman 2002), can measure improving the baseline performance. how friends inﬂuence each others in interacting with con- Unfortunately, our dataset is highly imbalanced (com- tent on the social network. Average path length (Fronczak, posed by 89491 news for conspiracy, and only 22650 news Fronczak, and Hołyst 2004), also known as Wiener index, for science). With such imbalance, standard evaluation met- gives us indications of the virality of the content, in terms of rics (such as precision, accuracy, and recall) can be mislead- distance of propagation from the seed page. Long cascad- ing, because they do not account for the uneven class fre- ing news, reshared many times from interacting friends, will quencies. Even computing averages of these metrics using

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text: path

text: length

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text: times

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text: quencies.

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text: computing

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text: these

text: metrics

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text: (cid:32) (cid:18) (cid:19) (cid:33) against the switching√costs adversary is upper bounded by: worst case deterministic adversary. In this experiment, arm √ O T2/3 K ln K ln δ1 1/3 2pliasytehres binetsot apnicdkginivgetsh1e fworroenvgerayrmevse,nthreouﬁnrdst. Taromtraiclwk athyes (cid:15) gives 0.38 whereas the third gives 1 for every round multiple of 3. The remaining arms always give 0. As shown by the 4 Experiments ﬁgure, this simple adversary is already powerful enough to make the algorithms attain their upper bound. We tested DP-EXP3-Lap, EXP3 together with the non- τ private EXP3 against a few different adversaries. The privacy parameter (cid:15) of DP-EXP3-Lap is set as deﬁned in corollary Stochastic adversary This adversary draws the gains of 3.2. This is done so that the regret of DP-EXP3-Lap and the ﬁrst arm i.i.d from Bern (0.55) whereas all other gains are EXP3 are compared with the same privacy level. All the drawn i.i.d from Bern (0.5). τ other parameters of DP-EXP3-Lap are taken as deﬁned in corollary 3.1 while the parameters of EXP3τ are taken as Fully oblivious adversary. For the best arm k, it ﬁrst deﬁned in corollary 3.2. · draws a number p uniformly in [0.5, 0.5 + 2 ε] and gen- For all experiments, the horizon is T = 218 and the number erates the gain g ∼ Bern (p). For all other arms, p is drawn of arms is K = 4. We performed 720 independent trials and from [0.5−ε, 0.5k,+t ε]. This process is repeated at every round. reported the median-of-means estimator2 of the cumulative In our experiments, ε = 0.05 regret. It partitions the trials into a equal groups and return 0 the median of the sample means of each group. Proposition An oblivious adversary. This adversary is identical to the 4.1 is a well known result (also in (Hsu and Sabato 2013; Lerasle and Oliveira 2011)) g√iving the accuracy of this estima- fully oblivious one for every round multiple of 200. Between tor. Its convergence is O(σ/ N), with exponential probabil- two multiples of 200 the last gain of the arm is given. ity tails, even though the random variable x may have heavy- tails. In comparison, the empirical mean can not provide The Switching costs adversary This adversary (deﬁned at such guarantee for any σ > 0 and conﬁdence in [0, 1/(2e)] Figure 1 in (Dekel et al. 2014)) deﬁnes a stochastic processes (Catoni 2012). (including simple Gaussian random walk as special case) for Proposition 4.1. Let x be a random variable with mean µ generating the gains. It was used to prove that any algorithm ∞ O and variance σ2 < . Assume that we have N independent against this adversary must incur a regret of (T2/3). (cid:112) sample of x and let µˆ be the median-of-means computed − − using a0|gr−oup|s.≤With probability at least 1 e a0/4.5, µˆ Discussion Figure 1 shows our results against a variety satisﬁes µˆ µ σ 6a /N. of adversaries, with respect to a ﬁxed oracle. Overall, the 0 We set the number of groups to a = 24, so that the performance (in term of regret) of DP-EXP3-Lap is very 0 conﬁdence interval holds w.p. at least 0.995. competitive against that of EXP3 while providing a signif- (cid:80) We also reported the deviation of each algorithm using the icant better privacy. This means that DP-EXP3-Lap allows Gini’s Mean Difference (GMD hereafter) (Gini and Pearson us to get privacy for free in the bandit setting against an − − 1912). GMD computes the deviation as N (2j N adversary not more powerful than the oblivious one. j=1 The performance of EXP3 is worse than that of DP- 1)x with x the j-th order statistics of the sample (that τ (j) ≤ (j)≤ ≤ EXP3-Lap against an oblivious adversary or one less pow- is x x . . . x ). As shown in (Yitzhaki and (1) (2) (N) erful. However, the situation is completely reversed against others 2003; David 1968), the GMD provides a superior the more powerful switching cost adversary. In that setting, approximation of the true deviation than the standard one. EXP3 outperforms both EXP3 and DP-EXP3-Lap con- To account for the fact that the cumulative regret of our τ ﬁrming the theoretical analysis. We can see EXP3 as the algorithms might not follow a symmetric distribution, we τ algorithm providing us privacy for free against switching cost computed the GMD separately for the values above and below adversary and adaptive m-bounded memory one in general. the median-of-means. At round t, we computed the cumulative regret against the 5 Conclusion ﬁxed oracle who plays the best arm assuming that the end of the game is at t. The oracle uses the actual sequence of We have provided the ﬁrst results on differentially pri- gains to decide his best arm. For a given trial, we make sure vate adversarial multi-armed bandits, which are optimal up that all algorithms are playing the same game by generating to logarithmic factors. One open question is how differen- the gains for all possible pair of round-arm before the game tial privacy affects regret in the full reinforcement learning starts. problem. At this point in time, the only known results in the MDP setting obtain differentially private algorithms for Deterministic adversary. As shown by (Audibert and Monte Carlo policy evaluation (Balle, Gomrokchi, and Pre- cup 2016). While this implies that it is possible to obtain Bubeck 2010), the expected regret of any agent against an policy iteration algorithms, it is unclear how to extend this to oblivious adversary can not be worse than that against the the full online reinforcement learning problem. 2Used heavily in the streaming literature (Alon, Matias, and Acknowledgements. This research was supported by the Szegedy 1996) SNSF grants “Adaptive control with approximate Bayesian ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: against

text: the

text: switching√costs

text: adversary

text: is

text: upper

text: bounded

text: by:

text: worst

text: case

text: deterministic

text: adversary.

text: In

text: this

text: experiment,

text: arm

text: O

text: K

text: ln

text: K

text: ln

text: δ1

text: binetsot

text: fworroenvgerayrmevse,nthreouﬁnrdst.

text: Taromtraiclwk

text: athyes

text: gives

text: whereas

text: the

text: third

text: gives

text: for

text: every

text: round

text: multiple

text: of

text: The

text: remaining

text: arms

text: always

text: give

text: As

text: shown

text: by

text: the

label: section

text: Experiments

text: ﬁgure,

text: this

text: simple

text: adversary

text: is

text: already

text: powerful

text: enough

text: to

text: make

text: the

text: algorithms

text: attain

text: their

text: upper

text: bound.

text: We

text: tested

text: together

text: with

text: the

text: non-

text: τ

text: private

text: against

text: a

text: few

text: different

text: adversaries.

text: The

text: privacy

text: parameter

text: of

text: is

text: set

text: as

text: deﬁned

text: in

text: corollary

text: Stochastic

text: adversary

text: This

text: adversary

text: draws

text: the

text: gains

text: of

text: This

text: is

text: done

text: so

text: that

text: the

text: regret

text: of

text: and

text: the

text: ﬁrst

text: arm

text: i.i.d

text: from

text: Bern

text: whereas

text: all

text: other

text: gains

text: are

text: are

text: compared

text: with

text: the

text: same

text: privacy

text: level.

text: All

text: the

text: drawn

text: i.i.d

text: from

text: Bern

text: τ

text: other

text: parameters

text: of

text: are

text: taken

text: as

text: deﬁned

text: in

text: corollary

text: while

text: the

text: parameters

text: of

text: are

text: taken

text: as

text: Fully

text: oblivious

text: adversary.

text: For

text: the

text: best

text: arm

text: k,

text: it

text: ﬁrst

text: deﬁned

text: in

text: corollary

text: draws

text: a

text: number

text: p

text: uniformly

text: in

text: ε]

text: and

text: gen-

text: For

text: all

text: experiments,

text: the

text: horizon

text: is

text: T

text: and

text: the

text: number

text: erates

text: the

text: gain

text: g

text: Bern

text: (p).

text: For

text: all

text: other

text: arms,

text: p

text: is

text: drawn

text: of

text: arms

text: is

text: K

text: We

text: performed

text: independent

text: trials

text: and

text: from

text: ε].

text: This

text: process

text: is

text: repeated

text: at

text: every

text: round.

text: reported

text: the

text: median-of-means

text: of

text: the

text: cumulative

text: In

text: our

text: experiments,

text: ε

text: regret.

text: It

text: partitions

text: the

text: trials

text: into

text: a

text: equal

text: groups

text: and

text: return

text: the

text: median

text: of

text: the

text: sample

text: means

text: of

text: each

text: group.

text: Proposition

text: An

text: oblivious

text: adversary.

text: This

text: adversary

text: is

text: identical

text: to

text: the

text: is

text: a

text: well

text: known

text: result

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text: (Hsu

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text: Sabato

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text: g√iving

text: the

text: accuracy

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text: estima-

text: fully

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text: for

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text: round

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text: of

text: Between

text: tor.

text: Its

text: convergence

text: is

text: O(σ/

text: N),

text: with

text: exponential

text: probabil-

text: two

text: multiples

text: of

text: the

text: last

text: gain

text: of

text: the

text: arm

text: is

text: given.

text: ity

text: tails,

text: even

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text: the

text: random

text: variable

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text: have

text: heavy-

text: tails.

text: In

text: comparison,

text: the

text: empirical

text: mean

text: can

text: not

text: provide

text: The

text: Switching

text: costs

text: adversary

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text: adversary

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text: such

text: guarantee

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text: σ

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text: conﬁdence

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text: Figure

text: in

text: (Dekel

text: et

text: al.

text: deﬁnes

text: a

text: stochastic

text: processes

text: (Catoni

text: (including

text: simple

text: Gaussian

text: random

text: walk

text: as

text: special

text: case)

text: for

text: Proposition

text: Let

text: x

text: be

text: a

text: random

text: variable

text: with

text: mean

text: µ

text: generating

text: the

text: gains.

text: It

text: was

text: used

text: to

text: prove

text: that

text: any

text: algorithm

text: O

text: and

text: variance

text: σ2

text: Assume

text: that

text: we

text: have

text: N

text: independent

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text: this

text: adversary

text: must

text: incur

text: a

text: regret

text: of

text: sample

text: of

text: x

text: and

text: let

text: µˆ

text: be

text: the

text: median-of-means

text: computed

text: using

text: probability

text: at

text: least

text: e

text: µˆ

text: Discussion

text: Figure

text: shows

text: our

text: results

text: against

text: a

text: variety

text: satisﬁes

text: µˆ

text: µ

text: σ

text: /N.

text: of

text: adversaries,

text: with

text: respect

text: to

text: a

text: ﬁxed

text: oracle.

text: Overall,

text: the

text: We

text: set

text: the

text: number

text: of

text: groups

text: to

text: a

text: so

text: that

text: the

text: performance

text: (in

text: term

text: of

text: regret)

text: of

text: is

text: very

text: conﬁdence

text: interval

text: holds

text: w.p.

text: at

text: least

text: competitive

text: against

text: that

text: of

text: while

text: providing

text: a

text: signif-

text: We

text: also

text: reported

text: the

text: deviation

text: of

text: each

text: algorithm

text: using

text: the

text: icant

text: better

text: privacy.

text: This

text: means

text: that

text: allows

text: Gini’s

text: Mean

text: Difference

text: (GMD

text: hereafter)

text: (Gini

text: and

text: Pearson

text: us

text: to

text: get

text: privacy

text: for

text: free

text: in

text: the

text: bandit

text: setting

text: against

text: an

text: GMD

text: computes

text: the

text: deviation

text: as

text: N

text: N

text: adversary

text: not

text: more

text: powerful

text: than

text: the

text: oblivious

text: one.

text: The

text: performance

text: of

text: is

text: worse

text: than

text: that

text: of

text: DP-

text: with

text: x

text: the

text: j-th

text: order

text: statistics

text: of

text: the

text: sample

text: (that

text: τ

text: (j)

text: (j)≤

text: against

text: an

text: oblivious

text: adversary

text: or

text: one

text: less

text: pow-

text: is

text: x

text: x

text: x

text: As

text: shown

text: in

text: (Yitzhaki

text: and

text: (N)

text: erful.

text: However,

text: the

text: situation

text: is

text: completely

text: reversed

text: against

text: others

text: David

text: the

text: GMD

text: provides

text: a

text: superior

text: the

text: more

text: powerful

text: switching

text: cost

text: adversary.

text: In

text: that

text: setting,

text: approximation

text: of

text: the

text: true

text: deviation

text: than

text: the

text: standard

text: one.

text: outperforms

text: both

text: and

text: con-

text: To

text: account

text: for

text: the

text: fact

text: that

text: the

text: cumulative

text: regret

text: of

text: our

text: τ

text: ﬁrming

text: the

text: theoretical

text: analysis.

text: We

text: can

text: see

text: as

text: the

text: algorithms

text: might

text: not

text: follow

text: a

text: symmetric

text: distribution,

text: we

text: τ

text: algorithm

text: providing

text: us

text: privacy

text: for

text: free

text: against

text: switching

text: cost

text: computed

text: the

text: GMD

text: separately

text: for

text: the

text: values

text: above

text: and

text: below

text: adversary

text: and

text: adaptive

text: m-bounded

text: memory

text: one

text: in

text: general.

text: the

text: median-of-means.

text: At

text: round

text: t,

text: we

text: computed

text: the

text: cumulative

text: regret

text: against

text: the

text: Conclusion

text: ﬁxed

text: oracle

text: who

text: plays

text: the

text: best

text: arm

text: assuming

text: that

text: the

text: end

text: of

text: the

text: game

text: is

text: at

text: t.

text: The

text: oracle

text: uses

text: the

text: actual

text: sequence

text: of

text: We

text: have

text: provided

text: the

text: ﬁrst

text: results

text: on

text: differentially

text: pri-

text: gains

text: to

text: decide

text: his

text: best

text: arm.

text: For

text: a

text: given

text: trial,

text: we

text: make

text: sure

text: vate

text: adversarial

text: multi-armed

text: bandits,

text: which

text: are

text: optimal

text: up

text: that

text: all

text: algorithms

text: are

text: playing

text: the

text: same

text: game

text: by

text: generating

text: to

text: logarithmic

text: factors.

text: One

text: open

text: question

text: is

text: how

text: differen-

text: the

text: gains

text: for

text: all

text: possible

text: pair

text: of

text: round-arm

text: before

text: the

text: game

text: tial

text: privacy

text: affects

text: regret

text: in

text: the

text: full

text: reinforcement

text: learning

text: starts.

text: problem.

text: At

text: this

text: point

text: in

text: time,

text: the

text: only

text: known

text: results

text: in

text: the

text: MDP

text: setting

text: obtain

text: differentially

text: private

text: algorithms

text: for

text: Deterministic

text: adversary.

text: As

text: shown

text: by

text: (Audibert

text: and

text: Monte

text: Carlo

text: policy

text: evaluation

text: (Balle,

text: Gomrokchi,

text: and

text: Pre-

text: cup

text: While

text: this

text: implies

text: that

text: it

text: is

text: possible

text: to

text: obtain

text: Bubeck

text: the

text: expected

text: regret

text: of

text: any

text: agent

text: against

text: an

text: policy

text: iteration

text: algorithms,

text: it

text: is

text: unclear

text: how

text: to

text: extend

text: this

text: to

text: oblivious

text: adversary

text: can

text: not

text: be

text: worse

text: than

text: that

text: against

text: the

text: the

text: full

text: online

text: reinforcement

text: learning

text: problem.

text: heavily

text: in

text: the

text: streaming

text: literature

text: (Alon,

text: Matias,

text: and

text: Acknowledgements.

text: This

text: research

text: was

text: supported

text: by

text: the

text: Szegedy

text: SNSF

text: grants

text: “Adaptive

text: control

text: with

text: approximate

text: Bayesian

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: of tasks, especially the CNN for image feature extraction. For aural information, it is reﬁned by some deep models including Deep Belief Network (DBN), Deep Autoencoder (DAE), Restricted Boltzmann Machines (RBMs) and Deep Bottleneck Features (DBNF)[14][18][27]. (2) The other merit of deep models is the well-performed fusion, conquering the biggest disad- vantage of traditional methods. Multimodal DBN (MDBN)[23], Multimodal DAEs (MDAEs)[17], Multimodal Deep Boltzmann Machine (MDBM)[24] are all unsupervised DNN that perform fusion. However, the aforementioned deep models have two primary shortcomings: (1) They don’t balance the modal fusion and temporal fusion, or even haven’t temporal fusion. Many methods simply concatenate the features of frames in a single video to do temporal fusion. Besides, in the modal fusion, such methods don’t consider the correlation among different frames. (2) The architecture of these models isn’t end-to-end, the training and testing getting cumbersome. Inspiring by Multimodal Recurrent Neural Networks [11] for image captioning, we propose an end- to-end DNN architecture, Auxiliary Multimodal LSTM (am-LSTM), for AVSR and lipreading. The am-LSTM overcomes the two main weaknesses mentioned before. It is composed of LSTMs, pro- jection and recognition, and is trained once. Therefore the modal fusion and temporal fusion are accomplished at the same time, i.e. the modal and temporal fusion are mixed and combined in terms of balance of the two fusion processes. To avoid overﬁtting and make the DNN converge faster, we use the connection strategy like Deep Residual Learning [8]–auxiliary connection. Early stopping [20] and dropout [25] regularization are used to ﬁght against overﬁtting as well. Because of the CNN and LSTM, am-LSTM is also a type of well-known CNN-RNN architecture. We conducted the experiments in three AVSR datasets: AVLetters (Patterson et al., 2002), AVLet- ters2 (Cox et al., 2008) and AVDigits (Di Hu et al., 2015). The results suggest that am-LSTM is better than classic AVSR models and some deep models before. 2 Related work Traditional AVSR Systems. Humans understand the multimodal world in a seemingly effortless manner, although there are vast information processing resources dedicated to the corresponding tasks by the brain [12]. Whereas it is difﬁcult for computers to understand multimodal information, because different modalities have their character in nature, same information can be expressed in different modalities in very dissimilar way, let alone distinct information. Audio is one dimensional temporal signal, video is three dimensional temporal signal and text carries semantic information. Multimodal fusion is the most important part of multimodal learning. There exists three kinds of fusion strategies: early fusion, late fusion and hybrid fusion. For early fusion, it sufﬁces to con- catenate all monomodal features into a single aggregate multimodal descriptor. In late fusion, each modality is classiﬁed independently. Intergration is done at the decision level and is usually based on heuristic rules. Hybrid fusion lies in-between early and late fusion methods and are speciﬁcally geared towards modeling multimodal time-evolving data [12] [4]. The representative work in the early years in this ﬁeld is multistream HMMs (mHMMs) [16] which were conﬁrmed adaptability and ﬂexibility for modeling sequence and temporal data [21]. But probabilistic models have some explicit limitations, especially strong priori. Researchers are also interested in the manner of mapping data in dissimilar space to one space jointly. Deep Learning for AVSR. Deep learning provides a powerful toolkit for machine learning. In AVSR, DNN also displays obvious increasement of accuracy. [18] uses pre-trained CNN to extracted visual features and denoising autoencoders to improve aural features, and then mHMMs to do fusion and classﬁcation. As I mentioned in , it mainly utilizes the feature extraction of DNN. Some other methods are multimodal fusion based on unsupervised DNN. One type is based on DAEs, the representative work is MDAEs [17]. Others based on Boltzmann Machines, [23] uses MDBN, [24] uses MDBM. Generally, DAEs-based models are easy to trained, but lack of theoretical support and ﬂexibility. RBM-based models are hard to train because of the partial function [5], but sufﬁcient support from probabilistic models and simple to extend. 3 The Proposed Model 2

text: of

text: tasks,

text: especially

text: the

text: CNN

text: for

text: image

text: feature

text: extraction.

text: For

text: aural

text: information,

text: it

text: is

text: reﬁned

text: by

text: some

text: deep

text: models

text: including

text: Deep

text: Belief

text: Network

text: (DBN),

text: Deep

text: Autoencoder

text: (DAE),

text: Restricted

text: Boltzmann

text: Machines

text: (RBMs)

text: and

text: Deep

text: Bottleneck

text: Features

text: The

text: other

text: merit

text: of

text: deep

text: models

text: is

text: the

text: well-performed

text: fusion,

text: conquering

text: the

text: biggest

text: disad-

text: vantage

text: of

text: traditional

text: methods.

text: Multimodal

text: DBN

text: Multimodal

text: DAEs

text: Multimodal

text: Deep

text: Boltzmann

text: Machine

text: are

text: all

text: unsupervised

text: DNN

text: that

text: perform

text: fusion.

text: However,

text: the

text: aforementioned

text: deep

text: models

text: have

text: two

text: primary

text: shortcomings:

text: They

text: don’t

text: balance

text: the

text: modal

text: fusion

text: and

text: temporal

text: fusion,

text: or

text: even

text: haven’t

text: temporal

text: fusion.

text: Many

text: methods

text: simply

text: concatenate

text: the

text: features

text: of

text: frames

text: in

text: a

text: single

text: video

text: to

text: do

text: temporal

text: fusion.

text: Besides,

text: in

text: the

text: modal

text: fusion,

text: such

text: methods

text: don’t

text: consider

text: the

text: correlation

text: among

text: different

text: frames.

text: The

text: architecture

text: of

text: these

text: models

text: isn’t

text: end-to-end,

text: the

text: training

text: and

text: testing

text: getting

text: cumbersome.

text: Inspiring

text: by

text: Multimodal

text: Recurrent

text: Neural

text: Networks

text: for

text: image

text: captioning,

text: we

text: propose

text: an

text: end-

text: to-end

text: DNN

text: architecture,

text: Auxiliary

text: Multimodal

text: LSTM

text: (am-LSTM),

text: for

text: AVSR

text: and

text: lipreading.

text: The

text: am-LSTM

text: overcomes

text: the

text: two

text: main

text: weaknesses

text: mentioned

text: before.

text: It

text: is

text: composed

text: of

text: LSTMs,

text: pro-

text: jection

text: and

text: recognition,

text: and

text: is

text: trained

text: once.

text: Therefore

text: the

text: modal

text: fusion

text: and

text: temporal

text: fusion

text: are

text: accomplished

text: at

text: the

text: same

text: time,

text: i.e.

text: the

text: modal

text: and

text: temporal

text: fusion

text: are

text: mixed

text: and

text: combined

text: in

text: terms

text: of

text: balance

text: of

text: the

text: two

text: fusion

text: processes.

text: To

text: avoid

text: overﬁtting

text: and

text: make

text: the

text: DNN

text: converge

text: faster,

text: we

text: use

text: the

text: connection

text: strategy

text: like

text: Deep

text: Residual

text: Learning

text: connection.

text: Early

text: stopping

text: and

text: dropout

text: regularization

text: are

text: used

text: to

text: ﬁght

text: against

text: overﬁtting

text: as

text: well.

text: Because

text: of

text: the

text: CNN

text: and

text: LSTM,

text: am-LSTM

text: is

text: also

text: a

text: type

text: of

text: well-known

text: CNN-RNN

text: architecture.

text: We

text: conducted

text: the

text: experiments

text: in

text: three

text: AVSR

text: datasets:

text: AVLetters

text: (Patterson

text: et

text: al.,

text: AVLet-

text: (Cox

text: et

text: al.,

text: and

text: AVDigits

text: (Di

text: Hu

text: et

text: al.,

text: The

text: results

text: suggest

text: that

text: am-LSTM

text: is

text: better

text: than

text: classic

text: AVSR

text: models

text: and

text: some

text: deep

text: models

text: before.

text: Related

text: work

text: Traditional

text: AVSR

text: Systems.

text: Humans

text: understand

text: the

text: multimodal

text: world

text: in

text: a

text: seemingly

text: effortless

text: manner,

text: although

text: there

text: are

text: vast

text: information

text: processing

text: resources

text: dedicated

text: to

text: the

text: corresponding

text: tasks

text: by

text: the

text: brain

text: Whereas

text: it

text: is

text: difﬁcult

text: for

text: computers

text: to

text: understand

text: multimodal

text: information,

text: because

text: different

text: modalities

text: have

text: their

text: character

text: in

text: nature,

text: same

text: information

text: can

text: be

text: expressed

text: in

text: different

text: modalities

text: in

text: very

text: dissimilar

text: way,

text: let

text: alone

text: distinct

text: information.

text: Audio

text: is

text: one

text: dimensional

text: temporal

text: signal,

text: video

text: is

text: three

text: dimensional

text: temporal

text: signal

text: and

text: text

text: carries

text: semantic

text: information.

text: Multimodal

text: fusion

text: is

text: the

text: most

text: important

text: part

text: of

text: multimodal

text: learning.

text: There

text: exists

text: three

text: kinds

text: of

text: fusion

text: strategies:

text: early

text: fusion,

text: late

text: fusion

text: and

text: hybrid

text: fusion.

text: For

text: early

text: fusion,

text: it

text: sufﬁces

text: to

text: con-

text: catenate

text: all

text: monomodal

text: features

text: into

text: a

text: single

text: aggregate

text: multimodal

text: descriptor.

text: In

text: late

text: fusion,

text: each

text: modality

text: is

text: classiﬁed

text: independently.

text: Intergration

text: is

text: done

text: at

text: the

text: decision

text: level

text: and

text: is

text: usually

text: based

text: on

text: heuristic

text: rules.

text: Hybrid

text: fusion

text: lies

text: in-between

text: early

text: and

text: late

text: fusion

text: methods

text: and

text: are

text: speciﬁcally

text: geared

text: towards

text: modeling

text: multimodal

text: time-evolving

text: data

text: The

text: representative

text: work

text: in

text: the

text: early

text: years

text: in

text: this

text: ﬁeld

text: is

text: multistream

text: HMMs

text: (mHMMs)

text: which

text: were

text: conﬁrmed

text: adaptability

text: and

text: ﬂexibility

text: for

text: modeling

text: sequence

text: and

text: temporal

text: data

text: But

text: probabilistic

text: models

text: have

text: some

text: explicit

text: limitations,

text: especially

text: strong

text: priori.

text: Researchers

text: are

text: also

text: interested

text: in

text: the

text: manner

text: of

text: mapping

text: data

text: in

text: dissimilar

text: space

text: to

text: one

text: space

text: jointly.

text: Deep

text: Learning

text: for

text: AVSR.

text: Deep

text: learning

text: provides

text: a

text: powerful

text: toolkit

text: for

text: machine

text: learning.

text: In

text: AVSR,

text: DNN

text: also

text: displays

text: obvious

text: increasement

text: of

text: accuracy.

text: uses

text: pre-trained

text: CNN

text: to

text: extracted

text: visual

text: features

text: and

text: denoising

text: autoencoders

text: to

text: improve

text: aural

text: features,

text: and

text: then

text: mHMMs

text: to

text: do

text: fusion

text: and

text: classﬁcation.

text: As

text: I

text: mentioned

text: in

text: it

text: mainly

text: utilizes

text: the

text: feature

text: extraction

text: of

text: DNN.

text: Some

text: other

text: methods

text: are

text: multimodal

text: fusion

text: based

text: on

text: unsupervised

text: DNN.

text: One

text: type

text: is

text: based

text: on

text: DAEs,

text: the

text: representative

text: work

text: is

text: MDAEs

text: Others

text: based

text: on

text: Boltzmann

text: Machines,

text: uses

text: MDBN,

text: uses

text: MDBM.

text: Generally,

text: DAEs-based

text: models

text: are

text: easy

text: to

text: trained,

text: but

text: lack

text: of

text: theoretical

text: support

text: and

text: ﬂexibility.

text: RBM-based

text: models

text: are

text: hard

text: to

text: train

text: because

text: of

text: the

text: partial

text: function

text: but

text: sufﬁcient

text: support

text: from

text: probabilistic

text: models

text: and

text: simple

text: to

text: extend.

text: The

text: Proposed

text: Model


text: 2 1.8 1.6 1.4 1.2 1 0.8 0.6 -15 -10 -5 0 5 10 1/ν (λ-λ )L c − × ∗ Figure 2: The scaling plot, (λ λc)L1/ν-m1/m1c, is presented for (+) N = 16, ( ) 18, ( ) (cid:3) 20, and ( ) 22; here, we adopted the scaling theory [26, 31] for the ﬁrst-order phase transition. The scaling parameters, λc = 0.8758 and ν = 0.5, are taken from the existing literatures, Refs. [25] and [26, 31], respectively; namely, there is no adjustable parameter involved in the scaling analysis. 11 m/ m c1 1 ##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##

font: default

text: (λ-λ

text: )L

text: c

label: caption

text: Figure

text: The

text: scaling

text: plot,

text: (λ

text: is

text: presented

text: for

text: N

text: and

text: here,

text: we

text: adopted

text: the

text: scaling

text: theory

text: for

text: the

text: ﬁrst-order

text: phase

text: transition.

text: The

text: scaling

text: parameters,

text: λc

text: and

text: ν

text: are

text: taken

text: from

text: the

text: existing

text: literatures,

text: Refs.

text: and

text: respectively;

text: namely,

text: there

text: is

text: no

text: adjustable

text: parameter

text: involved

text: in

text: the

text: scaling

text: analysis.

text: m/

text: m

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: [26] C. J. Hamer, M. Aydin, J. Oitmaa and H.-X. He, J. Phys. A: Math. Gen. 23 (1990) 4025. [27] Y.-W. Dai, S. Y. Cho, M. T. Batchelor and H.-Q. Zhou, Phys. Rev. E 89 (2014) 062142. [28] E. R. Gagliano and C. A. Balseiro, Phys. Rev. Lett. 59 (1987) 2999. [29] M. E. Fisher and A. N. Berker, Phys. Rev. B 26 (1982) 2507. [30] M. S. S. Challa, D. P. Landau and K. Binder, Phys. Rev. B 34 (1986) 1841. [31] M. Campostrini, J. Nespolo, A. Pelissetto and E. Vicari, Phys. Rev. E 91 (2015) 052103. [32] M. Campostrini, J. Nespolo, A. Pelissetto and E. Vicari, Phys. Rev. Lett. 113 (2014) 070402. [33] M.A. Novotny, J. Appl. Phys. 67 (1990) 5448. [34] Y. Nishiyama, Phys. Rev. E 78 (2008) 021135. [35] D. Podolsky and S. Sachdev, Phys. Rev. B 86 (2012) 054508. [36] V. Agostini, G. Carlino, M. Caselle and M. Hasenbusch, Nucl. Phys. B 484 (1997) 331. [37] R. Fiore, A. Papa and P. Provero, Nucl. Phys. B Proc. Suppl. 119 (2003) 490. 16

label: reference

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text: and

text: H.-X.

text: He,

text: J.

text: Phys.

text: A:

text: Math.

text: Gen.

text: Y.-W.

text: Dai,

text: S.

text: Y.

text: Cho,

text: M.

text: T.

text: Batchelor

text: and

text: H.-Q.

text: Zhou,

text: Phys.

text: Rev.

text: E

text: E.

text: R.

text: Gagliano

text: and

text: C.

text: A.

text: Balseiro,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: E.

text: Fisher

text: and

text: A.

text: N.

text: Berker,

text: Phys.

text: Rev.

text: B

text: M.

text: S.

text: S.

text: Challa,

text: D.

text: P.

text: Landau

text: and

text: K.

text: Binder,

text: Phys.

text: Rev.

text: B

text: M.

text: Campostrini,

text: J.

text: Nespolo,

text: A.

text: Pelissetto

text: and

text: E.

text: Vicari,

text: Phys.

text: Rev.

text: E

text: M.

text: Campostrini,

text: J.

text: Nespolo,

text: A.

text: Pelissetto

text: and

text: E.

text: Vicari,

text: Phys.

text: Rev.

text: Lett.

text: M.A.

text: Novotny,

text: J.

text: Appl.

text: Phys.

text: Y.

text: Nishiyama,

text: Phys.

text: Rev.

text: E

text: D.

text: Podolsky

text: and

text: S.

text: Sachdev,

text: Phys.

text: Rev.

text: B

text: V.

text: Agostini,

text: G.

text: Carlino,

text: M.

text: Caselle

text: and

text: M.

text: Hasenbusch,

text: Nucl.

text: Phys.

text: B

text: R.

text: Fiore,

text: A.

text: Papa

text: and

text: P.

text: Provero,

text: Nucl.

text: Phys.

text: B

text: Proc.

text: Suppl.


text: 27] by means of the numerical diagonalization method. The method allows us to evaluate the dynamical susceptibilities, Eqs. (12) and (17), via the continued- fraction-expansion method [28]. In Fig. 1, we present a schematic drawing for a spectral function. To be speciﬁc, we present the Hamiltonian for the (2 + 1)-dimensional q = 3 Potts model (cid:18) (cid:19) (cid:88) (cid:88) (cid:88) H − 2 2π − − N − − N 2 2πL = cos (L L ) λ (R+ + R ) H cos i . (8) (cid:104) (cid:105) 3 3 i j i i 3 3 ij i=1 i=1   Here, the operator        0 0 0  Li = 0 1 0 , (9) 0 0 2 (cid:80) | (cid:105) is placed at each square-lattice point i = 1, 2, . . . , N; namely, the base l i | (cid:105) | (cid:105) (l = 0, 1, 2) satisﬁes L l = l l . The summation (cid:104) (cid:105) runs over all possible i i i i i ij (cid:104) (cid:105) ± ±| (cid:105) nearest neighbor pairs ij . The operator R induces the transition R l = i i i | ± (cid:105) l 1 mod 3 , and the parameter λ denotes the corresponding coupling constant. i This model exhibits the ﬁrst-order phase transition at λ = λ = 0.8758(14) c (H = 0) [25], which separates the ordered (λ < λ ) and disordered (λ > λ ) c c phases. An inﬁnitesimal magnetic ﬁeld H = 20/L2.5 (L: linear dimension of the ﬁnite-size cluster) stabilizes [3] the ground state, and the power 2.5 comes from the putative scaling theory [26] for the (2 + 1)-dimensional q = 3 Potts model. Note that the ﬁrst-order phase transition also obeys the remedied scaling theory [29, 30, 31, 32]. The rest of this paper is organized as follows. In the next section, we present the numerical results. The simulation algorithm is presented as well. In Sec. 3, we address the summary and discussions. 2. Numerical results 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: by

text: means

text: of

text: the

text: numerical

text: diagonalization

text: method.

text: The

text: method

text: allows

text: us

text: to

text: evaluate

text: the

text: dynamical

text: susceptibilities,

text: Eqs.

text: and

text: via

text: the

text: continued-

text: fraction-expansion

text: method

text: In

text: Fig.

text: we

text: present

text: a

text: schematic

text: drawing

text: for

text: a

text: spectral

text: function.

text: To

text: be

text: speciﬁc,

text: we

text: present

text: the

text: Hamiltonian

text: for

text: the

text: q

text: Potts

text: model

label: equation

text: H

text: 2π

text: N

text: N

text: cos

text: (L

text: L

text: λ

text: (R+

text: R

text: H

text: cos

text: i

text: i

text: j

text: i

text: i

text: ij

text: Here,

text: the

text: operator

text: Li

text: is

text: placed

text: at

text: each

text: square-lattice

text: point

text: i

text: N;

text: namely,

text: the

text: base

text: l

text: i

text: (l

text: satisﬁes

text: L

text: l

text: l

text: l

text: The

text: summation

text: runs

text: over

text: all

text: possible

text: i

text: i

text: i

text: i

text: i

text: ij

text: nearest

text: neighbor

text: pairs

text: ij

text: The

text: operator

text: R

text: induces

text: the

text: transition

text: R

text: l

text: i

text: i

text: i

text: l

text: mod

text: and

text: the

text: parameter

text: λ

text: denotes

text: the

text: corresponding

text: coupling

text: constant.

text: i

text: This

text: model

text: exhibits

text: the

text: ﬁrst-order

text: phase

text: transition

text: at

text: λ

text: λ

text: c

text: (H

text: which

text: separates

text: the

text: ordered

text: (λ

text: λ

text: and

text: disordered

text: (λ

text: λ

text: c

text: c

text: phases.

text: An

text: inﬁnitesimal

text: magnetic

text: ﬁeld

text: H

text: (L:

text: linear

text: dimension

text: of

text: the

text: ﬁnite-size

text: cluster)

text: stabilizes

text: the

text: ground

text: state,

text: and

text: the

text: power

text: comes

text: from

text: the

text: putative

text: scaling

text: theory

text: for

text: the

text: q

text: Potts

text: model.

text: Note

text: that

text: the

text: ﬁrst-order

text: phase

text: transition

text: also

text: obeys

text: the

text: remedied

text: scaling

text: theory

text: The

text: rest

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: the

text: next

text: section,

text: we

text: present

text: the

text: numerical

text: results.

text: The

text: simulation

text: algorithm

text: is

text: presented

text: as

text: well.

text: In

text: Sec.

text: we

text: address

text: the

text: summary

text: and

text: discussions.

text: Numerical

text: results

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In this section, we present the numerical results for the Potts model (8). To begin with, we explain the simulation algorithm. 2.1. Numerical algorithm We employed the exact diagonalization method to simulate the Potts model ≤ (8) for a rectangular cluster with N 22 spins. In order to treat a variety of N = 16, 18, . . . systematically, we implemented the screw-boundary condition { } [√33]. According to Ref. [33], an alignment of spins li with the ﬁrst- and N-th-neighbor interactions reduces to a rectangular cluster under the screw- boundary condition. Based on this idea, we express the Hamiltonian as (cid:16) (cid:17) (cid:88) (cid:88) √ H − N − − N 3 2πL = H (1) + H N λ (R+ + R ) H cos i . (10) D D i i 2 3 i=1 i=1 Here, the diagonal matrix H (v) denotes the v-th-neighbor interaction for D { } (cid:104){ }| |{ }(cid:105) an alignment l ; that is, the diagonal element is given by l H (v) l = i (cid:80) i D i (cid:104){ }| |{ }(cid:105) |{ }(cid:105) |{ }(cid:105) l TPv l with the translation operator, P l = l , and the Potts i i i i+1 (cid:104){ }| |{ (cid:48)}(cid:105) − interaction, l T l = N δ (cid:48). The above formulae are mathemati- i i i=1 li,li cally closed; however, for an eﬃcient simulation, Eqs. (9) and (10) of Ref. [34] may be of use. We performed the numerical diagonalization for the Hamiltonian matrix (10) by means of the Lanczos method. The single-magnon mass gap m is given by 1 − m = E E , (11) 1 1 0 with the ground-state (E ) and ﬁrst-excited (E ) energy levels within the 0 1 zero-momentum sector. Because√the N spins form a rectangular cluster, the linear dimension is given by L = N, which sets a foundermental length scale in the subsequent scaling analyses. 2.2. Single-magnon mass gap m 1 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: this

text: section,

text: we

text: present

text: the

text: numerical

text: results

text: for

text: the

text: Potts

text: model

text: To

text: begin

text: with,

text: we

text: explain

text: the

text: simulation

text: algorithm.

text: Numerical

text: algorithm

text: We

text: employed

text: the

text: exact

text: diagonalization

text: method

text: to

text: simulate

text: the

text: Potts

text: model

text: for

text: a

text: rectangular

text: cluster

text: with

text: N

text: spins.

text: In

text: order

text: to

text: treat

text: a

text: variety

text: of

text: N

text: systematically,

text: we

text: implemented

text: the

text: screw-boundary

text: condition

text: According

text: to

text: Ref.

text: an

text: alignment

text: of

text: spins

text: li

text: with

text: the

text: ﬁrst-

text: and

text: N-th-neighbor

text: interactions

text: reduces

text: to

text: a

text: rectangular

text: cluster

text: under

text: the

text: screw-

text: boundary

text: condition.

text: Based

text: on

text: this

text: idea,

text: we

text: express

text: the

text: Hamiltonian

text: as

text: H

text: N

text: N

text: H

text: H

text: N

text: λ

text: (R+

text: R

text: H

text: cos

text: i

text: D

text: D

text: i

text: i

text: Here,

text: the

text: diagonal

text: matrix

text: H

text: (v)

text: denotes

text: the

text: v-th-neighbor

text: interaction

text: for

text: D

text: an

text: alignment

text: l

text: that

text: is,

text: the

text: diagonal

text: element

text: is

text: given

text: by

text: l

text: H

text: (v)

text: l

text: i

text: i

text: D

text: i

text: l

text: TPv

text: l

text: with

text: the

text: translation

text: operator,

text: P

text: l

text: l

text: and

text: the

text: Potts

text: i

text: i

text: i

text: interaction,

text: l

text: T

text: l

text: N

text: δ

text: The

text: above

text: formulae

text: are

text: mathemati-

text: i

text: i

text: li,li

text: cally

text: closed;

text: however,

text: for

text: an

text: eﬃcient

text: simulation,

text: Eqs.

text: and

text: of

text: Ref.

text: may

text: be

text: of

text: use.

text: We

text: performed

text: the

text: numerical

text: diagonalization

text: for

text: the

text: Hamiltonian

text: matrix

text: by

text: means

text: of

text: the

text: Lanczos

text: method.

text: The

text: single-magnon

text: mass

text: gap

text: m

text: is

text: given

text: by

text: m

text: E

text: E

text: with

text: the

text: ground-state

text: (E

text: and

text: ﬁrst-excited

text: (E

text: energy

text: levels

text: within

text: the

text: zero-momentum

text: sector.

text: Because√the

text: N

text: spins

text: form

text: a

text: rectangular

text: cluster,

text: the

text: linear

text: dimension

text: is

text: given

text: by

text: L

text: N,

text: which

text: sets

text: a

text: foundermental

text: length

text: scale

text: in

text: the

text: subsequent

text: scaling

text: analyses.

text: Single-magnon

text: mass

text: gap

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In this section, we investigate the single-magnon mass gap m (11) with 1 the scaling theory [26, 31]. The ﬁrst-order phase transition obeys the properly remedied scaling theory. − In Fig. 2, we present the scaling plot, (λ λ )L1/ν-m /m , for (+) N = 16, c 1 1c × ∗ (cid:3) | ( ) 18, ( ) 20, and ( ) 22. Here, the symbol m denotes m = m [31], 1c 1c 1 λ=λc and the scaling parameters, λ = 0.8758 and ν = 0.5, are taken from Refs. [25] c and [26, 31], respectively. We stress that there is no ad hoc ﬁtting parameter involved in the scaling analysis. The data in Fig. 2 seem to collapse into a scaling curve, indicating that the simulation data already enter the scaling regime. Encouraged by this ﬁnding, we turn to the analysis of the spectral properties. (cid:48)(cid:48) 2.3. Hierarchical spectral peaks m via χ (ω) 2,3,... A Based on the ﬁnite-size scaling [26, 31] demonstrated in the preceding sec- tion, we analyze the dynamical susceptibility (cid:48)(cid:48) −(cid:61)(cid:104) | † − H − | (cid:105) χ (ω) = 0 A (ω + E + iη) 1A 0 , (12) A 0 | (cid:105) with the ground-state energy (vector) E ( 0 ) and the energy-resolution 0 parameter η. Here, the perturbation (cid:32)operato(cid:33)r is set to (cid:88) 2 P N A = J , (13) i i=1   with        0 i i  − Ji = i 0 0 , (14) − i 0 0 P − | (cid:105)(cid:104) | and the projection operator = 1 0 0 . We calculated the dynamical susceptibility (12) with the continued-fraction expansion [28]. The dynamical susceptibility (spectral function) obeys the scaling formula 5

text: In

text: this

text: section,

text: we

text: investigate

text: the

text: single-magnon

text: mass

text: gap

text: m

text: with

text: the

text: scaling

text: theory

text: The

text: ﬁrst-order

text: phase

text: transition

text: obeys

text: the

text: properly

text: remedied

text: scaling

text: theory.

text: In

text: Fig.

text: we

text: present

text: the

text: scaling

text: plot,

text: (λ

text: λ

text: /m

text: for

text: N

text: c

text: and

text: Here,

text: the

text: symbol

text: m

text: denotes

text: m

text: m

text: λ=λc

text: and

text: the

text: scaling

text: parameters,

text: λ

text: and

text: ν

text: are

text: taken

text: from

text: Refs.

text: c

text: and

text: respectively.

text: We

text: stress

text: that

text: there

text: is

text: no

text: ad

text: hoc

text: ﬁtting

text: parameter

text: involved

text: in

text: the

text: scaling

text: analysis.

text: The

text: data

text: in

text: Fig.

text: seem

text: to

text: collapse

text: into

text: a

text: scaling

text: curve,

text: indicating

text: that

text: the

text: simulation

text: data

text: already

text: enter

text: the

text: scaling

text: regime.

text: Encouraged

text: by

text: this

text: ﬁnding,

text: we

text: turn

text: to

text: the

text: analysis

text: of

text: the

text: spectral

text: properties.

text: Hierarchical

text: spectral

text: peaks

text: m

text: via

text: χ

text: (ω)

text: A

text: Based

text: on

text: the

text: ﬁnite-size

text: scaling

text: demonstrated

text: in

text: the

text: preceding

text: sec-

text: tion,

text: we

text: analyze

text: the

text: dynamical

text: susceptibility

text: H

text: χ

text: (ω)

text: A

text: (ω

text: E

text: iη)

text: A

text: with

text: the

text: ground-state

text: energy

text: (vector)

text: E

text: and

text: the

text: energy-resolution

text: parameter

text: η.

text: Here,

text: the

text: perturbation

text: is

text: set

text: to

text: P

text: N

text: A

text: J

text: i

text: with

text: i

text: i

text: Ji

text: i

text: i

text: P

text: and

text: the

text: projection

text: operator

text: We

text: calculated

text: the

text: dynamical

text: susceptibility

text: with

text: the

text: continued-fraction

text: expansion

text: The

text: dynamical

text: susceptibility

text: (spectral

text: function)

text: obeys

text: the

text: scaling

text: formula


text: ≈ the two-magnon-spectrum-threshold peak ω/m 2; instead, the bound-state 1 hierarchy m becomes hardly observable. 3,4 The result indicates that a naive external disturbance such as the speciﬁc- heat-type perturbation B does not create the bound states higher than m 1,2 very eﬃciently. It is signiﬁcant to set up the perturbation operator, which does not commute with any terms of the Hamiltonian. Note that the exact diagonalization method allows us to survey various types of the (oﬀ-diagonal) perturbation operators so as to observe m clearly. 3,4,... 3. Summary and discussions The hierarchy m /m for the (2 + 1)-dimensional q = 3 Potts model (8) 2,3,4 1 was investigated with the numerical diagonalization method. The method al- (cid:48)(cid:48) lows us to calculate the dynamical susceptibilities χ , Eqs. (12) and (17), A,B (cid:48)(cid:48) via the continued-fraction expansion [28]. Through the probe χ , we obtained A (m /m , m /m , m /m ) = [1.80(3), 2.5(1), 3.05(25)]. The particles m ac- 2 1 3 1 4 1 3,4 quire intrinsic peak widths, ∆m /m = 0.3 and ∆m /m = 0.35, respectively; 3 1 4 1 these spectra are embedded within the two-magnon spectrum. According to Refs. [18, 19], the hierarchy m /m of the Potts model and that of the 2,3,4 1 pure gauge theory are alike. For instance, as for the Z -symmetric gauge group 2 [36], there was reported a hierarchy, (m (cid:48)/m , m (cid:48)(cid:48)/m , m (cid:48)/m ) = (0+) 0+ (0+) 0+ (2+) 0+ [1.88(2), 2.59(4), 3.23(7)], quite reminiscent of ours, Eq. (16). (cid:48)(cid:48) As a reference, we calculated χ (17); here, the operator B coincides with B the second term of the Hamiltonian (8), and hence, it would be relevant to (cid:48)(cid:48) the experimental study. It turned out that the probe χ is insensitive to the B hierarchy m , indicating that the choice of the perturbation operator is 3,4,... vital to observe m . In this sense, the exact diagonalization method has an 3,4,... advantage in that we are able to treat various perturbation operators so as to observe the hierarchy m clearly. 3,4,... Acknowledgment This work was supported by a Grant-in-Aid for Scientiﬁc Research (C) from Japan Society for the Promotion of Science (Grant No. 25400402). 9

text: the

text: two-magnon-spectrum-threshold

text: peak

text: ω/m

text: instead,

text: the

text: bound-state

text: hierarchy

text: m

text: becomes

text: hardly

text: observable.

text: The

text: result

text: indicates

text: that

text: a

text: naive

text: external

text: disturbance

text: such

text: as

text: the

text: speciﬁc-

text: heat-type

text: perturbation

text: B

text: does

text: not

text: create

text: the

text: bound

text: states

text: higher

text: than

text: m

text: very

text: eﬃciently.

text: It

text: is

text: signiﬁcant

text: to

text: set

text: up

text: the

text: perturbation

text: operator,

text: which

text: does

text: not

text: commute

text: with

text: any

text: terms

text: of

text: the

text: Hamiltonian.

text: Note

text: that

text: the

text: exact

text: diagonalization

text: method

text: allows

text: us

text: to

text: survey

text: various

text: types

text: of

text: the

text: (oﬀ-diagonal)

text: perturbation

text: operators

text: so

text: as

text: to

text: observe

text: m

text: clearly.

text: Summary

text: and

text: discussions

text: The

text: hierarchy

text: m

text: /m

text: for

text: the

text: q

text: Potts

text: model

text: was

text: investigated

text: with

text: the

text: numerical

text: diagonalization

text: method.

text: The

text: method

text: al-

text: lows

text: us

text: to

text: calculate

text: the

text: dynamical

text: susceptibilities

text: χ

text: Eqs.

text: and

text: A,B

text: via

text: the

text: continued-fraction

text: expansion

text: Through

text: the

text: probe

text: χ

text: we

text: obtained

text: A

text: (m

text: /m

text: m

text: /m

text: m

text: /m

text: The

text: particles

text: m

text: ac-

text: quire

text: intrinsic

text: peak

text: widths,

text: ∆m

text: /m

text: and

text: ∆m

text: /m

text: respectively;

text: these

text: spectra

text: are

text: embedded

text: within

text: the

text: two-magnon

text: spectrum.

text: According

text: to

text: Refs.

text: the

text: hierarchy

text: m

text: /m

text: of

text: the

text: Potts

text: model

text: and

text: that

text: of

text: the

text: pure

text: gauge

text: theory

text: are

text: alike.

text: For

text: instance,

text: as

text: for

text: the

text: Z

text: -symmetric

text: gauge

text: group

text: there

text: was

text: reported

text: a

text: hierarchy,

text: (m

text: m

text: m

text: quite

text: reminiscent

text: of

text: ours,

text: Eq.

text: As

text: a

text: reference,

text: we

text: calculated

text: χ

text: here,

text: the

text: operator

text: B

text: coincides

text: with

text: B

text: the

text: second

text: term

text: of

text: the

text: Hamiltonian

text: and

text: hence,

text: it

text: would

text: be

text: relevant

text: to

text: the

text: experimental

text: study.

text: It

text: turned

text: out

text: that

text: the

text: probe

text: χ

text: is

text: insensitive

text: to

text: the

text: B

text: hierarchy

text: m

text: indicating

text: that

text: the

text: choice

text: of

text: the

text: perturbation

text: operator

text: is

text: vital

text: to

text: observe

text: m

text: In

text: this

text: sense,

text: the

text: exact

text: diagonalization

text: method

text: has

text: an

text: advantage

text: in

text: that

text: we

text: are

text: able

text: to

text: treat

text: various

text: perturbation

text: operators

text: so

text: as

text: to

text: observe

text: the

text: hierarchy

text: m

text: clearly.

text: Acknowledgment

text: This

text: work

text: was

text: supported

text: by

text: a

text: Grant-in-Aid

text: for

text: Scientiﬁc

text: Research

text: (C)

text: from

text: Japan

text: Society

text: for

text: the

text: Promotion

text: of

text: Science

text: (Grant

text: No.


text: χ’’(ω) (k=0) lowest two-magnon bound state elementary second-lowest magnon two-magnon third-lowest bound state two-magnon bound state two-magnon spectrum 0 m1 m2 2m1 m3 m4 ω Figure 1: A schematic drawing of a dynamical susceptibility for the Potts model (8) in the ordered phase within the zero-momentum (k = 0) sector is presented. There appear hierarchical peaks with the mass gaps m1,2,...: The elementary excitation m1 corresponds to the single magnon, which forms a series of bound states m2,3,.... The two-magnon spectrum extends above ω > 2m1. The excitations m1,2,... may have a relevance to the glueball spectrum for the (pure) gauge ﬁeld theory [18, 19]. 10 ##LTFigure## ##LTLine## ##LTLine##

text: χ’’(ω)

text: lowest

text: two-magnon

text: bound

text: state

text: elementary

text: second-lowest

text: magnon

text: two-magnon

text: third-lowest

text: bound

text: state

text: two-magnon

text: bound

text: state

text: two-magnon

text: spectrum

text: ω

text: Figure

text: A

text: schematic

text: drawing

text: of

text: a

text: dynamical

text: susceptibility

text: for

text: the

text: Potts

text: model

text: in

text: the

text: ordered

text: phase

text: within

text: the

text: zero-momentum

text: (k

text: sector

text: is

text: presented.

text: There

text: appear

text: hierarchical

text: peaks

text: with

text: the

text: mass

text: gaps

text: The

text: elementary

text: excitation

text: corresponds

text: to

text: the

text: single

text: magnon,

text: which

text: forms

text: a

text: series

text: of

text: bound

text: states

text: The

text: two-magnon

text: spectrum

text: extends

text: above

text: ω

text: The

text: excitations

text: may

text: have

text: a

text: relevance

text: to

text: the

text: glueball

text: spectrum

text: for

text: the

text: (pure)

text: gauge

text: ﬁeld

text: theory

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 16 sented here includes a diﬀusive term. As discussed by characterise each of these regimes, and we have demon- Sureshkumar and Beris [56], this can have a stabilising strated that the upturn in the drag coeﬃcient occurs at a eﬀect, suppressing possible numerical instabilities. Here ﬁxed value of the maximum of these two numbers across we have made an eﬀort to minimise the eﬀect of such the full range of medium porosities. We have also stud- a term by getting as close as is numerically possible to ied the creeping ﬂow of an Oldroyd B ﬂuid past a linear → → the dual limit ∆x, ∆y/(cid:96) 0, (cid:96)/L , L 0, which en- array of cylinders conﬁned to a channel bounded by solid x y sures that the diﬀusive lengthscale (cid:96) is well resolved yet walls, where we have found that the ﬂow remains steady small compared to any features of the ﬂow ﬁeld. This ap- for all Weissenberg numbers explored. proach suppresses spurious numerical instabilities whilst All the simulations in this work have assumed a purely keeping physical ones (if present). A second explanation two-dimensional ﬂow, with translational invariance into could be due to the presence of small but non-zero iner- the page in the sketches of Fig. 1. In experimental prac- tia in Ref. [33]. As discussed by Hoda et al. [57], weak tice, the upturn in the drag coeﬃcient is often accompa- inertial eﬀects can provide a mechanism by which elas- nied by the onset of three dimensional ﬂows (which are tic disturbances can be ampliﬁed. Finally it is conceiv- furthermore often time-dependent). To capture these ef- able that the time-dependent states observed in Ref. [33] fects, we would need to perform simulations in fully three are the result of a nonlinear instability. This mecha- dimensions. The two dimensional results reported here nism has been proposed to explain observed instabilities should clearly be treated with caution for Weissenberg in Poiseuille ﬂow of viscoelastic ﬂuids, which are believed numbers exceeding O(1), where three dimensional eﬀects to be linearly stable [35, 58]. A ﬁnal possibility concerns may pertain. spatial resolution. While the number of SPH particles James [9] recently considered the ratio of shear- to cannot necessarily be directly compared to the number extensionally-induced ﬁrst normal stresses in viscoelastic of ﬁnite-diﬀerence grid points, we note that the second ﬂow past a biperiodic array of cylinders. They demon- largest particle resolutions in Ref. [33] would roughly strated an O(1) lower bound on this, implying that shear correspond to the lowest resolution grid in Fig. 18 for generated ﬁrst normal stresses N cannot be neglected. 1 which we see ﬂuctuations. Wagner and McKinley [59] recently examined the re- VIII. CONCLUSIONS sponse of an Oldroyd-B ﬂuid to a ﬂow with a character- parameter α(t) varying sinusoidally in time, intended to In this work, we ﬁrst studied two dimensional creep- mimic the variation experienced by a given ﬂuid element ing ﬂow of the Oldroyd B, FENE-CR and FENE-P vis- as it moves through a periodic ﬂow cell of the kind stud- coelastic ﬂuids past a biperiodic square array of cylinders. ied in this work. For large Deborah numbers De (deﬁned Our aim has been to understand the dramatic upturn re- as the ratio of the polymer relaxation time to the time ported experimentally in the drag coeﬃcient as a function required to pass one unit cell), they demonstrated that of increasing Weissenberg number. attempts to predict the ﬁrst normal stress N1 just above By performing simulations across a wide range of val- and below the cylinder using the local Weissenberg num- ues of the porosity of the ﬂow geometry, from ‘dilute’ ber fails to reﬂect the previous ﬂow history and drasti- to near-touching cylinders, we have demonstrated two cally underestimates the value of N1. In view of these qualitatively diﬀerent mechanisms that may separately recent works, in future numerical studies of the kind per- underpin this thickening eﬀect. The ﬁrst operates in the formed here it would clearly be interesting to consider highly porous case of widely spaced cylinders, and in- more explicitly the role of normal stresses. volves a strong stretching of the polymer chains in the ex- ACKNOWLEDGEMENTS tensional wake aft of each cylinder. The second operates in the regime of more densely packed obstacles, and in- EJH and SMF gratefully acknowledge ﬁnancial sup- volves a strong deformation of the polymer chains as they port for this work from Schlumberger Gould Research. squeeze into the vertical gap between vertically adjacent We also thank the referees for their valuable comments cylinders. Two diﬀerent Weissenberg numbers separately on the paper. ∗ [9] D. F. James, J. Non-Newtonian Fluid Mech. 232, 33 (2016). † [10] A. Clarke et al., Soft Matter 11, 3536 (2015). [1] R. G. Larson and P. S. Desai, Annu. Rev. Fluid Mech. 47, 47[11(2]0A15. )C. larke, A. M. Howe, J. Mitchell, S. Staniland and L. A. Hawkes, SPE Journal, 2015, 174654-PA. [2] G. M. Homsy, Annu. Rev. Fluid Mech. 19, 271 (1987). [12] J. Mitchell, K. Lyons, A. M. Howe, and A. Clarke, Soft Matter 12, 460 (2016). [3] D. Bensimon et al., Rev. Mod. Phys. 58, 977 (1986). [13] H. Darcy, Les fontaines publiques de la ville de Dijon (Dalmont, Paris, 1856). [4] C. Chmielewski, Journal of Rheology 36, 1105 (1992). [14] P. C. Carman, Chemical Engineering Research and Design 15, 32 (1937). [5] L. Skartsis, Journal of Rheology 36, 589 (1992). [15] K. K. Talwar, Journal of Rheology 36, 1377 (1992). [6] B. Khomami and L. D. Moreno, Rheologica Acta 36, 367 (19[1967]).K. K. Talwar and B. Khomami, Journal of Non-Newtonian Fluid Mechanics 57, 177 (1995). [7] A. M. Howe, A. Clarke, and D. Giernalczyk, Soft Matter 11,[1674]19A.(2S0o1u5v)a.liotis and A. N. Beris, Computer Methods in Applied Mechanics and Engineering 129, 9 (1996). [8] D. F. James, R. Yip, and I. G. Currie, J. Rheol. 56, 1249 (20[1182]).A. W. Liu, Ph.D. thesis, Massachusetts Institute of Technology, 1997. ##LTLine## ##LTLine## ##LTLine##

text: sented

text: here

text: includes

text: a

text: diﬀusive

text: term.

text: As

text: discussed

text: by

text: characterise

text: each

text: of

text: these

text: regimes,

text: and

text: we

text: have

text: demon-

text: Sureshkumar

text: and

text: Beris

text: this

text: can

text: have

text: a

text: stabilising

text: strated

text: that

text: the

text: upturn

text: in

text: the

text: drag

text: coeﬃcient

text: occurs

text: at

text: a

text: eﬀect,

text: suppressing

text: possible

text: numerical

text: instabilities.

text: Here

text: ﬁxed

text: value

text: of

text: the

text: maximum

text: of

text: these

text: two

text: numbers

text: across

text: we

text: have

text: made

text: an

text: eﬀort

text: to

text: minimise

text: the

text: eﬀect

text: of

text: such

text: the

text: full

text: range

text: of

text: medium

text: porosities.

text: We

text: have

text: also

text: stud-

text: a

text: term

text: by

text: getting

text: as

text: close

text: as

text: is

text: numerically

text: possible

text: to

text: ied

text: the

text: creeping

text: ﬂow

text: of

text: an

text: Oldroyd

text: B

text: ﬂuid

text: past

text: a

text: linear

text: the

text: dual

text: limit

text: ∆x,

text: L

text: which

text: en-

text: array

text: of

text: cylinders

text: conﬁned

text: to

text: a

text: channel

text: bounded

text: by

text: solid

text: x

text: y

text: sures

text: that

text: the

text: diﬀusive

text: lengthscale

text: is

text: well

text: resolved

text: yet

text: walls,

text: where

text: we

text: have

text: found

text: that

text: the

text: ﬂow

text: remains

text: steady

text: small

text: compared

text: to

text: any

text: features

text: of

text: the

text: ﬂow

text: ﬁeld.

text: This

text: ap-

text: for

text: all

text: Weissenberg

text: numbers

text: explored.

text: proach

text: suppresses

text: spurious

text: numerical

text: instabilities

text: whilst

text: All

text: the

text: simulations

text: in

text: this

text: work

text: have

text: assumed

text: a

text: purely

text: keeping

text: physical

text: ones

text: (if

text: present).

text: A

text: second

text: explanation

text: two-dimensional

text: ﬂow,

text: with

text: translational

text: invariance

text: into

text: could

text: be

text: due

text: to

text: the

text: presence

text: of

text: small

text: but

text: non-zero

text: iner-

text: the

text: page

text: in

text: the

text: sketches

text: of

text: Fig.

text: In

text: experimental

text: prac-

text: tia

text: in

text: Ref.

text: As

text: discussed

text: by

text: Hoda

text: et

text: al.

text: weak

text: tice,

text: the

text: upturn

text: in

text: the

text: drag

text: coeﬃcient

text: is

text: often

text: accompa-

text: inertial

text: eﬀects

text: can

text: provide

text: a

text: mechanism

text: by

text: which

text: elas-

text: nied

text: by

text: the

text: onset

text: of

text: three

text: dimensional

text: ﬂows

text: (which

text: are

text: tic

text: disturbances

text: can

text: be

text: ampliﬁed.

text: Finally

text: it

text: is

text: conceiv-

text: furthermore

text: often

text: time-dependent).

text: To

text: capture

text: these

text: ef-

text: able

text: that

text: the

text: time-dependent

text: states

text: observed

text: in

text: Ref.

text: fects,

text: we

text: would

text: need

text: to

text: perform

text: simulations

text: in

text: fully

text: three

text: are

text: the

text: result

text: of

text: a

text: nonlinear

text: instability.

text: This

text: mecha-

text: dimensions.

text: The

text: two

text: dimensional

text: results

text: reported

text: here

text: nism

text: has

text: been

text: proposed

text: to

text: explain

text: observed

text: instabilities

text: should

text: clearly

text: be

text: treated

text: with

text: caution

text: for

text: Weissenberg

text: in

text: Poiseuille

text: ﬂow

text: of

text: viscoelastic

text: ﬂuids,

text: which

text: are

text: believed

text: numbers

text: exceeding

text: where

text: three

text: dimensional

text: eﬀects

text: to

text: be

text: linearly

text: stable

text: A

text: ﬁnal

text: possibility

text: concerns

text: may

text: pertain.

text: spatial

text: resolution.

text: While

text: the

text: number

text: of

text: SPH

text: particles

text: James

text: recently

text: considered

text: the

text: ratio

text: of

text: shear-

text: to

text: cannot

text: necessarily

text: be

text: directly

text: compared

text: to

text: the

text: number

text: extensionally-induced

text: ﬁrst

text: normal

text: stresses

text: in

text: viscoelastic

text: of

text: ﬁnite-diﬀerence

text: grid

text: points,

text: we

text: note

text: that

text: the

text: second

text: ﬂow

text: past

text: a

text: biperiodic

text: array

text: of

text: cylinders.

text: They

text: demon-

text: largest

text: particle

text: resolutions

text: in

text: Ref.

text: would

text: roughly

text: strated

text: an

text: lower

text: bound

text: on

text: this,

text: implying

text: that

text: shear

text: correspond

text: to

text: the

text: lowest

text: resolution

text: grid

text: in

text: Fig.

text: for

text: generated

text: ﬁrst

text: normal

text: stresses

text: N

text: cannot

text: be

text: neglected.

text: which

text: we

text: see

text: ﬂuctuations.

text: Wagner

text: and

text: McKinley

text: recently

text: examined

text: the

text: re-

text: VIII.

text: CONCLUSIONS

text: sponse

text: of

text: an

text: Oldroyd-B

text: ﬂuid

text: to

text: a

text: ﬂow

text: with

text: a

text: character-

text: parameter

text: α(t)

text: varying

text: sinusoidally

text: in

text: time,

text: intended

text: to

text: In

text: this

text: work,

text: we

text: ﬁrst

text: studied

text: two

text: dimensional

text: creep-

text: mimic

text: the

text: variation

text: experienced

text: by

text: a

text: given

text: ﬂuid

text: element

text: ing

text: ﬂow

text: of

text: the

text: Oldroyd

text: B,

text: FENE-CR

text: and

text: FENE-P

text: vis-

text: as

text: it

text: moves

text: through

text: a

text: periodic

text: ﬂow

text: cell

text: of

text: the

text: kind

text: stud-

text: coelastic

text: ﬂuids

text: past

text: a

text: biperiodic

text: square

text: array

text: of

text: cylinders.

text: ied

text: in

text: this

text: work.

text: For

text: large

text: Deborah

text: numbers

text: De

text: (deﬁned

text: Our

text: aim

text: has

text: been

text: to

text: understand

text: the

text: dramatic

text: upturn

text: re-

text: as

text: the

text: ratio

text: of

text: the

text: polymer

text: relaxation

text: time

text: to

text: the

text: time

text: ported

text: experimentally

text: in

text: the

text: drag

text: coeﬃcient

text: as

text: a

text: function

text: required

text: to

text: pass

text: one

text: unit

text: cell),

text: they

text: demonstrated

text: that

text: of

text: increasing

text: Weissenberg

text: number.

text: attempts

text: to

text: predict

text: the

text: ﬁrst

text: normal

text: stress

text: just

text: above

text: By

text: performing

text: simulations

text: across

text: a

text: wide

text: range

text: of

text: val-

text: and

text: below

text: the

text: cylinder

text: using

text: the

text: local

text: Weissenberg

text: num-

text: ues

text: of

text: the

text: porosity

text: of

text: the

text: ﬂow

text: geometry,

text: from

text: ‘dilute’

text: ber

text: fails

text: to

text: reﬂect

text: the

text: previous

text: ﬂow

text: history

text: and

text: drasti-

text: to

text: near-touching

text: cylinders,

text: we

text: have

text: demonstrated

text: two

text: cally

text: underestimates

text: the

text: value

text: of

text: In

text: view

text: of

text: these

text: qualitatively

text: diﬀerent

text: mechanisms

text: that

text: may

text: separately

text: recent

text: works,

text: in

text: future

text: numerical

text: studies

text: of

text: the

text: kind

text: per-

text: underpin

text: this

text: thickening

text: eﬀect.

text: The

text: ﬁrst

text: operates

text: in

text: the

text: formed

text: here

text: it

text: would

text: clearly

text: be

text: interesting

text: to

text: consider

text: highly

text: porous

text: case

text: of

text: widely

text: spaced

text: cylinders,

text: and

text: in-

text: more

text: explicitly

text: the

text: role

text: of

text: normal

text: stresses.

text: volves

text: a

text: strong

text: stretching

text: of

text: the

text: polymer

text: chains

text: in

text: the

text: ex-

text: ACKNOWLEDGEMENTS

text: tensional

text: wake

text: aft

text: of

text: each

text: cylinder.

text: The

text: second

text: operates

text: in

text: the

text: regime

text: of

text: more

text: densely

text: packed

text: obstacles,

text: and

text: in-

text: EJH

text: and

text: SMF

text: gratefully

text: acknowledge

text: ﬁnancial

text: sup-

text: volves

text: a

text: strong

text: deformation

text: of

text: the

text: polymer

text: chains

text: as

text: they

text: port

text: for

text: this

text: work

text: from

text: Schlumberger

text: Gould

text: Research.

text: squeeze

text: into

text: the

text: vertical

text: gap

text: between

text: vertically

text: adjacent

text: We

text: also

text: thank

text: the

text: referees

text: for

text: their

text: valuable

text: comments

text: cylinders.

text: Two

text: diﬀerent

text: Weissenberg

text: numbers

text: separately

text: on

text: the

text: paper.

text: D.

text: F.

text: James,

text: J.

text: Non-Newtonian

text: Fluid

text: Mech.

text: A.

text: Clarke

text: et

text: al.,

text: Soft

text: Matter

text: R.

text: G.

text: Larson

text: and

text: P.

text: S.

text: Desai,

text: Annu.

text: Rev.

text: Fluid

text: Mech.

text: )C.

text: larke,

text: A.

text: M.

text: Howe,

text: J.

text: Mitchell,

text: S.

text: Staniland

text: and

text: L.

text: A.

text: Hawkes,

text: SPE

text: Journal,

text: G.

text: M.

text: Homsy,

text: Annu.

text: Rev.

text: Fluid

text: Mech.

text: J.

text: Mitchell,

text: K.

text: Lyons,

text: A.

text: M.

text: Howe,

text: and

text: A.

text: Clarke,

text: Soft

text: Matter

text: D.

text: Bensimon

text: et

text: al.,

text: Rev.

text: Mod.

text: Phys.

text: H.

text: Darcy,

text: Les

text: fontaines

text: publiques

text: de

text: la

text: ville

text: de

text: Dijon

text: (Dalmont,

text: Paris,

text: C.

text: Chmielewski,

text: Journal

text: of

text: Rheology

text: P.

text: C.

text: Carman,

text: Chemical

text: Engineering

text: Research

text: and

text: Design

text: L.

text: Skartsis,

text: Journal

text: of

text: Rheology

text: K.

text: K.

text: Talwar,

text: Journal

text: of

text: Rheology

text: B.

text: Khomami

text: and

text: L.

text: D.

text: Moreno,

text: Rheologica

text: Acta

text: K.

text: Talwar

text: and

text: B.

text: Khomami,

text: Journal

text: of

text: Non-Newtonian

text: Fluid

text: Mechanics

text: A.

text: M.

text: Howe,

text: A.

text: Clarke,

text: and

text: D.

text: Giernalczyk,

text: Soft

text: Matter

text: and

text: A.

text: N.

text: Beris,

text: Computer

text: Methods

text: in

text: Applied

text: Mechanics

text: and

text: Engineering

text: D.

text: F.

text: James,

text: R.

text: Yip,

text: and

text: I.

text: G.

text: Currie,

text: J.

text: Rheol.

text: W.

text: Liu,

text: Ph.D.

text: thesis,

text: Massachusetts

text: Institute

text: of

text: Technology,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##