Disentanglement Analysis with Partial Information Decomposition
Authors: Seiya Tokui
Issei Sato
Venue: ICLR 2022 Poster (2022)
Keywords: disentangled representations
variational autoencoders
deep generative models

Abstract: We propose a framework to analyze how multivariate representations disentangle ground-truth generative factors. A quantitative analysis of disentanglement has been based on metrics designed to compare how one variable explains each generative factor. Current metrics, however, may fail to detect entanglement that involves more than two variables, e.g., representations that duplicate and rotate generative factors in high dimensional spaces. In this work, we establish a framework to analyze information sharing in a multivariate representation with Partial Information Decomposition and propose a new disentanglement metric. This framework enables us to understand disentanglement in terms of uniqueness, redundancy, and synergy. We develop an experimental protocol to assess how increasingly entangled representations are evaluated with each metric and confirm that the proposed metric correctly responds to entanglement. Through experiments on variational autoencoders, we find that models with similar disentanglement scores have a variety of characteristics in entanglement, for each of which a distinct strategy may be required to obtain a disentangled representation.

Published as a conference paper at ICLR 2022
DISENTANGLEMENT ANALYSIS WITH PARTIAL INFOR MATION DECOMPOSITION
Seiya Tokui
Issei Sato
The University of Tokyo tokuiêg ecc. wtw.ee.....tooooooooooo u-tokyo.
The University of Tokyo
satoêg ecc. u-tokyo.ac.... jp
ABSTRACT
We propose framework to analyze how multivariate representations disentan- gle ground-truth generative factors. A quantitative analysis of disentanglement has been based metrics designed to compare how one variable explains each generative factor. Current metrics, however, may fail to detect entanglement that involves more than two variables, e.g., representations that duplicate and rotate generative factors in high dimensional spaces. In this work, we establish a frame- work to analyze information sharing in multivariate representation with Par- tial Information Decomposition and propose a new disentanglement metric. This framework enables us understand disentanglement in terms of uniqueness, re- dundancy, and synergy. We develop an experimental protocol to assess how in- creasingly entangled representations are evaluated with each metric and confirm that the proposed metric correctly responds to entanglement. Through experi- ments on variational autoencoders, we find that models with similar disentangle- ment scores have variety of characteristics in entanglement, for each of which a distinct strategy may be required to obtaín a disentangled representation.
INTRODUCTION
Disentanglement is a guiding principle for designing a learned representation separable into parts that individually capture the underlying factors of variation. The concept is originally concerned as an inductive bias towards obtaining representations aligned with the underlying factors of variation
in data (Bengio et al. (013) and has been applied controlling otherwise unstructured represen-
tations of data from several domains, e.g., images (Karras et al. 2019 Esser et al. 2019). text (Hu
al. 2017). and audio (Hsu et al. 2019) name just few.
While the concept is appealing, defining disentanglement is not clear. After Higgins et al. (2017) generative learning methods with regularized total correlation have been proposed (Kim & Mnih 2018 Chen et al.) 2018); however, it is still not clear if independence of latent variables is essential for better disentanglement (Higgins et al., 2018). Furthermore, it is not obvious to measure disen- tanglement given true generative factors. Towards understanding disentanglement, it is crucial to define disentanglement metrics, for which several attempts have been made (Higgins et al. 2017
Kim & Mnih 2018 Chen et 2018 [astwood & Williams 2018 Do & Tran 2020 Zaidi al. 2020 however, there are still problems to be solved.
Current disentanglement metrics may fail to detect entanglement involving more than two variables.
In these metrics, one first measures how each variable explains generative factor and then com- pares or contrasts them among variables. With such a procedure, we may overlook multiple variables conveying information of one generative factor. For example, let (zz, Z2) be representation consisting of two vectors, where each variable in zı disentangles a distinct generative factor and z2 rotation of 2 not axis-aligned with the factors. Since any dimension of 2a alone may convey little information of one generative factor, these metrics do not detect that multiple variables en- code one generative factor redundantly. Although this simple example, this kind of information sharing may arise in learned representations as well, if not the variables are linearly correlated.
In this work, we present a disentanglement analysis framework aware of interactions among multiple variables. Our key idea is to decompose the information of representation into entangled and disen- tangled components using Partial Information Decomposition (PID), which a framework mod- information theory to analyze information sharing among multiple random variables (Williams)

---

Published as a conference paper at ICLR 2022
U1, U2)
[(u; U2)
u(uzvi U2)
R(u;vi, 02)
U(u u2 \ (v))
[(u;; U2)
U: unique information
R: redundant information
C: complementary information
Figure I: Information diagram of three variable system in PID. Each circle represents mutual infor- mation, and each area separated by them represents a decomposed term in PID. When we substitute a generative factor for u, a latent variable for V1, and the other latent variables for U2, the unique information (u; 01 (222) represents the information of the factor disentangled by the latent variable. See Figure 5 in Appendix for the alternative form similar to the ones we will use in Section
& Beer 2010). As illustrated in Figure the mutual information [(u; v1, 02) = E log p(u.o p(u))((r..a) between a random variable u and a pair of random variables (v), 02) is decomposed into four redundant information
nonnegative terms: unique information U(u;v, \ v2) and U(u; 02
R(,;"), 02), and complementary (or synergistic) information C(u; U1, 02). While these partial in- formation terms have agreed-upon concrete definitions yet (Bertschinger et al., 2014; Finn & Lizier 2018: Lizier et al.] 2018 Finn & Lizier 2020, Siiterraans] (020)). we can derive universal lower and upper bounds of the partial information terms only with well-defined mutual information terms. We apply the PID framework to representations learned from data by letting u be generative factor and , 12 be one and the remaining latent variables, respectively. The unique information of a latent variable intuitively corresponds to the amount of disentangled information, while the redun- dant and complementary information correspond to different types of entangled information. We can quantify disentanglement and multiple types of entanglement through the framework, which enriches our understanding on disentangled representations.
Our contributions are summarized as follows.
PID-based disentanglement analysis framework: We propose a disentanglement anal- ysis framework that captures interactions among multiple variables with PID. With this framework, one can distinguish two different types of entanglement, namely redundancy and synergy, which provide insights on how representation entangles generative factors.
We derive lower and upper bounds of partial information terms. We formulate a disentanglement metric, called UNIBOUND, us- ing the lower bound of unique information. We design entanglement attacks, which inject entanglement to a given disentangled representation, and confirm through experiments us- ing them that UNIBOUND effectively captures entanglement involving multiple variables.
Tractable bounds of partial information terms:
Detailed analyses of learned representations: We analyze representations obtained by variational autoencoders (VAEs). We observe that UNIBOUND sometimes disagrees with interactions may dominate learned represen-
other metrics, which indicates tations. We also observe that different types of entanglement arise in models learned with different methods. This observation provides us an insight that we may require distinct approaches to remove them for disentangled representation learning.
PROBLEM FORMULATION AND NOTATIONS
Let x be a random variable representing a data point, drawn uniformly from a dataset D. Assume available for each data point; in other words, we can access the subset D(y) C D of the data points with any fixed generative factors y. Let z (zi,-.., 21.) be latent representation consisting of L random variables. An inference model is provided as the conditional distribution p(z|z). Our goal is to evaluate how well the latent variables z disentangle each generative factor in y. The inference model can integrate out the input as p(-ly) Ep(rly) [p]+[=)] = Piyl Eee Ply) p(+[=); therefore, we only use and z in most of our discussions.
that the true generative factors y (V1 ........ ... y))
NNte that this \ is not set difference operator. It is just a common notation used in the PID literature to emphasize the unique information not symmetric and resembles the set difference depicted in Fig
2

---

Published as a conference paper at ICLR 2022
We denote the mutual information between random variables and v by [[; v) = E log p(u.0)
p(u)p(e)
the entropy of by H(u) -Eolog p(u)], and the conditional entropy of given v by (ulp)
- E p(u[))] We denote vector of zeros by 0, vector of ones by 1, and an identity matrix by I. We denote by (H, Σ) the Gaussian distribution with mean j and covariance Σ and denote its
density function by (M; 2).
2 RELATED WORK
The importance of representing data with multiple variables conveying distinct information has been recognized at least since the '80s (Barlow 1989. Barlow et al.) 1989 Schmidhuber 1992). The minimum entropy coding principle (Watanabe 1981). which aims representing data by random variables z with the sum of minimum marginal entropies De H (ze), is found to be useful for un- supervised learning to remove the inherent redundancy in the sensory stimuli. The resulting rep- resentation minimizes the total correlation and is called factorial coding. Recent advancements in disentangled representation learning based on VAEs (Kingma & Welling 2014) are guided by the same principle as minimum entropy coding (Kim & Mnih 2018| et al. 2018 Gao et al., 2019).
Understanding better representations, which tackled from the coding side as above, is also ap- proached from the generative perspective. It is often expected that data generated from genera- tive factors through a process that entangles them into high dimensional sensory space (DiCarlo & Cox )007). As generative factors are useful as the basis of downstream learning tasks, obtaining disentangled representations from data is a hot topic of representation learning (Bengio et al.) 2013). Towards learning disentangled representations, it is arguably important quantitatively measure disentanglement. In that regard, Higgins et al. (2017) established a standard evaluation procedure using controlled datasets with balanced and fully aunttaanotttted ground-truth factors. A variety of met- rics have then been proposed on the basis of the procedure. Among them, Higgins et al. (2017) and Kim & Mnih (2018) propose metries based on the deviation of each latent variable conditioned by a generative factor. In contrast, Mutual Information Gap (MIG) (Chen et al. 2018)) and its variants (Do Tran 2020 Zaidi et al. 2020) are based on mutual information between latent variable and generative factor. We extend the latter direction, considering marti-variable interactions.
Barlow (1989) discussed redundancy by comparing the population and the individual variables by their entropies, i.e., total correlation. It is though less trivial to measure redundancy as an informa- tion quantity. The PID framework (Williams & Beer 2010) provides an approach to understanding redundancy among multiple random variables work provides some desirable relationships between decomposed information terms, while it leaves some degrees of freedom to determine all of them, for which several definitions have been proposed (Williams & Beer 2010 Bertschinggr et al.) 2014 Finn & Lizier 2018: 2020 Sigtermans 2020).
constituent of mutual information. The frame-
The PID framework has been applied machine learning models. For example, Tax al. (2017) measured the PID terms for restricted Boltzmann machines using the definition of Williams & Beer (210)) Yu et al. (202T) took alternative route, similar our approach, where they measured lin- ear combinations of PID terms by corresponding linear combinations of mutual information terms. These work aims at analyzing the learning dynamics of models in supervised settings. In contrast, we use the PID framework for analyzing disentanglement in unsupervised representation learning.
3 PARTIAL INFORMATION DECOMPOSITION FOR DISENTANGLEMENT
In this section, we analyze the current metrics and introduce our framework. In Section introduce PID of the system we concern. Section 3.2, we investigate the current metrics in terms of vartil- interactions. In Section 3.3 and 3.4 construct our disentanglement metric with bounds for PID terms. We provide a method of computing the bounds Section 3.5
3.1 PARTIAL INFORMATION DECOMPOSITION
We tackle the problem of evaluating disentanglement of a latent representation z relative to the true generative factors y from information-theon generative factor yk is captured by the latent representation z. The information of yk captured by z measured using mutual information I(yk: z) (zz) Hzllyx).
perspective. Let us consider evaluating how one
In a desirably disentangled representation, we expect one of the latent variables 2, to exclusively capture the information of the factor yk- To evaluate given representation, are interested in
3

---

Published as a conference paper at ICLR 2022
understanding how the information is distributed between latent variable 24 and the remaining representation 2, = (2 e This is best described by the PID framework, where the mutual information is decomposed into the following four terms.
[(x::2] z) = Ryyx: 2e, z,e) ) + (yxx 24 \ z.e)) + Uyn: z. \ =e) + C(uki Zxpr).
(1)
Here, the decomposed terms represent the following non-negative quantities.
Redundant information R(yk: ze, z(p)) is the information of yk held by both 24 and 2
Unique information (yys ze xxee) is the information of yis held by a and not held by aie- The opposite term U(y Ziee) ze) also defined by exchanging the roles of ž and z,e.
Complementary information (or synergistic information) C(ys; 2e, 2,,2xe) e) is the information of yı held by = (ze, A,e that is not held by either Eς 2 alone.
The following identities, combined with [ partially characterize each term.
Tu::: =&) = Ryv: Ze, Rvee) (v): ze \ Aee), T(::: = [.vx: =e, mxe) + U1KE ze). (2)
The decomposition this system is illustrated Figure 2a
We expect disentangled representations to concentrate the information of to single latent vari- able zg, and let the other variables z not convey the information in either unique, redundant, or synergistic ways. This is understood in terms of PID as maximizing the unique information (yx: ze
zee) while minimizing the other parts of the decomposition.
The above formulation is incomplete as one degree of freedom remains determine the four terms with the three equalities. Instead of stepping into searching for suitable definitions of these terms, we build discussions applicable any such definitions that fulfill the above incomplete requirement
3.2 UNDERSTANDING DISENTANGLEMENT METRICS FROM INTERACTION PERSPECTIVE
Current disentanglement metrics are typically designed to measure how each latent variable captures factor and compare it among latent variables, i.e., output high score when only one latent variable captures the factor well.
For example, the Beta metric (Higgins al. 2017) is computed by estimating the mean abso- lute difference (MAD) of two i.i.d. variables following ((2)))).) = Ep(rly, )P(z((z) for each by Monte-Carlo sampling and training a linear classifier that predicts from the noisy estimations of the differences. The Factor metric (Kim & Mnih 2018) is computed similarly, except that the MAD is replaced with variance (following normalization by population), and majority-vote -vote clas- sifier is used to eliminate a failure mode of ignoring one of the factors and to avoid depending on hyperparameters. These metrics have the same goal of finding a mapping between and k by com- paring the deviation of when fixing yk- Since the deviation is computed for each separately, these metrics do not count how each latent variable 2 interacts with the other variables z,.-
Another example is MIG (Chen et al., 2018), which compares mutual information T(y 24) for all l and uses the gap between the maximum and the second maximum among them. More precisely, MIG defined by the following formula.
1 K
MIG
max min (1y z4) (3rs 2)) E. EzL
(3)
(1N)
A=1
Here, dividing each summand by (yx) balances the contribution of each factor when they discrete. The difference in mutual information is rewritten as a difference in unique information as
[ynn =e) I (ynn U(y #e =e:)) Ld (yn ze \ z2) U( ze. z.r)
(4)
In that sense, MIG effectively captures the pairwise interactions between latent variables. This metric still ignores interplays between more than two variables. Figure 2b reveals that some of the redundant information R R(y: ze, Z:e) is positively evaluated in MIG, which should have been considered signal of entanglement. Note that there are several extensions to MIG (Do Tran 2020
Li et al. 2020 Zaidi et al., 2020); see Appendix B for detailed discussions on them.
2 Moot studies on PID only deal with discrete systems, while deep representations often include continuous variables. There have been attempts to define and analyze PID for continuous systems (Barrett 2015 Schick- Poland et al. [202 Pakman al. 2021). Note that the domain of variables, which our framework depends not limited with the PID framework.
4

---

Published as a conference paper at ICLR 2022
I (y)iz)
[(y:; Z(ze)
Ulysi2 20 mxee)
Rlyss Zg, Z(e)
21 (3x: Zee zz)
Clyxi Ee, z.ee)
(a) PID for systems with yk, and 2
[(nni zz)
I [
7 Tu:: 23,e)
I [ 2p)
MIG
U(yk: 21
Zpr
R (Us: Ze, 2x.
U(yk: Ze ) zz)
UNIBOUND
L/ Mi: ze mg))
RR 24, Zxee)
U(yse Z, ze)
Clyss Ze, 2x
(b) Side-by-side comparison of positive and negative terms in UNIBOUND and MIG
Figure 2: Information diagrams depicted by bands (the style borrowed from Figure 8.1 of MacKay (2003)) White boxes represent mutual information, which
compute. (a) The bands depict the decomposition used in the PID-based disentanglement evaluation. (b) This diagram superposes the decomposition for systems with (yk: 2e, zxe) and (yk: ze, =e- where zg is the latent variable chosen by MIG evaluation. The green boxes are positively evaluated in MIG (the top colored line) and UNIBOUND (the bottom colored line), while the red boxes are negatively evaluated in them. Observe that MIG positively evaluates a part of the redundancy, namely Ryy ze, Zie) - Ryns ze, ze ), as it does not take into account the interactions among strict supersets of { ze
3.3 UNTBOUND: NOVEL DISENTANGLEMENY METRIC
We can lower bound the unique information in any possible PID definitions by computable compo- nents, as we did in 4 obtaining
4 To bound (yx: ze z:e) instead of Uyys ze \/ (e we replace zy with 2,6-
U(ys xxe) [1xx \ (z,)) U(yi Ziee =e)] [1vs x2) I (2)e 2())
(5)
where we use H4 (0. } as the difference in mutual information is not guaranteed to be negative. The decomposed terms evaluated by the bound illustrated in the lower part Figure 2b It effectively excludes, from the positive term, the effect of interaction between 2 and any other latent variables. similar way to MIG, we summarize this bound over all the generative factors to obtain the metric call UNIBOUND.
UNIBOUND 1 K
max (Tyes Tyy:; m.)) Σ H(yk)
Dividing each summand by the entropy H (yv) has the same role as in MIG; it makes the evaluation fair between factors and eases the comparison as the metric normalized when y/s discrete.
3.4 OTHER BOUNDS FOR PARTIAL INFORMATION TERMS
The UNIBOUND metric is handy quantity to compare representations by single scalar, while PID itself may provide more ideas on how given representation entangles or disentangles the factors. To fully leverage the potential, derive bounds for all the terms of interest, including redundancy and synergy terms, from both lower and upper sides.
Let (1ni 26 [yyi ze) [1ns zee) (wxxz) z)) be the interaction information of triple derive the following bounds from Eq
(yk, ze, (.). Using nonnegativity of PID terms,
[1 20) (yni 2,)4 < U(yn zz xx)) < [gys zx) [ zei
[I 2x))) R(yn: z4, zxe)) S min 24). I(Nuiz,))
(7)
[- (T ze; 2x) C(uui z4, mx)) & min(T(ye: ze), T(n me)) [T 26; =6,
5

---

Published as a conference paper at ICLR 2022
Note that all six bounds are computed by arithmetics on [(yx: 22), [yx: z,,), and [(::: z). We can summarize each lower bound in similar way as we did in and summarize the corresponding upper bound using the same é for each & as the lower bound. While these bounds only determine the terms as intervals, they provide us enough insight into the type of entanglement dominant in the representation (redundancy synergy).
3.5 ESTIMATING BOUNDS BY EXACT LOG- MOG-MARGINAL DENSITIES
When the dataset is fully annotated with discrete generative factors and the inference distribution p(z[z) and its marginals p( 2 p p(z(, z are all tractable (e.g., mean field variational models), we can compute the bounds in similar way as is done by Chen et al. (2018) for MIG. Let Zş be either of Zr, zie or z. We denote by D(yx) the subset of the dataset with specific value of yt- The mutual information [y)s zs) = -H (zslvv) ) (zs) can then be computed by the following formula.
1 (yxizz) Ep(ya.ss) ks))
log [D(Uk))
Σ
p(zs|z)
log
1 D
Σ Erlasla) FED
(8)
ZE (ED(U))
Assuming that each generative factor yk is discrete and uniform, employ stratified sampling over p(3/e zz). We approximate the expectation over p(zs(gk) by sampling from the subset D(yk) and then sampling zs from p(z|zz) to avoid quadratic computational cost. Following Chen et al. (2018). we used the sample size of 10000 in experiments. We use log-sum-ce function to compute log Σ p(zs|a)) m²) log Σ exp(log cxp(logp(zs)p)) [z)) for numerical stability. The PID bounds and the UNIBOUND metric are computed by combining these estimations.
When the inference distribution p(z(ar) factorized, its log marginal logs(z))z) computed by just adding up the log marginal of each variable as Etes log log.((zz]ja p()))z)). Otherwise, we need to ex- plicitly derive the marginal distribution and compute the log density. For example, when p(z]xr is a Gaussian distribution with mean μ and non-diagonal covariance E, p(zs) (z) Gaussian distri- bution with mean (1: hes and covariance (Eiy)a.jss Such case arises for the attacked model we will describe in the next section.
Let M be the sample size for the expectations, and N = |D] be the size of the dataset. Then, the computation of Eq requires O(MN) evaluations of the conditional density (z)))r))
ENTANGLEMENT ATTACKS
To confirm that the proposed framework effectively captures interactions among multiple latent variables, we apply it to adversarial representations that entangle any generative factors. Instead of making a single artificial model, we modify a given model that disentangles factors well into a noised version that entangles the original variables. We call this process entanglement attack.
Let z E R be the representation defined by a given model. Our goal is to design a transform from to attacked representation 2 so that disentanglement metrics fail to capture entanglement unless they correctly handle multi-variable interactions.
As mentioned in Section metrics that do not take into account the interaction among multiple latent variables may underestimate redundant information. To crystallize such a situation, we first design an entanglement attack to inject redundant information into multiple variables. For complete- ness, we also design a similar attack to inject synergistic information as well.
4.1 REDUNDANCY ATTACK
Let U be an I x L orthonormal matrix and e E RR be a random vector following the standard normal distribution (O, Using hyperparameter 2 0, we define the redundancy attack by
red
Z aUz +
The coefficient adjusts the degree of entanglement. When 0. the new representation just appends noise elements to a given representation, which does not affect the disentanglement. In- creasing makes the additional dimensions less noisy, resulting in rrd redundantly encoding the information of factors conveyed by the original representation. The mixing matrix chooses how the information of individual variables in is distributed to the additional dimensions. If we choose

---

Published as a conference paper at ICLR 2022
U the dimensions are not mixed; thus considering one-to-one interaction between pairs of variables is enough to capture the redundancy. To mix the dimensions, we can use U = - instead, which is orthonormal matrix that mixes each variable with the others.
To evaluate mutual information terms for MIG and UNIBOUND metrics after the attack, we need an explicit formula for the inference distribution p p( (x)). When the original model has Gaussian inference distribution p(z|x) = (z; (z),,, E(x)), the attacked model is a summation of linear transformation of and standard normal noise, which results in Gaussian distribution
pl x3) = N -N(2 g
( (aUu(z))
(
E(r) (+)(())
I+ (
4.2 SYNERGY ATTACK
For completeness, we also define an attack that entangles representation by increasing synergistic information. Using the same setting with an L x orthonormal matrix U and a noise vector e ~ (0, I), we construct the synergy atluck by
(aUc+ (oUC+Z) (
(10)
Here again, the coefficient a adjusts the degree of entanglement, When a O, the attacked version just extends the original representation with independent noise elements. By increasing the noise are mixed with the parts conveying the information of factors. The attacked vector 25y fully conveys the information of factors in regardless of
recover the original representation by z - 1:22, Note that most existing metrics correctly react to this attack since the information of individual variables is destroyed by the noise. The upper bound of the unique information is one of the quantities that positively evaluate synergistic information and is expected to ignore the attack.
5 EVALUATION
We evaluated the metrics in a toy model in Section 5.1 and in VAEs trained on datasets with the true generative factors in Section 5.2 We also performed detailed analyses of VAEs by plotting the PID terms of each factor, which is deferred to Appendix due to the limited space.
5.1 EXACT ANALYSIS WITH TOY MODEL
We consider a toy model with attacks defined in Section 4 as a sanity check. Suppose that data are generated by factors y NN I), and we have latent representation that disentangles them up to noise as z ~ (yy, o II)), where a > O. With this simple setting, we can analytically compute MIG and UNIBOUND. For example, when set 0.1 and K
can derive the scores after the redundancy attack as MIG = log 101 1+101 and UNIBOUND = log 101 x 1+1.010))
1+0.65aa
where c = log(2re). As a function of o, UNIBOUND decreases faster than MIG. The difference comes from the information distributed among the added dimensions of the attacked vector; while UNIBOUND counts all the entangled information, MIG only deals with one of the added dimensions. Indeed, the amount the untracked entanglement remains in MIG after taking the limit of a + oo as MIG + log 65, while UNIBOUND + O.
We can also compute the scores after the synergy attack as MIG UNIBOUND
log [++ a=+0.01): where both metrics correctly capture the injected synergy. Refer to Appendix Clfor the derivations of the exact scores for arbitrary parameter choice.
5.2 EMPIRICAL ANALYSIS WITH ANNOTATED DATASETS
We used the DSPRITES and 3DSHAPES datasets for our analysis. The DSPRITES dataset consists of 737, 280 binary images generated from five generative factors (shape, size, rotation, and x/y co- ordinates). The 3DSHAPES dataset consists 480, 000 color images generated from six generative factors (ffoor/wall/ojeect hues, scale, shape, and orientation). All the images have 64 x 64 pixels. We used all the factors for y, encoded as a set of discrete (categorical) random variables.
We used variants of VAEs methods for disentangled representation learning: 3-VAE (Higgins) jet al.) 2017),, FactorVAE (Kim & Mnih 2018). 3-TCVAE (Chen et al.] 2018). and dointVAr (Dupont 2018). We trained all the models with six latent variables, of which in JointVAE is
7

---

Published as a conference paper at ICLR 2022
BVAE FactorVAE TCVAE JointVAE
2.0
BetaWAE
1.0
Tactorong
0.8
0.5
normalized information
0.6
DSPRITES
TectorViC KoWG
PiAt 10
0.5
0.5
JniBound
0.4
0.7
MG
UniBound
0.2
0.0
0.3
U
URC URC URC UUR. R
C
BetaViiE
FactoriAE
1.0
mebri BeTDWAE
meer PactorikE
0.8
BVAF FactorVAF TCVAF JointVAF
0.0
0.5
3.0
normalized information
3.6
Ttudil
0.6
3DSHAPES
9.2
0.6
0.6
1.0
3.0
mebric MBS
metik a UnBoound
0.1
0.8
MIS
UniBound
3.
0.2
0.3
0.5
0.0 U URC URC URC
3.
(a) Disentanglement scores
(bD terms
(c) Redundancy attacks
Figure 3: Experimental results for VAEs trained with DSPRITES (top row) and 3DSHAPES (bottom row). (a) Disentanglement scores. (b) Estimated PID terms. Three orange bars each plot represent the possible values of unique (U), redundant (R), and complementary (C) information, respectively. The top and bottom of each orange area correspond to the upper and lower bounds of the term, computed with Eq 7 (c) Disentanglement scores of 3-VAE and 3-TCVAE after redundancy attack with varying strength. See Appendix H for a larger version of the plots.
a three-way categorical variable for DSPRITES and a four-way categorical variable for 3DSHAPES. We trained each model eight times with different random seeds and chose the best half of them for each metric to avoid cluttered results due to training instability. We optimized network weights with Adam (Kingma Ba 2015). We used the standard convolutional networks used the literature for the encoder and the decoder. See Appendix D for the details of architectures and hyperparameters.
For disentanglement metrics, we compare BetaVAE metric (Higgins et al. 2017). FactorVAE metric (Kim & Mnih 2018). MIG (Chen et al. 2018), and the proposed UNIBOUND metric.
We first compared the models with each metric as shown in Figure 3a The trend is basically similar between UNIBOUND and the other metrics, while they disagree some cases. For example, Factor- VAE achieves higher MIG score for DSPRITES than 3-VAE, while its UNIBOUND score is low. As we saw Figure 2h. such case occurs when part of the redundancy Ryy; 2τ, z - ze, ze) is large. This observation indicates that Factor VAF effectively forces each variable to encode infor- mation of distinet factors (i.e., one- ve-ee redundancy small), while it fails avoid entangling the information over multiple variables (i.e., one-vs-all redundancy is large).
We can confirm that the redundancy is indeed large in FaE by computing the PID bounds. In Figure 3bl plot the aggregated bounds of (f(y); #ę z.e). R(y 24, 2,e), and C(yss ze, z.e). The plot reveals that Factor tends to encode the factors redundantly into multiple latent variables.
To understand the tasks and models more deeply, we evaluated the PID bounds for each factor in Figure 4 We can see that the factors are captured by the models in various ways. For example, 3- TCVAE succeeds to disentangle the position and scale factors in DSPRITES, while it encodes orien- tation synergistically. It may reflect the inherent difficulty of disentangling this factor in DSPRITES, as the image of each shape corresponding to orientation is chosen arbitrarily. These observations help to choose what kind of inductive biases to introduce into the models.
8

---

Published as a conference paper at ICLR 2022
DSPRITES
3DSHAPES
2.. C.4 K.C (d) JointVAE
(a) 3-VAE
(b) FactorVAB
(c) 8-TCVAE
Figure 4: Estimated PID terms of each factor in DSPRITES and 3DSHAPES. As in Figure 3b three orange bars in each plot show the range between lower and upper bound estimations of umique information (U), redundant information (R), and complementary information (C). See Figure in Appendix for larger version and Table 5 for qualitative interpretations.
FactorVAE, on the other hand, tends to encode factors redundantly. This indicates FactorVAE suc- ceeds to make individual variables encode each factor, while fails to prevent other variables from encoding the same information. JointVAE also encodes factors redundantly. While it fails disen- tangle all the factors, this is the only model that encodes the shape and orientation to single latent variables. This be viewed the effect of introducing discrete variable into the representation.
We can understand large redundancy as the models failing to make variables independent enough. The lower bound of redundancy Eq. 7 related to independence as
(T 22; ee [T2;; 2.e) (2z; mels)))+ S (
(11)
Therefore, a high redundancy lower bound indicates large mutual information [(x;; Z(e); i.e., the latent variables are highly dependent. We conjecture that FactorVAE, which approximates the total correlation DXL (p(z) II. p(zz)) by critic, fails to make the critic capture the dependency in enough in our experiments. Since the MIG score relatively high, the critic succeeds to capture pairwise dependency, while it fails to capture higher dimensional dependency. The high redundancy JointVAE can also be explained by the lack of independence; see Appendix for details.
As in Figure some metries have large deviations from the median. This is caused by the random- ness in training rather than in evaluation; see Appendix I for detailed analyses.
As we did for the toy model, we performed entanglement attacks in Figure 3c assess its effect on each metric in learned representations. We selected the best training trials of /- VAE and 3-TCVAE. Here, we only plot the results with the redundancy attack, as each metric already behaves well against the synergy attack. The plot reveals that BetaVAE and FactorVAE metrics do not detect the redundancy injected by the attack. MIG slightly decreases with the attack, which is not significant against the score variation between learning methods strongly reacts against the attack, indicating that it effectively detects the injected redundancy.
observed in Figure Ba UNIBOUND
6 CONCLUSION
We established framework of disentanglement analysis using Partial Information Decomposition. We formulated new disentanglement metric, UNIBOUND, using the unique information bounds, and confirmed with entanglement attacks that the metric correctly responses to entanglement caused by multi-variable interactions which are not captured by other metrics. UNIBOUND sometimes disagrees with other metrics VAEs trained with controlled datasets, which indicates that multi- variable interactions arise not only in artifitial settings but in learned representations. We found that VAEs trained with different methods induce representations with variety of ratios between PID terms, even if their disentanglement scores are close. a major future work to develop learning methods of disentangled representations the basis of these observations.

---

Published as a conference paper at ICLR 2022
ACKNOWLEDGMENTS
We thank members of Issei Sato Laboratory and researchers Preferred Networks for fruitful dis- cussions, and thank the reviewers for helpful comments to improve the work.
REFERENCES
H.B. Barlow. Unsupervised Learning. Neural Computation, 1(3):295 311, 1989. ISSN 0899-7667. doi: 10.1 (22.2/ccco.99.......2..
H.B. Barlow, T.P. Kaushal, and G.J. Mitchison, Finding Minimum Entropy Codes. Neural Compu- tation, 1(3):412-433 423422243,, 1989. ISSN 0899-7667. doi: 10.1 http://eco..999.1..333
Adam B. Barrett. Exploration of synergistic and redundant information sharing in static and dynam- ical gaussian systems. Phys. Rev. 91:052802, 2015. doi: 10.1 103/Poyseee....9111 0528022.
Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1589) 2013. doi: 10. 10.1109/TPAM..2013..50
Nils Bertschinger, Johannes Rauh, Eckehard Olbrich, Jürgen Jost, and Nihat Ay. Quantifying unique information. Entropy, 16(4):1161-2183 2014. ISSN 1099-4300. doi: 10. 10.3399/e16044111.
Ricky T. Q. Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disen- tanglement in variational autoencoders. Advances in Neural Information Processing Systems, volume 31, 2018.
James J. DiCarlo and David D. Cox. Untangling invariant object recognition. Trends in Cognitive Sciences, 118):333 331,341,,, 2007. ISSN 1364-6613, doi: tit.::.j.ic....7.66666
Kien Do and Truyen Tran. Theory and evaluation metrics for learning disentangled representations. In International Conference on Learning Representations, 2020.
Emilien Dupont. Learning disentangled joint continuous and discrete representations. In Advances in Neural Information Processing Systems, volume 31, 2018.
Cian Eastwood and Christopher K. I. Williams. framework for the quantitative evaluation of disentangled representations. In International Conference on Learning Representations, 2018.
Patrick Esser, Johannes Haux, and Björn Ommer. Unsupervised robust disentangling of latent char- acteristics for image synthesis. In Proceedings of the Inil. Conf. on Computer Vision (ICCV), 2019.
Conor Finn and Joseph T. Lizier. Pointwise partial information decomposition using the specificity and ambiguity lattices. Entropy, 20(4), 2018. ISSN 1099-43000. doi: 10.3390/e20040299..
Conor Finn and Joseph Lizier. Generalised measures of multivariate information content. Entropy, 22(2), 2020. ISSN 1099-4300. doi: 10.3390/e22020216
Shuyang Gao, Rob Brekelmans, Greg Ver Steeg, and Aram Galstyan. Auto-encoding total corre- lation explanation. In Proceedings the Twenty-Second International Conference Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pp. 1157- 1166, 2019.
Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: vae: Learning basic visual concepts with constrained variational framework. In International Conference on Learning Representations, 2017.
Irina Higgins, David Amos, David Pfau, Sébastien Racanière, Lofe Matthey, Danilo J. Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. CoRR, abs/1812.02230.. 2018.
10

---

Published as a conference paper at ICLR 2022
Wei-Ning Hsu, Yu Zhang, Ron J. Weiss, Yu-An Chung, Yuxuan Wang. Yonghui Wu, and James Glass. Disentangling correlated speaker and noise for speech synthesis via data augmentation and adversarial factorization. In ICASSP 2019 2019 IEEE International Conference Acousties, Speech and Signal Processing (ICASSP), pp- 5901-5905, 2019. 8683561.
doi: 10.1 N0.1109/CASSPP2011.
Zhiting Hu, Zichao Yang, Xiaodan Liang, Ruslan Salakhutdinov, and Eric Xing. Toward trolled generation of text. In Proceedings of the 34th International Conference on Machine Learn- ing, volume 70. pp. 1587-1596, 2017.
Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR), 2019.
Hyunjik Kim and Andriy Mnih. Disentangling by factorising, In Proceedings of the 35th Interna- tional Conference on Machine Learning, volume 80, pp. 2649-2658, 2018.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd Interna- tional Conference on Learning Representations, ICLR, 2015.
Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In 2nd International Conference on Learning Representations, ICI.R, 2014.
Zhiyuan Li, Jaideep Vitthal Murkute, Prashnna Kumar Gyawali, and Linwei Wang. Progressive learning and disentanglement of hierarchical representations. In International Conference on Learning Representations, 2020.
Joseph T. Lizier, Nils Bertschinger, Jürgen Jost, and Michael Wibral. Information decomposition of target effects from multi-source interactions: Perspectives previous, current and future work. Entropy, 20(4), 2018. ISSN 1099-4300. doi: 10.3390/620040307..
David JC MacKay, Information theory, inference and learning algorithms. Cambridge university
press, 2003.
Ari Pakman, Dar Gilboa, and Elad Schneidman. Estimating the unique information of continuous variables in recurrent networks. CoRR, abs/2102.00218, 2021.
Kyle Schick-Poland, Abdullah Makkeh, Aaron J. Gutknecht, Patricia Wollstadt, Anja Sturm, and Michael Wibral. A partial information decomposition for discrete and continuous variables. CoRR, abs/2106.12393, 2021.
Jürgen Schmidhuber, Learning factorial codes by predictability minimization. Neural Computation, 4(6):863 879, 1992. doi: 10.1162/neco. 1992.4.66999..
David Sigtermans. path-based partial information decomposition. Entropy, 22(9), 2020. ISSN 1099-4300. doi: 10.3390ee2200052..
Tycho M.S. Tax, Pedro A.M. Mediano, and Murray Shanahan. The partial information decom- position of generative neural network models. Entropy, 19(9), 2017. ISSN 1099-4300. doi: 10.3390/e1990447..
Satosi Watanabe. Pattern recognition quest for minimum entropy. Pattern Recognition, 13(5): 381-387, 1981. ISSN 0031-3203. doi: 10.1016/0031-220 1-2003(8 1990944-7.
Paul L. Williams and Randall D. Beer. Nonnegative decomposition of multivariate information.
CoRR, abs/1004.2515. 2010.
Shujian Yu, Kristoffer Wickstrøm, Robert Jenssen, and José C. Príncipe. Understanding convo- lutional neural networks with information theory: An initial exploration. IEEE Transactions on Neural Networks and Learning Systems, 32( 32(1)::35-444 2021. doi: 10.1109/TNNLS.222. 2968509.
Julian Zaidi, Jonathan Boilard, Ghyslain Gagnon, and Marc-André Carbonneau. Measuring disen- tanglement: A review of metrics. CoRR, abs/2012.09277. 2020.

---

Published as a conference paper at ICLR 2022
A
ADDITIONAL DIAGRAMS
I [lu v1, 02)
I(uvr))
I [lu 02)
U(u \ 02)
R(u; UI, U2)
U(u 02 v1)
C(u; U1, U2)
Figure 5: Alternative diagram of Figure ! in the band style following Figure 8.1 of MacKay 2003
B RELATIONSHIPS TO OTHER INFORMATION- THERRETTI METRICS
There have been several metrics proposed in the literature for information information-thhortiicc measurement of disentanglement.
2020 proposed four metrics, namely WSEPIN, WINDIN, RMIG, and JEMMIG. Among them, WSEPIN is computed based on the information gap [(x; ze m,) [2:; z) [x; ze). In the PID perspective, this quantity upper bounds the unique information of
For example. Do & Tran
x held by ze, i.e., [(r; ze ()) 2 ((;; ze Zie)- It is similar to the upper bound of unique informa-
tion that derived in 7 as
[vv =e) [TT ae 0 S T(N:: ze) 17 (U [wN:: z) - [
(12)
In that sense, WSEPIN is an upper bound of unique information, where is replaced with æ and the nomnegativity of redundant information is ignored. As WSEPIN does not handle generative factors separately, it is useful when the generative factors are unknown, while our approach provides more detailed information of (dis)entanglement when the generative factors are available.
RMIG is an extension to MIG, where the conditioning between yk and x is inverted so that one can compute the quantity with uncontrolled dataset. This approach may be extended for our framework as well, in a similar way to applying the inversion to MIG. We did not use this extension as we only deal with controlled datasets in this paper to contrast our approach with a wider variety of metrics.
JEMMIG is also an extention to MIG, which provides a measurement of how each variable captures only one generative factor. This aspect is also studied by [Easwwood & Williams) (2018) and Li cet al. (2020). As used a set of simple latent variables each of which either one- dinensnnnsi real variable or a categorical variable with a small number of arms (three or four depending on the dataset), expect that each variable does not capture much information of multiple generative factors. Indeed, we did not observe any latent variable selected for more than two generative factors, and when latent variable is selected for two generative factors, the overall disentanglement score is low so that the effect of duplicated selection is ignorable. We still consider it an interesting future work to extend our approach for analyzing entanglement of multiple generative factors into a single latent variable.
C DERIVATIONS FOR THE EXACT ANALYSIS WITH THE Toy MODEL
In this section, we provide sketch of analytically computing the metrics for the toy model we used We use the following formulae.
in Section 5.1
Let be Gaussian vector with covariance Σ. Then, its entropy is given by (u) 1
2 log(2nr El), where |-| is the matrix determinant. When we remove the l-th element from the remaining vector u, follows Gaussian distribution whose covariance matrix Ex is obtained by removing the f-th row and column from Σ. The determinant of Σ is the cofactor of Σ at the L-th diagonal element, i.e., Exe = (E With this formula, derive the entropy of u.e as H(u,) H (u)
+ log Σ -)ee--
A B C
Suppose that an invertible matrix Σ is written by blocks Σ
If A and its
D
Schur complement S D CA -BB are invertible, the determinant and inverse of Σ are
12

---

Published as a conference paper at ICLR 2022
Table 1: Exact values metrics for toy model. We use - 21 11 for the attacks. We do not normalize the metrics by (3ts) as y is isotropic and continuous; normalization by it just scales the results by a constant scalar. For the attacked models, we show the limit of a - oo to understand how well each metric reacts completely entangled representations.
Metric
(1 2010 o 119))
MIG
loog(1 + 22)
log
o2 22
log 1 a=+0²
+ log
1+
U lower bound (UNIBOUND)
log(1 + )
2 log ("
(I+a (1+a"
log a at
U upper bound
log (1 + +
log (1(44a((+00))
+ 0
loog
- 0
#=
written as
E- A AA-14 AA CCA
-A-
Σ |A| SS)
(13)
-S- -1
S-1
Similarly, and its Schur complement := A BD - 'C are invertible, then
-T-'BD- ---D-CTT- D + DD
T-1
Σ D T Σ-
(14)
Provided that z is a Gaussian vector with covariance E, we can derive the entropies of the attacked vectors. Recall that the redundancy and synergy attacks are defined as follows.
red
I aU 1
()(()
2
TThH
( aU
(
where is an orthonormal matrix, a > O, and 6 ~ NNO, 1). Here, 0 is the zero matrix. These again follow Gaussian distributions, whose covariance matrices are computed follows.
Cov(z =
I
Σ
I aU =
Σ
aEU
aU
0
aUE
I+
aU
Σ
0
0
(a"+IE
aU
Cov(z
I
0
I
aUTT
aUT
I
Using Eq 13 and Eq 14 we obtain their determinants and inverses.
Ia1 E-1 aU
-aT)
Cov(z [
Cov(zrod)-1_
Σ Σ-1
-ax
[Cov(z*m) = Σ
Cov(Z"")
-aU I-1
I U
Therefore, we obtain the following entropies for each [l....,LI.
1
H ( = log (2Te
(
log(2Tee (2Te) E (a² + (E ))))))
2
2
1
log (2 (1 + u²u Eu:)),
H (z
log (TT Σ)),
2
1
(
log log(2 (2 2 -ee)),
(
log (2Te) Σ )et)
2
1
H
log (mTe),
H (Z log (mm Σ (1 + a v Σ vv()).
2
(( = H (m
(2ge 2
(15)
Here, u, and Va are the 8-th columns of U and U, respectively. When use U-I these written ut V, 2 where (ei....,ez) the standard basis of RR.
1
R 11
2
13

---

Published as a conference paper at ICLR 2022
Table 2: Encoder and decoder architectures used in the DSPRITES and 3DSHAPES experiments. Data flow top to bottom. Conv4x4s2pl represents spatial convolution layer with kernel size 4x4, stride 2x2, and pixel padding at each side of the image. ConvT represents the transposed convo- lution layer. FC stands for fully-connected layer. The last fully-connected layer of the encoder outputs the parameters of the variational posterior distribution, the number of which depends on the model definition as follows. We used six latent variables in all models, which are all Gaussian ex- cept for JointVAE where one of them is replaced with a categorical variable. As Gaussian variables parameterized by mean and standard deviation while the categorical variables are parameter- ized by logits, the final feature dimensionality is 12 for the models except for JointVAE where the dimensionality is 13 for DSPRITE and 14 for 3DSHAPES.
Encoder
Decoder
Conv4x4s2pl, 32 channels
FC, 256 features
Conv4x4s2pl, 32 channels
FC, 64x4x4 features
Conv4x4s2pl, 64 channels
ConvT4x4s2pl, 64 channels
Conv4x4s2pl, 64 channels
ConvT4x4s2p1, 32 channels
FC, 256 features
ConvT4x4s2pl, 32 channels
FC, * features
ConyT4x4422p 1 or 3 channels
Recall that, in the toy model, the representation z is drawn from o where the generative factors y follow the standard Gaussian distribution. Therefore, the marginal distribution of the
representation is p(z) (z; O, (1 + o²)1). When the representation is conditioned by a single factor yk, it follows p( p(zlyx)
N(2 ykek, (1 + of) exee) ). By substituting the covariance matrices of these distributions to E in Eq 15 and computing their differences, we obtain the mutual
information terms as follows.
(140))
log
I+a² o²
(k: =0=00 f), (k: + f),
2 log 1+ a1- RP)
(k = E),
[yk: atod)
2
0
I log 1+02(1+02- Rz)
(1+0
(k + ()
2
log Ita0a
Lta + (1+a²) 2
(k: = {),
1+
[yye; 2(ee)
2
Iyyx 2red 2 =
log
1
log lto"
(I: # f),
2
a²
2
log o ta
+0
(k ===00 f),
Tyye 27(h)
2
T(yk; (ym = 0,
(16)
(k: # f),
[y): 2
0 log
(k: 1k=00 E), (k: + E), 0z
I(yx:
2 (t
1 log
(1+o (1+03)
(k: t),
3 log (1+o (++0))+++++++)
(k: # e)
12 log g²
Tyyx; 2re0)
Tyy:; 23n)
The metrics TableTarre are computed from these quantities. In addition, we can compute other partial information terms as follows.
1 1+ (1=(1+++))
Ryys: # gndd)
log
Clyni # ((a)) O,
2
1+
1
(1 (1+0 (1+ +
c R(ys
2g A 0,
Clys: 22th 2.(x)
log o2
2
We can observe that the redundancy and synergy terms effectively increase with the corresponding attacks.
D MODEL ARCHITECTURES AND TRAINING HYPERPARAMETERS
The architectures of the encoder and decoder used in the experiments are listed in Table 2 We used ReLU nonlinearity at each convolutional layer except for the final output of each network. In addition, we used for the critic in FactorVAE a feedforward network with five hidden layers each of which consists of 1,000 leaky ReLU units with the slope coefficient of 0.2.
14

---

Published as a conference paper at ICLR 2022
Table 3: Training hyperparameters used for DSPRITES experiments.
Model
3-VAE
FactorVAE
JointVAE
B-TCVAE
Batch size
64
64
64
2,048
Iterations
300,000
300,000
300,000
30,000
Adam
5 10
1 x 10-4
5 x 10-
1x 10
Adam Pl. D2
0.9, 0.999
0.9, 0.999
0.9. 0.999
0.9. 0.999
Critic Adam a
1 x 10-1
Critic Adam B1
0.5, 0.9
Discrete variable capacity C.
1.1
Continuous variable capacity C²
40
Regularization coefficient
B =
35
7 150
Table 4: Training hyperparameters used for 3DSHAPES experiments.
Model
B-VAE
FactorVAE
JointVAE
B-TCVAE
Batch size
64
64
64
2,048
Iterations
500,000
500,000
500,000
50,000
Adam a
10
1 x 10-4
x 10
x 10
Adam B1, 32
0.9, 0.999
0.9, 0.999
0.9, 0.999
0.9, 0.999
Critic Adam
1 x 10
Critic Adam B1,02
0.5, 0.9
Discrete variable capacity C.
1.1
Continuous variable capacity C=
40
Regularization coefficient
B =4 4
20
7
150
3 = 4
The hyperparameters used in training are listed in Table 3 and Table listed in the tables, we used the values suggested in the original papers.
4 For hyperparameters not
E FULL RESULTS FOR FACTOR-WISH WISE PID ANALYSES
We illustrate the estimated bounds of PID terms for each factor in Figure 6 whose small version appeared in Figure We summarize these results Table This table is made by observing and categorizing the plots in Figure into some patterns follows. Note that we ignore the error bars here.
1. If the lower bound of the unique information is larger than the upper bounds of the redun- dant and complementary information, mark the plot as disentangled. In this case, the model is determined as successfully disentangling the factor regardless of the concrete definition of PID. Note that the model does not necessarily learn the factor completely; see the figure for how much the information of the factor is uniquely captured by latent variable.
2. If the upper and lower bounds of the redundant information larger than those of the complementary information by a margin, mark the plot as redundant. In this case, the model entangles the factor in redundant way. As we analyzed in Section 5.2 this indicates that the latent variables are highly dependent.
3. If the upper and lower bounds of the complementary information are larger than those of the redundant information by a margin, mark the plot as synergistic. In this case, the model entangles the factor in a synergistic way. This may occur even if the latent variables are independent. We may require additional inductive biases to help the model disentangle the factor.
4. Otherwise, mark the plot as flat.
F POSSIBLE EXPLANATIONS FOR HIGH REDUNDANCY IN JOINTVAE
We observed in Table 5 and Figure 6 that JointVAE suffers from high redundancy in all the factors. To explain this phenomenon, review the training scheme of JointVAE (Dupont 2018). The
15

---

Published as a conference paper at ICLR 2022
floor
aoitre leee information
0.8
auttralzee informelion
0.8
3-VAE
0.4
0.4
omeetation position
ftoor
nortnelizn informelion
0.8
nottnalieed informelion
0.8
Factor
0.4
0.4
+
shage
scale
orientation
position
position
floor hue
wall hue
object hue
scale
orientation
norteelized Informetion
0.8
normelized Informetion
0.8
B-TCVAE
0.4
0.4
URC
shape
scale
orientation
position
position
floor hue
wall hue
object hue
scale
shape
orientation
normelized Informetion
0.0
notnelized informmtion
o.
Joint
0.4
0.4
00
URC U C URE URC (a) DSPRITES
(b) 3DSHAPES
Figure 6: Large version of Figure
16

---

Published as a conference paper at ICLR 2022
Table 5: Qualitative summary of PID decomposition for model-factor pairs. Each pair is explained by the following terms: disentangled: the unique information is larger than other terms; synergistic, redundant: the corresponding PID term is large; flat: no term exceeds others much, which indicates that all three terms are small (i.e., multiple variables contain distinct information of the factor) or both redundancy and synergy are large. Note that these qualitative analyses are only applicable to our experimental settings. In particular, high redundancy of VAE (marked by in the table)
may be caused by a capacity hyperparameter. See Appendix
G for details.
Dataset
Factor
8-VAE
FactorVAE redundant redundant flat
B-TCVAE flat disentangled
JointVAE
shape
flat synergistic
redundant
scale
redundant
DSPRITES
orientation
synergistic
synergistic
redundant"
position
synergistic
redundant redundant disentangled
disentangled
redundant"
position
synergistic
redundant"
floor hue
disentangled disentangled disentangled
redundant
wall hue
redundant redundant redundant
redundant
3DSHAPES object hue
redundant redundant redundant flat
redundant"
scale
flat
synergistic
redundant
shape
flat flat
flat synergistic
redundant"
orientation
redundant disentangled
redundant*
Table 6: KL terms and total correlation of latent variables learned by JointVAE. The values in parentheses show the standard deviation.
Dataset
(pl. ((i))
DKI. (p(22.1.
0, I)) Total correlation
DSPRITES
1.10 (+0.00)
40.00 (+0.04)
25.02 (#0..4)
3DSHAPES
(+0000)
39.99 (#0.05)
25.98 (+0.56)
VAi model consists of one categorical variable 21 and L-1 Gaussian variables Z2:L. Let (21)
be the uniform categorical distribution and pa(xlzz be the decoder to be learned simultancously. Then, the objective function of JointVAE is
f == Ep(z(z) [log pa((<<a)))) [z]] - Dra (==
- (e))
- cab,
which involves three hyperparameters: the regularization coefficient y, the capacity of the discrete variable Cı, and the capacity of continuous variables c2. Throughout training, 7 is kept constant, while the capacities C₁ and C2 are gradually increased and saturated at the predefined maximum values (C. and Cg. respectively) in the middle of training. As the capacities are positive at the end of training, this indicates that the KL terms, which include the total correlation of the latent variables (Kim& Mnih 2018 Chen et al. 2018), are large in the trained model. Therefore, as we discussed in Section 5.2, this model does not add pressure on the representation to be less redundant, which may explai the high redundancy See Appendix G the results with varying C.. whose results also support the above hypothesis as low capacities induce lower redundancy. This training scheme designed to align the amount information captured by discrete and continuous variables, which seem to be successful as observed, i.c.. it effectively captures the shape factor in both datasets compared to the other methods.
G INFLUENCE OF REGULARIZATION HYPERPARAMETERS ON DISENTANGLEMENT
We trained each model with varying hyperparameters (B for 3-VAE and (-TCVVE, for Factor- VAE, and the capcity of continuous variables C, for JointVAE) and investigated how the metrics as well as PID terms are affected by the regularization strengthe
To confirm that large capacity actually causes large dependency, we measured the KL terms and the total correlation of the trained JointVAE models. We summarize the results in Table 0 These values indicate that the KT. terms are actually close the capacity hyperparameters (see Table 3 and Table 4p. and more than half them are occupied hy the total correlation.
Forr VAF, we chose the final capacity of continuous variables instead of the regularization coefficient expect the capacity he more relevant disentanglement. The former controls the KI. divergence
17

---

Published as a conference paper at ICLR 2022
2.0
4.0
6.0
8.0
0.8
normalized information
0.6
6.4
0.4
0.4
malr L-ündBound
24
0.4
0.2
0.0
URC
URC
UR UR
igguooization coofficients
rejuraliation cofffidiens
(a) 3-VAE. The KL term coefficient 9 is varied.
10.0
20.0
30.0
40.0
mm mcChic-BetaAA
normalized information
0.8
0.6
0.
0.4
metiic +MG
0.5
0.5
0.2
8.
30
20
30
20
8.0
30
0.0
URC
URC
URC URC
(b) Factor VAE. The multiplier of total correlation 7 is varied.
2.0
4.0
6.0
8.0
Vebric-fattorVE
normalized information
0.8
0.5
2.8
0.6
0.4
MIC
metr U-villuun
0.4
24
0.2
0.0
URC URC
URC U
reguralizaiio confficintss reguraization coefficienss
(c) 8-TCAAE The multiplier of total correlation is varied.
10.0
20.0
30.0
40.0
0.8
normalized information
6.4
0.8
0.6
0.4
Patri-- MG
0.2
84
10
20
30
40
0.0
32
20
30
42
0.0
URC URC URC URC
(d) JointVAE, The final capacity of continuous variables Cs is varied. Note that higher capacity induces lower regularization effect.
Figure 7: Metrics and PID estimations for varying regularization coefficients on DSPRITES. For each model, the left panel shows how each metric reacts to changing the hyperparameter. The right panel shows the PID esitmations of resulting representations. In all plots, the horizontal axis corresponds to the regularization hyperparameter.
18

---

Published as a conference paper at ICLR 2022
2.0
4.0
6.0
8.0
matre-BeetaMM
normalized information
0.8
0.6
0.4
52
02
0.4
meti: MIG
0.8
metr Unlound
0.6
0.2
24
0.0
URC URC
URC UR C
reguratzation conficintts regiralzationn confficeets
(a) 3-VAE. The KL term coefficient $ is varied.
5.0
10.0
15.0
20.0
normalized information
0.8
0.6
0.2
0.2
0.4
HIG
0.6
0.2
0.4
0.0
URC C URC URC URC
requralzation coufficients
reqursitaation coafficienta
(b) VAC. The multiplier of total correlation 7 varied.
2.0
4.0
6.0
8.0
metri-BBebiit
normalized information
0.8
0.6
84
0.4
0.4
metric-N1S
0.5
mctr ceUniBound
0.6
0.2
84
0.0
URC URC
URC URC
Iegunatzation reginaioation
(c) 3-TCVAE The multiplier of total correlation is varied.
10.0
20.0
30.0
40.0
1.0
1.0
matricetactertie
normalized information
0.8
0.8
0.5
0.6
0.4
0.2
0.2
0.0
20
URC URC URC UR C
coemitilents
repuratiocion contictercs
(d) Join final capacity of continuous variables C, varied. Note that higher capacity induces lower regularization effect.
Figure Metrics and PID estimations for varying regularization coefficients on 3DSHAPES..
19

---

Published as a conference paper at ICLR 2022
The results for DSPRITES and 3DSHAPES illustrated in Fig and Fig respectively. From the left panels, we can observe that the UNIBOUND metric is positively correlated with the regular- ization strength It indicates that the regularization method introduced by each model positively contributes to disentanglement panels. They show some interesting effects of regularization against the types of entanglement. For example, in JointVAE. the representation has low redundancy when the capacity of continuous vari- ables is small, and the redundancy grows significantly when we increase the capacity. It indicates that a large capacity makes the model enforce each latent variable to capture details of the input images, ignoring how the information is redundantly captured by other variables.
the PID perspective. We also estimated PID bounds in the right
H LARGE FIGURES OF EXPERIMENTAL RESULTS
We put large versions of Figure and Figure in Figure 9 and Figure respectively, for finer rendering.
I TRAINING AND EVALUATION STABILITY
We illustrate the disentanglement scores of models trained with eight training seeds in Figure 10 and Figure [1] As each disentanglement metric involves sampling during evaluation, the evaluated score has some randomness even if fix the training random seed. This plot reveals that the deviation caused by randomness evaluating disentanglement metrics is much smaller than the deviation caused by randomness in training each model.
between the aggregated posterior of continuous variables and their prior at the end of training, while the latter controls the strength of enforcing the KI. divergence close the capacity. See the original paper (Dupont 2018 more details. In JointVAF, the hyperparameter controls the final KI. term; hence, smaller hyperparameter should in- duce more disentangled representation.
20

---

Published as a conference paper at ICLR 2022
metric BetaVAE
metric FactorVAE
metric BetaVAE
metric FactorVAE
0.B
0.8
metric score
metric score
0.6
model
0.6
model
AVAE
BVAE
0.4
FactorvAE
0.4
FactorvAE
0.2
TCVAE
0.2
TCVAE
JointVAE
JointVAE
0.0
metric MIG
metric UniBound
metric MIG
metric = UniBound
1.0
1.0
metric score
0.8
metric score
0.8
0.6
0.6
0.4
0.4
(a) Disentanglement
BVAE FactorVAE TCVAE
JointVAE
BVAE
FactorVAE TCVAE JointVAE
0.8
0,8
normalizeg information
normalized information
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
U
R
C
U
R
C
U
R
C
URC
URC
R c
U
C
U
R
C
(b) PID terms
BetaVAE
FactorVAE
BetaVAE
FactorVAE
1.0
1.0
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.6
BVAE
0.6
TCVAF
0.6
0.6
0.5
0.5
UniBound
0.8
MIG
0.5
UniBound
0.5
0.6
0.5
0.6
0.3
0.4
0.2
0.5
0.2
0.3
0.1
0.4
0.1
1
3
1
3
(c) Redundancy attacks
Figure 9: Large versions of Fig
(Left) Results for DSPRITES. (Right) Results for 3DSHAPES.
21

---

Published as a conference paper at ICLR 2022
metric BetaVAE
metric FactorVAE
metric BetaVAE
mebric FactorVAE
dieertangeeeen soone
divertannleeeent mettenn score
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
1.0
metric
dbbentangleneenn score
disente Wieettngleeeee mene Stare
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0
(a) B- VAE
(b) FactorVAE
metric BetavAA
metric FactorVAE
metric BetaVAE
metric FactorvAE
dbentenglemee score
deentenglemeet score
metric MIG
metric UniBound
metric MIG
metric UniBound
Obeettanglemen Score
dbeettengeeeee sccree
(c) 3-TCVAE
(d) JointVAE
Figure 10: Disentanglement scores before aggregating across different training seeds for DSPRITES.
We optimized parameters for each model eight times with different random seeds, whose scores illustrated by the eight boxes in each plot.
22

---

Published as a conference paper at ICLR 2022
metric BetaVAE
metric FactorVAE
mebic BetaVAE
metric FactorvAE
dbentengleeee score
deeetenglemeet score
metric MIG
metric UniBound
metric MIG
metric UniBound
dbeettangeenee Scote
dbeentanglernee nentt sccree
(a) B VAE
(b) FactorVAE
metric BetaVAE
metric FactorVAE
metric BetaVAE
mebric FactorvAE
3.0
Sberttnglemenn soore
deertengleement score
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
1.0
libentangleeens scote
disentaagleeeent stare
0.8
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0
(c) 8-TCVAE
(d) JointVAE
Figure 11: Disentanglement scores before aggregating across different training seeds for 3DSHAPES.
23


Reviews:
[{"review_id": "jb4YDO2wkK1", "reviewer": "Reviewer_kftq", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "[Strengths]\n- Applying PID for disentanglement analysis looks quite interesting, and the derivation generally looks thorough.\n- The paper proposes a new disentanglement metric to address the weakness of existing metrics, which do not capture multi-variable interactions properly.\n- The paper is well backed-up by a comprehensive supplementary material that seems to contain all the details required for reproducibility. The paper is fairly clearly written and easy to follow.\n\n[Comments]\n- It would be better to quantitatively and/or visually support the claim “Current metrics, however, may fail to detect entanglement that involves more than two variables, e.g., representations that duplicate and rotate generative factors in high dimensional spaces,” which is the most important motivation of this work.\n- Though simulating some entanglement attacks is worth trying, I am not sure whether these attacks actually and frequently happen in disentanglement learning. It would be better to find and show the cases similar with the redundancy and synergy attacks by analyzing learned features.\n- Unsupervised disentanglement learning with VAEs has been known to provide noisy results that are extremely affected by random seeds, architectures, and hyper-parameters (Locatello, 2019). Because the experiments merely relied on unsupervised VAEs with a single fixed architecture and few hyperparameter settings, I am not sure whether the results obtained under such a limited setup are reliable and whether this work could attract the attention of readers in the disentanglement learning field. For example, is the trend of the results in Figure 3 consistent for other architectures and settings? How did the authors set the regularization coefficients for disentanglement learning (beta, gamma) and other hyperparameters? Furthermore, I think drawing a conclusion described in Table 1 may be dangerous because the examined VAEs were tested under a very limited setup. I would suggest the authors to include (i) additional results under various settings and/or (ii) some results using semi- or weakly-supervised disentanglement models (Locatello, 2020a; Locatello, 2020b; Chen, 2020; Feng, 2018; Szabó, 2018; Kingma, 2014).\n-I am not sure whether enforcing nonnegativity in the RHS of eq. (5) using the max{0,*} operation is theoretically allowed. What happens if the max{0,*} operation is removed in the UniBound metric?\n- The results of JointVAE using the proposed metric seem weird. Though I checked the authors’ claims (This can be viewed as the effect of introducing a discrete variable into the representation; The high redundancy in JointVAE can also be explained by the lack of independence), they were unclear for me. In particular, the claim “As the capacities are positive at the end of training, this indicates that the KL terms, which include the total correlation of the latent variables (Kim & Mnih, 2018; Chen et al., 2018), are large in the trained model” may be incorrect because the KL term includes not only the TC term but also the data-latent MI and dimension-wise KL terms, meaning that a large KL does not directly indicate a large TC. It would be better to add more clear explanations (e.g., by actually measuring the TC terms). Furthermore, it would be helpful to see whether these results are also observed with other discrete-variable VAEs such as CascadeVAE (Jeong, 2019).\n- It would be better to include some qualitative results (latent traversal, t-SNE embedding space visualization) for visually understanding how feature spaces change according to the value of the metrics.\n- Abstract: asses -> assess\n\n(Locatello, 2019) Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations, ICML'19\n\n(Locatello, 2020a) Weakly-supervised disentanglement without compromises, ICML'20\n\n(Locatello, 2020b) Disentangling factors of variations using few labels, ICLR'20\n\n(Chen, 2020) Weakly supervised disentanglement by pairwise similarities, AAAI'20\n\n(Feng, 2018) Dual Swap Disentangling, NeurIPS’18\n\n(Szabó, 2018) Challenges in Disentangling Independent Factors of Variation, ICLRW'18\n\n(Kingma, 2014) Semi-supervised Learning with Deep Generative Models, NeurIPS’14\n\n(Jeong, 2019) Learning Discrete and Continuous Factors of Data via Alternating Disentanglement, ICML’19\n", "summary_of_the_paper": "The authors leveraged partial information decomposition (PID) for analyzing multi-variable interactions in latent representations. The PID framework shows that (i) the mutual information between a latent variable and a generative factor can be divided into unique, redundant, and synergistic information terms, and (ii) the uniqueness term corresponds to the degree of disentanglement. The authors also introduced a disentanglement metric by modifying MIG and conducted experiments with VAE-based models on two datasets.", "summary_of_the_review": "The main idea (applying PID for disentanglement analysis) looks interesting, and the underlying technical contributions seem good. However, the empirical results are very weak and not impressive. I thus find it difficult to argue for acceptance of the work.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "OBqts77mTlN", "reviewer": "Reviewer_25MQ", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "\nGood:\n\nThe paper is well written, the explanations are very clear. Not only the theory, but the experiments as well, e.g. Table 1 captures the high level experimental results nicely that otherwise would have been hard to decipher from the raw data (in the appendix)\n\nThe ideas behind the metric are intuitively appealing. The 3 main information terms are well defined and explained. The changes to MIG metric makes sense in this light, as it better captures which term should be counted positively and negatively in the metric (Fig 2)\n\nThe paper shows 2 adversarial attacks, where the latent vectors are perturbed with noise so each attack targets the redundancy and synergy term respectively.\n- It is shown theoretically on a toy example that UniBound responds better to redundancy attacks, i.e. converges to zero faster with more noise. For synergy attacks they are the same\n- It is shown on 2 datasets that UniBound performs better vs. other metricks against redundancy attacks.\n\nSeveral disentangling methods were compared not only with the proposed metric, but analysed w.r.t the a detailed decomposition of terms.\n\nBad:\n\n\"and the inference distribution p(z|x) and its marginals p(zl |x), p(z\\l |x) are all tractable (e.g., mean field variational models)\"\nPerhaps this is a strong assumption.\n\nMinor:\nThe notation could be a bit better.\n- The mutual information has a different font than than U, R, C. They should have the same font, as they are just a a renaming of the same function, but applied on different variables.\n- The above is confusing, when the 'interaction information of a triple' I is introduced with the same font as R, R and C.\n", "summary_of_the_paper": "The paper proposes a novel disentanglement metric 'Unibound'. To construct the metric the method uses partial information decomposition, which decomposes the the mutual information between the latent variables and the labels into a sum of other information terms with specific roles:\n- Redundant information\n- Unique information\n- Complementary information\n\nEach of these terms are precisely defined and an intuitive explanation os given.\nThe proposed UnibBund metric is defined similarly to MIG based on the above terms. The main difference is that UnibBund uses a better composition than MIG, that:\n- better captures our intuitive notion of disentanglement\n- handles adversarial representation attacks better", "summary_of_the_review": "The proposed metric in the paper is well designed and supported by:\n- intuition\n- theory\n- experiments vs. adversarial attacks and compared with other metrics\n\n--------- UPDATE ---------\n\nI have read the other reviews and rebuttals. The authors clarified some details and notations. I keep my score and recommend the paper for publication.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "NxXH5PQdKwH", "reviewer": "Reviewer_ezka", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n1) The problem that the authors want to address is important.\n2) The proposed metric is reasonable.\n\nCons:\n1) The paper misses important references and discussion of prior work.\n2) The presentation is not clear.\n3) Experiments are limited to only 2 toy datasets and do not really highlight the advantage of the proposed metric.\n4) The proposed metric seems not practical compared to existing metrics. \n", "summary_of_the_paper": "This paper proposes a metric for evaluating disentanglement inspired by Partial Information Decomposition (PID). It also proposes 2 “entanglement” attacks to highlight the advantages of the proposed metric.", "summary_of_the_review": "1) Novelty:\n- The idea of an information-theoretic analysis framework for disentangled representations was proposed by Do & Tran [1] but is not discussed in the paper.\n- The metric UniBound proposed in the paper has a very similar formula to those of the WSEPIN and WINDIN metrics in [1] but with the input image x replaced by the label y. I think the authors should discuss this similarity in their paper.\n- The authors should also discuss and compare their metrics with other up-to-date improvements of MIG such as JEMMIG [1] or MIG-sup [2]. Please check [3] for more detail of these metrics.\n\n2) Clarity in the presentation:\n- I think the authors abuse notations in their paper which causes difficulty in reading. For example, if $\\mathcal{R}$, $\\mathcal{U}$, $\\mathcal{C}$, $\\mathcal{I}$ are just mutual information, the authors should write all of them as $I$. I think the variables inside these terms are enough to define their meanings. \n- I don’t understand the meaning of the symbol $\\backslash$ in the mutual information. Is it equivalent to the symbol $\\|$ for conditional probability? If they are equivalent, I think the authors should write $\\|$ to make the mutual information more familiar to readers. Otherwise, they need to write explicit formulas of $\\mathcal{U}(y_k; z_{\\ell} \\backslash z_{\\ell '})$ and $\\mathcal{U}(y_k; z_{\\ell '} \\backslash z_{\\ell})$ since I cannot guess what they are.\n- The derivation of the proposed metric UniBound in Eq. 5 only uses Unique Information ($\\mathcal{U}$) and neither Redundant Information ($\\mathcal{R}$) nor Synergetic Information ($\\mathcal{C}$). I wonder how the authors can relate their metric to “redundancy” and “synergy”?\n- How the lower bound and upper bound of each term are related to disentanglement is not well discussed. I am almost unable to deduce anything when looking at Fig. 3b.\n\n3) Correctness and practicality of the method:\n- I don’t really understand why the latent variable $z$ is concatenated with a new vector in case of Redundancy and Synergy attacks? Doesn’t it double the length of $z$ and make $z$ no longer a suitable input for the decoder?\n- I am concerned about the practicality of the proposed metric as it is computed by summing over many small probability density values ($\\log\\sum_{x} p(z|x)$ in Eq. 8), which often leads to numerical instability if $z$ is high-dimensional. This is serious because incorrect results can cause incorrect interpretations. MIG and its improvements (JEMMIG, MIG-sup), however, do not suffer from this problem since these metrics use the conditional probability of a single factor $p(z_{i}|x)$, not all factors. This problem was analyzed in [1] (Appdx. A5). I would like to hear the authors’ explanation about this. I also would like to know the dimensionality of the latent code z used in their experiments as I cannot see this value in Table 3, Appdx. C.\n- Clipping $I(y_{k}, z_{\\ell}) - I(y_{k}, z_{\\backslash \\ell})$ to be >= 0 makes the metric unable to compare between models that have $I(y_{k}, z_{\\ell}) \\ge I(y_{k}, z_{\\backslash \\ell})$. This is problematic because $I(y_{k}, z_{\\backslash \\ell})$ is likely to be larger than $I(y_{k}, z_{\\ell})$. An example of this is the UniBound values for JointVAE in Fig. 3a.\n- In Fig. 3a, UniBound produces very different results for FactorVAE on dSprite and on 3DShapes. I would like to hear the explanation for this.\n\n[1] Theory and Evaluation Metrics for Learning Disentangled Representations, Do & Tran, ICLR 2020\n\n[2] Progressive Learning and Disentanglement of Hierarchical Representations, Li et al., ICLR 2020\n\n[3] Measuring Disentanglement - A Review of Metrics, Zaidi et al., 2021", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "5ZZ7lBzG0V", "reviewer": "Reviewer_mDLP", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: \nAnalyzing latent space and quantifying disentanglement is an open question in the representation learning field. The authors used PID to generalize the concept of information gap for multivariate analysis to assess the level of entanglement and disentanglement, which is rather novel and quite valuable. Deriving upper and lower bounds for the introduced partial information terms as a function of interaction information, $\\mathcal{I}(.)$, is original. The entanglement attacks design is not new but is well-justified for this work. \n\nStrengths: \n- The manuscript is well organized and well written. \n- The paper attempts to make a connection between the existing literatures in information gap theory and partial information decomposition to improve the entanglement analysis.\n- The experimental studies illustrate that the proposed metric successfully reveals the disentanglement decrement under entanglement attacks, while the comparable measures fail.\n\n\nLimitations: \n- Ablation study is a critical study for this topic. I expected to see detailed ablation studies, beyond the suggested entanglement attacks.\n- There is no analysis for robustness of partial information measures (the proposed bounds). \n- I think the traversal analysis is missing in this work, especially in entanglement attacks. It would be quite informative if the authors report some visual results.\n- I think the recent work by Do and Tran, ICLR2020, which formulates the disentanglement as another set of information-theoretic metrics is quite relevant to this work. It would be informative if the authors elaborate on those metrics as well and show which dimension(s) of information is missing in each framework.\n\nAdditional Comment/Questions:\n- For each VAE model, how does sampling affect the disentanglement metrics? Was the sensitivity of partial information calculation to the number of samples studied?\n- I wonder about entanglement attacks that may withstand being revealed by the proposed partial information measures. Any thoughts?\n- If I understand correctly, for JointVAE, the upper and lower bounds of partial information are quite close to each other (the bound is very tight). What does this exactly mean? Why does it happen mainly for JointVAE? \n- Did you consider both continuous and discrete variables when reporting the disentanglement? \n- Batch size for TCVAE is quite large. Is this the case in the original implementation as well?\n- The qualitative summary in Table 1 can be reported in the appendix and instead the bounds for each partial information can be reported. Or at least those qualitative measures can be elaborated in the main text.\n", "summary_of_the_paper": "The paper proposes an information theoretic approach to analyze the disentanglement. Using Partial Information Decomposition (PID) framework, the authors attempt to characterize the interaction among latent factors, correlation/disentanglement, for more than two latent variables. They generalize the existing mutual information gap (MIG) metric for multiple latent variables, by decomposing mutual information term into unique, redundant, and complementary information. The authors discussed that the pairwise correlation analysis endorses a portion of the redundant information as a measure of disentanglement, which should have been considered as entanglement. Additionally, they derive bounds for the partial information terms and demonstrate the limitations of existing disentanglement measures, i.e., BetaVAE, FactorVAE, and MIG in quantifying multivariate correlation in the latent space. The authors also designed two entanglement attacks to assess the disentanglement measures under injected correlation to the latent representation. The authors used two datasets, dSprites and 3dshapes to empirically substantiate their findings. ", "summary_of_the_review": "I think this is a good submission and the proposed information-theoretic framework is persuasive. However, I am on the borderline between acceptance and rejection. The main contribution seems limited and the ablation study including entanglement attacks only considers a few simple cases of entanglements. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The topic of the paper is the use of partial information decomposition (PID) for the analysis of interactions in latent representations.

All reviewers ended up appreciating the paper after a good extensive discussion with the authors. The numerical investigation is somewhat on the short side. One reviewer asks for more ablation studies and one reviewer asks for more investigation on real datasets to show the advantage of the method.

The paper is borderline. The theoretical development is fine. But one could argue that the paper could benefit from some more work on the experiments. However, the main points of the method is in place and further validation of the method can be left for future contributions.
Decision: Accept (Poster)