Identifying Informative Latent Variables Learned by GIN via Mutual Information
Authors: Chen Zhang
Yitong Sun
Mingtian Zhang
Venue:  (2021)

Abstract: How to learn a good representation of data is one of the most important topics of machine learning.  Disentanglement of representations, though believed to be the core feature of good representations, has caused a lot of debates and discussions in recent. Sorrenson et al. (2020), using the techniques developed in nonlinear independent analysis theory, show that general incompressible-flow networks (GIN) can recover the underlying latent variables that generate the data, and thus can provide a compact and disentangled representation. However, in this paper, we point out that the method taken by GIN for informative latent variables identification is not theoretically supported and can be disproved by experiments.  We propose to use the mutual information between latent variables and the auxiliary variable to correctly identify informative latent variables.  We directly verify the improvement brought by our method in experiments on synthetic data.  We further show the advantage of our method on various downstream tasks including classification, outlier detection and adversarial attack defence.

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IDENTIFYING INFORMATIVE LATENT VARIABLES
LEARNED BY GIN VIA MUTUAL INFORMATION
Anonymous authors Paper under double-blind review
ABSTRACT
How learn good representation of data is of the most important topics of machine learning. Disentanglement of representations, though believed to be the core feature of good representations, has caused lot of debates and discussions in recent. Soorenson et al. (2020), using the techniques developed in nonlinear independent component analysis theory. show that general incompressible- networks (GIN) can recover the underlying latent variables that generate the data, and thus can provide a compact and disentangled representation. However, in this paper, we point out that the method taken by GIN for informative latent variables selection is not theoretically supported and can be disproved by experiments. We propose to use the mutual information between each learned latent variables and the auxiliary variable to correctly identify informative latent variables. We di- rectly verify the improvement brought by our method in experiments synthetic data. We further show the advantage of our method on various downstream tasks including classification, outlier detection and adversarial attack defence both synthetic and real data.
INTRODUCTION
Representation learning is arguably one of the most important area in machine learning. Many re- searchers believe that being able to extract useful and interpretable features is a crucial advantage of deep networks over other learning models. A data representation can be obtained either via supervised learning task or an unsupervised learning task. The former one includes the popular ImageNet pretrained backbone in computer vision tasks, while the latter one mainly consists of generative models like variants of VAES, GANs and flow-based models. Among all the genera- tive models, VAEs and flow-based models can naturally output the representation of data or even their density, which is convenient for the representation learning purpose. Moreover, supervision information of labels can be integrated into generative models to further improve their performance. General Incompressible-foov Networks (GIN, Sorrenson et al. (2020)), the model we considered in this paper, falls into this case.
Disentanglement is a widely discussed concept by many representation learning works. However. the best of our knowledge, has not been given a widely accepted definition (Bengio al. 2013 Higgins et al., 2018). Many disentangled representation learning algorithms focus on recovering the independent latent variables that generate the data (Burgess al., 2018 Chen al. 2018b). How- ever, Locatello et al. (2018) show that without more assumptions than independence among latent variables, impossible recover them from the observation of data. This result is equivalent to the con-identifiability of non-lineat independent component analysis (ICA) (Comon 1994). Actu- ally, any assumptions solely on the latent variables' distribution without referring to the observable data is not sufficient for the identifiability (Hyvarinen and Morioka 2016 Khemakhem al. 2020).
A set of sufficient conditions is proposed under the framework nonlinear ICA by Khemakhem et al. (2020). The core condition requires that the data, denoted by x, generated by latent vectors z through a generative model p(x z), and conditioned on an auxiliary variable u. the entries of are independent and follow some exponential family distributions. This be expressed by the

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following formulas.
Qi(zi)
*
(z.u(Z)u) II 1=1 (u)
exp
Σ Tes Tes(ci)essu)
(1)
= (z) 6.
In Eq. and 2 E IR" and x E Rd. Usually people believe that n < d; that is, the data are distributed near low dimensional manifold embedded in very high dimensional space. With both u and x being observable, the identifiability of z can be established. Moreover, Khemakhem et al.] also show that the latent variables and the generative function can be learned by optimizing the ELBO using a special VAE, called iVAE (i for identifiable).
However, iVAE assumes that all the latent variables correlated with and it requires the correct number of latent variables for iVAE to work, which is usually not known. Sorrenson et al. (2020) consider the generative model with Eq. replaced by
x = =2)
where I' is invertible, (z)x(Z)u) is a Gaussian density supported on Rn with free expectations and diagonal covariance, and p(c), supported on Rd-- is free from u. Entries of z are refered to as informative latent variables, while entries of are treated noise. They prove that is identifiable up to a permutation and scaling. Based on this identifiability result, [Sorrenson et al. propose to learn the informative latent variables using flow-based model, called the General Incompressible-fie -fwoow Networks (GIN). The output of GIN model, w = gxx). is a vector with its dimension equal to dim(z, E), since both / and y are invertible. We, hence, need method to identify the entries of w that estimate 2.
Sorrenson et al.) propose variance based criterion for the informative latent variables selection. Specifically, they select wi's with large variances as estimates to z and treat the remaining entries as solely related [Sorrenson et do not clearly explain the logic behind this criterion. For the following reasons, we think this criterion is not guaranteed to work all the time:
1. Note that volume-preservie transform cannot preserve the variances of the input data, because golume-preserving constrains the determinant of the Jacobian instead of the size of each singular value.
2. According to the identifiability theorem in Sorrenson et al. 2020)) each can only be estimated up an affine transform, which means that the variances of informative we's 's can be smaller than the variances of non- non-informative ones.
Nevertheless, they show that the variance based criterion correctly identify the ground truth informative latent variables experiments on synthetic data. However, their experiments, the noise latent variables all have very small variances. When reproducing their experiments, find that if we increase the variances of e, but which are still less than the variances of z), this criterion will fail.
On the other hand, the identifiability result in Sorrenson et al. (2020) implies that the entries of w can be categorized into two classes, entries one class are linearly related entries z, while entries in the other class have distributions that are irrelevant to u. This observation inspires us to scrutinise the notion of informative latent variables under the generative framework described by Eq. The key difference between informative and non- informaarte latent variables is not their variances, but whether they are correlated with the auxiliary variable u.
Following this new perspective, it is natural to choose the mutual information between wy's and as the criterion for identifying informative latent variables, which we call it the mutual informa- tion (MI) criterion. In this paper, we compare the MI criterion with the previous variance based criterion (VAR) using abundant experiments on various tasks, including disentanglement quality on synthetic data, classification performance real data, outlier detection capability, and robustness under adversarial attacks. In all these cases, the MI criterion shows superior performance over the VAR criterion.
The main content of the paper is organised as follows. In Section 2, we briefly review the iden- tifiability result of Sorrenson al. (2020), and show why the mutual information work for
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informative latent variables selection. In Section 3, compare the performance of the MI criterion with that of the VAR criterion by experiments on synthetic data generated following the theoretical assumptions. In Section 4, we demonstrate the impacts of different informative variables selection criteria on classification, outlier detection, and defence against adversarial attacks by experiments on EMNIST (Cohen al., 2017). In Section 5, we discuss some other interesting discoveries related to the representation learning under the framework of nonlinear ICA, And finally, we conclude our results and summarise some future working directions in the last section.
THEORY OF NONLINEAR ICA AND MUTUAL INFORMATION CRITERION
Since our work is improvement based on GIN, we first review the main result of Sorrenson al. (2220) with a detailed elaboration on the relation between the learned representation w and the auxiliary variable u, and then make use of this property derive our mutual information criterion. Due to the length restriction of the paper, we refer interested readers to Appendix A for formal assumptions and statements.
With the data generation model defined by Eg and The proof of the identifiability theorem in Sorrenson al. (2020) mainly consists of the following three steps:
1. Start with (x u) Pa (x u). where the conditional marginal data distributions are determined by Eq. and Eq. with two possibly different transforms, latent variables and
parameters; 0 (T, (22, E), in the generation process and Φ (g. X() in the estimation process.
2. Then eliminate terms in Eq. that are irrelevant to the auxiliary variable u.
3. And finally obtain the relation between the statistics T(z) and T(w),
T(z) = I. L''(w(((W) +c,
where
(11) (1o)
)1,2(01) )1,2(u0)
An.a(u1) (
(1,1(Unk) - (u,)
)1,((uak) )1,2(u0)
(a,k(un)) - An (No)]
(ws) (u,)
X1a(u1) (uo)
May(us)) Na. (uo)
L'=
(M (mn)
M12(mn)) N12(10)
and entries of c are functions of u.
Since the latent variables conditioned on u are assumed to be distributed as Gaussian, T(z) (z, z² Their main theorem shows that after appropriate rearrangement of entries of z,
nx nx(d=n) R)
L- D
0
0 D
O
0
where D is an n n diagonal matrix with non-zero entries. This means that after rearrangement, the first n entries of w are just scaling of z. For the rest d - n entries, denoted by Wa+lids their connection with z cannot be justified by Eq. since the corresponding coefficients are zeros. Their relation with u, however, is clear. Since entries in L L corresponding to Wa+lid are all zero, we know that Xy (w) must equal to M/ (uo) for i [n + 1, d]. j = 1,2 and E [1, nk). This means that the distribution of w; for E [n + 1, remains unchanged no matter what is the value of u.
Based on their result, we know that an informative w; equals to some 2j after affine transform and thus has non-zero mutual information with u, while non-informative w; has zero mutual information with u. Therefore, the mutual information between each w; and can be used as a reliable indicator for justifying whether w; is an informative latent variable. On the other hand, as we explained in the introduction, the composite mapping g ff does not necessarily preserve the scale of and e. Even if e have much smaller variances compared to z, it is not necessarily true that the variances of (w; )d 1 less than those of {w;) We demonstrate this by experiment in the next
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section. Therefore using the size of variances of {w, V for informative latent variables selection can result in serious problems.
We summarise the algorithm for the MI criterion in Alg. The mutual information between w; and u is computed based on the samples by directly calling the function provided in the python package sklearn. To choose the threshold for mutual information decay, one employ exist- ing techniques for truncation, i.e., maximum curvature, sharp drop, etc. The computational cost of mutual information is usually huge for high dimensional random vectors, but for that between con- tinuous scalar random variable and a discrete random variable, the computational cost is negligible compared to network training time.
Algorithm 1: Informative Latent Variables Selection (MI)
input Learned latent variables {w; [= auxiliary variable u and initialise T output: Set of indices of informative latent variables T
for i <-= to do
L Evaluate [ww;; u):
Choose thresholds 17 > 0 according to the decay of (Iw,; u) 3)
for i <- to n do
if I(w;; u) > 1 then Add i into T:
3 PERFORMANCE ON SYNTHETIC DATA
In this section, we demonstrate the performance of the MI criterion on synthetic data, compared to the VAR criterion. The synthetic data are constructed similarly with the GIN paper. The data space is R and so is the latent space. Within the 10 latent variables, there are only 2 informative ones. These two latent variables 21, 22 are distributed as a Gaussian mixture with components, each of which, labelled by the auxiliary variable u, has different expectation and diagonal covariance matrix. The extra 8 latent variables are treated as noise, which follows independent Gaussian distributions. An invertible function f _0 + I. maps each latent vector sampled according to the distribution described above to a data point x. our experiment, the statistics of the Gaussian mixture are randomly generated (see Appendix DD. the noise variables are distributed as the standard Gaussian, and the invertible function is simulated by a flow model with random parameters. Note that in our data generating process, the variance of the noise is still less than that of informative variables, but not as small as in Sorrenson et al. (2020). Figure shows 4 examples of the distributions of the 10 dimensional latent vectors projected to different choices of variable pairs. When they are projected to the correct informative dimensions, the illustration shows a typical Gaussian mixture shape, as shown in the left subfigure. The other three subfigures show the distribution of latent variables when only one or none of the projection directions is informative. Figure 2 x projected to different choices of dimensions.
shows the distribution of data
Projecioo (0.
Projectio
Projection
Projecioon
10
Figure 1: Four 2D-projections of the distribution of latent variables (2, Projection (i,j) means the 2D-projection to the i-th and j-th dimension. (0, 1) are the informative latent variables z. Points sampled from 5 different Gaussian components in the Gaussian mixture are dyed by different colours.
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Projection 7)
Projedim
Projection
Peopution C,
-9
Figure 2: Four 2D-projectioss of the data f(2,
To learn the representation from data, we train GIN model by minimising the following objective:
(g-1(x; 0) pi())) + log(or(u))) )
C(O, 14, or = E(C,))ED
(5)
(0
Note that here both the weights of the flow and the statisties of the Gaussian mixture (μ, o)) are tuned during training, Another form of objective suggested for the GIN model in the original paper to replace trainable (Ju. σ)) by batch statisties. Consistent with the report of the original paper, we find that setting (μ,, ()) to be trainable shows similar performance with equating them to batch statistics. However, trainable (n, or) match the theoretical framework better, avoid large variation caused by small batch size, and save the computational cost of evaluating the statistics for density estimation during downstream tasks. Therefore, in the main content the paper, we will always consider the objective given by Eq For the experiment results of the alternative way, see Appendix C Other specific choices of parameters for the training include: batch size at 240 and initial learning rate 10 with multiplicative decay rate 0.1.
After training, we compute both the variances of the features in the representation w and their mutual information with the auxiliary variable u, and plot them in descending order; Figure 3 If we select the two latent variables with the largest variances and plot them, we can see that their distribution is different from the informative latent variables shown in Figure [ According the
MI criterion, we select the two variables with the largest mutual information with u. They almost perfectly reproduce the distribution of the 2 informative latent variables. If we do not know the correct number of informative variables, we can estimate by truncating the number of features in the representation according to the magnitude of mutual information values. In Figure we can see that the sharp drop occurs exactly between the second and the third variables from left. This experiment shows that the GIN model can successfully learn the informative latent variables as indicated by the theory, but we need the MI criterion to correctly identify them.
Recmsery by
Rocovary by
180
The Irte. Latan
Sardar Deciation
Motual Coliooi
Info. Liteni
Bank
(a)
(b)
(c)
(d)
Figure 3: Performance of the VAR criterion and the MI criterion on the informative latent variables identification on synthetic data. a) The distribution of the 2 features with the largest variances; b) The standard deviations of the 10 features (black) of and the true latent variables (grey) in descending order. The numbers are the indices of features for convenience of comparison. c) The distribution the 2 features with the largest mutual information with u. d) The mutual information of the 10 features (black) of and the 10 true latent variables (grey) in descending order.
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We further demonstrate the importance of identifying the correct informative latent variables for the classification and outlier detection tasks. Note that in our experiments is the label in the classification task. For the classification task, we evaluate the posterior probability Ps(u x) for each test example using the features of the representation selected into the set S according to either MI criterion the VAR criterion. Specifically, the posterior distribution of label is defined as
(6) where Ws is the vector of selected features, p(wslu) is derived from p(w]u) that learned by the GIN model and p(u) is the distribution of labels (uniform distribution of 5 classes). In Figure we can see that using the two latent variables identified by our MI criterion, the classification tasks can be solved almost perfectly, while the original VAR criterion is not competent at all.
ps(u|x)= p(u ()))s)) p( p(Ws|u)p(u)
cod sarppess
100
Chasitaain Acueny
3a Accenayy 1
0371
1013
Indo Laten
MUR0C
asympratie bourd
Asyrnptotic hound
AUROC VAR AUROC
secensy
acuuacy
Top
Runge
Top Truncaiinn Range
23
Top
Figure 4: Performance on synthetic data classi- fication: MI criterion VAR criterion. The axis represents the number of features selected for the ps( x) calculation according to the MI and VAR criterion respectively; the y-axis for the accuracy. The asymptotic bound is the theoretical upper bound for the classification ac- curacy. subfigures show different trials.
Figure 5: Left: The illustration of out-of- distribution samples (in red, from a mixture of two Gaussians) samples from ground truth Gaussian mixture in the latent space projected to the informative dimensions. Right: The AU- ROC values for different number features se- lected by the MI criterion and the VAR criterion respectively.
For the outlier (or -nttof-diitrrbutiin sample) detection task, we sample outliers in the latent space from two Gaussians far apart from the high-density area of normal latent samples, shown in Figure []] and then send them through the generator to obtain the data outliers. The test set is constructed by mixing nearly equal number of normal samples and outlier samples. We choose the density evaluated on the chosen w; 's as the criterion for detection. The values of area under receiver operating characteristic (AUROC) in Figure [ reflect the effectiveness of the criterion in outlier detection (Hendrycks and Gimpel 2016). The MI criterion again shows significant advantages over the VAR criterion.
In both the classification and the outlier detection tasks, it seems that involving all the latent variables can still work well, though we may pay huge computational price when the dimension of data high. The situation is different in practice. When too many noise variables are involved in the density evaluation, the signal provided by the informative latent variables may not be able to outstand. This becomes clear in our experiments on EMNIST in the next section.
PERFORMANCE ON EMNIST
In this section, we investigate the performance of the MI criterion on the EMNIST dataset. The models are trained on the EMNIST-digits dataset with the same network architecture adopted by Sorrenson et al. 22200) and the objective function shown Eq 5 For this experiment, the data E 222x22 the digits images, the auxiliary variable u is the label of digits and the learned feature vector w 784 dimensional.
The real dataset by no means satisfies the exact assumptions required by the identifiability theorem. And hence experiments on real datasets can help us understand how much of reality these assump- tions capture. Since the underlying latent variables are not available for real datasets, cannot directly verify the performance of the representation learning and the informative latent variable se- lection algorithms by visualising their projected distribution. A common method is to visualise the

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sample generation quality by traversing the latent space to demonstrate the performance. But this is still very qualitative and arbitrary to subjective interpretation. Instead of these visualisation meth- ods, we choose three downstream tasks: classification, outlier detection and adversarial defence, to measure the performance of the algorithm.
The classification tasks are solved in similar way in last section by computing the posterior prob- ability of labels conditioned on the test samples using features selected by each criterion. The result shown in the left subfigure of Figure 6 With 30 latent variables identified by the MI criterion, the classification accuracy can reach 94.15%, while using VAR criterion cannot gives us satisfactory accuracy. Different from the phenomenon we observed on synthetic data, the accuracy degrades when we select too many features that have small mutual information with the auxiliary variable. This indicates that the noise in the representation extracted from data may hurt the performance of downstream tasks, and thus accurately selecting the informative variables is crucial. From the right subfigure, we can see that the mutual information curve has sharp decay point, which corresponds to the number of latent variables that gives the best classification performance.
Clasiitcaiinn Ascamacy
Value
bod 200e
MI Rucurnc
10
MI dansy
MI ROC
VAR accuracy
VARR docay
VAR ROC
Top Truncation Runge
Bank
Figure 6: Left: the classification accuracy on EMNIST-digits of
Figure 7: The ROC curves using
the MI criterion and the VAR criterion with different possible top truncation within the full range. Right: the decay curves of mu-
the density value (on the learned Gaussian mixture) threshold for outlier detection task, MI30
tual information and standard deviation of the estimated latent. The truncation for best accuracy: 30 for MI; 300 for VAR.
versus VAR300.
In contrast, recall that on synthetic data, involving noise variables does not hurt the classification accuracy. Our conjecture that the GIN model trained on the synthetic data almost perfectly disen- tangled the informative and noise latent variables, which can be seen from the sharp decay of mutual information in Figure [3] Since the noise features in the representation contain little information of u, they cannot impact the calculation of posterior probability p(u x). Therefore, in this case, involving noise variables does not hurt the performance. However, for the model trained on real data, the features it learned are not all informative at the same level. We can see this from the right subfigure of Figure where many latent variables have mutual information with u around the level of 0.1. These variables may not be able provide more information about the labels than the first. 30 variables, but they may affect the predicted labels. In particular, if the marginal distributions of these variables have small variances; that is, all components are clustered close with each other, then a small value shift on these variables will cause a huge change of density, which can dominate the overall density and change the predicted label. This also explains why the variance criterion works better when more than 200 variables are included: The variables with relatively large variances are not likely to strongly affect the overall density. This conjecture can be further investigated in the future. Nevertheless, it is worth noting that as the low mutual information entries are very unlikely to encode additional information than those with high mutual information, and thus we should not include them for downstream tasks in principle.
To investigate the outlier detection capability, we use the model trained on EMNIST-digits to distin- guish EMNIST-eetters from EMNIST-digits samples. Using the density as the detection threshold, we plot the ROC curves for the 30 variables selected by the MI criterion, referred to as MB30, and the 300 variables selected by the VAR criterion, referred as VAR300. The number of latent vari- ables are chosen at the best classification accuracy level. Figure shows that MI30 dominates the VAR300. The AUROC value of the MI30 is 0.763, while the value of VAR300 is 0.591.
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Adversarial examples are extreme case of outliers. These examples deviate from normal samples so little that human vision cannot identify the difference, but they can fool the classification model and change its prediction. Li al. 2019) shows that conditional variational autoencoders are more robust than discriminative models against adversarial attacks when used together with the Bayes rule for the classification purpose and carefully designed defence mechanism. Here we are trying to show if the GIN is a better model for adversarial robustness than other probabilistic models. Instead, we want to show that using the mutual information criterion can help GIN models achieve better adversarial robustness.
In the experiment, we attack the model with the widely-used fast gradient sign method (FGSM) [Gooffellow et al.] 2013). By attempting adversarial attack at different levels, we find out that with- out applying defence, the classifier based on ps(" (u x) is not robust no matter for MI30 or VAR300. For the model using MI30, the optimal attacking strength is 0.02. As shown in Table [ at that level, the prediction of the model can be changed by nearly 99% adversarial examples. At the same level, the model using VAR300 can only be successfully fooled by 56% examples. However, we cannot simply conclude that VAR300 should be preferred than MI30 because its clean accuracy is much lower and 0.02 is not the optimal attacking strength for VAR300. To defend the adversar- ial attack, we define the following rejection rule: If ps( (w(x)lu) < ( ko /u) (here k = by our choice) for all label u. the criterion reports an adversarial example and refuses classify it. Note that for MI30 and VAR300, their subset S of features are different. At the level = 0.02, this defence strategy on MI30 can recognise about 94% attacks and thus only 5.8% of all attacks can succeed. If the attacker wants to bypass the defence, it has to choose a much weaker attacking strength and the overall success rate is still quite low, This is the case for MI30 when 7) 0.001 as shown in Table However, if we use VAR300, such a defence strategy has almost no effect at level 7 = 0.001 and very limited effect level 1) = 0.02. An interesting observation on the no attack case (1) = 0) is that the error rate after applying the defence module is slightly lower than the error rate without defence. Though it does not mean that the overall classification accuracy is improved because around 1 0.916% 0.084% test samples
screened out and do not get their predicted labels, it does imply that these test samples are overall harder for the model, which is not confident in its predictions on them. For VAR300, did not observe such effect. All these results indicate that for defending adversarial attacks, features selected by the MI criterion are more reliable.
Table 1: Adversarial defence performance of MI30 and VAR300. Error rate is the frequency of incorrectly classified samples. Pass rate is the frequency of samples passing the adversarial sample detection. Passed error rate is the frequency of incorrectly classified samples within the samples which pass the detection. Error rate with defence is the product of pass rate and passed error rate.
w/o defence
w/ defence
MI/VAR
error rate
pass rate
passed error rate
error rate
n 0.0585/0. 1250
0.9155/0.9720
0.0490/0. 1325
0.0448/0. 1288 0.0903/0.2033
7 = 0.001
0.1130/0.2061
0.9129/0.9758
0.0989/0.2086
77 0.02
0.9899/0.5577
0.0594/0.3140
0.9827/0.7865
0.0583/0.2470
DISCUSSION
In the experiments on synthetic data, we have two interesting discoveries. First, when we use GIN to learn the synthetic data, not only the informative variables can be successfully recovered as pre- dicted by the theory, but also can the noise variables be learned. Actually, the features in the learned representation w very close to a permutation of the true latent variables (z, c). This demon- strated in Figure 9 by plotting the correlation matrix between w and (z, <). This result indicates that the identifiability theorem established by the current nonlinear ICA theory may be extended to a stronger version. In the identifiability theorem of linear ICA (Comon) 1994). the transforms allowed are restricted to linear ones while distributions of data are only required not to be Gaussian. Com- pared to the linear ICA theory, current nonlinear ICA theory requires stronger assumption on latent distributions, which have to be in an exponential family, but has no constraints on the transforms. Note that in the synthetic data experiments, both the generating and the learning models are flows.
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If we restrict the nonlinear transform certain types of neural networks, we may be able to drop the exponential family assumption on their latent variables but still get identifiability.
More than the correct recovery of noise variables, variables are approximately preserved, in particular when the variances of noise variables are much smaller (10-1 for the following example) than those of the informative variables; see Figure 8 This may be caused by the fact that the greatest and least singular values of the Jacobian of flow-based models both upper and lower bounded. Because of this phenomena, the variance criteria can find out the correct informative variables for low noise scenarios. Such a variance-preservee stable when the variance of informative variables and noise have closer scales. Therefore, the mutual information criterion has more reliable performance in informative latent variables selection.
even find that the variances of latent
property
Infommatien FINE
Recnunry by VAR
Viriace
Beewwer by hil
Starra Intrrmation
05
1-1
4-
02
(a)
(b)
(c)
(d)
(e)
Figure 8: a) Ground truth informative latent dims. b) The top-2 features in w by the VAR crite-
rion. c) Standard deviations of 2; and w, ss. d) The top-2 features by the MI criterion. e) Mutual information of z and w.'s.
Granad Trash Latn P
Trshh
Groord
05
Edimutell
Strmaee Latent
Extimaled
Figure 9: Four examples of correlation between ground truth latent (by rows), including both infor- mative and noise dimensions, and estimated latent (by cols) of four different trained GIN models.
6 CONCLUSION
Based on our theoretical analysis and extensive experiments on synthetic and real data, we conclude that for any GIN type of models, the mutual information between representations and the auxiliary variable is better criterion for informative latent variables selection than the previously proposed variance criterion. We also show that by selecting the correct features in the representation, the GIN model can perform better on the classification, outlier detection and adversarial defence tasks. The capability of GIN models perfectly recovering of all latent variables up to permutation and a bounded scaling synthetic data indicates the possibility of a stronger version of identifiability of nonlinear ICA using flow-based models. More importantly, the perspective that the auxiliary variable provides a context for the informative latent variables selection eliminates the boundary between informative latent variables and noise. It is thus possible that different auxiliary variables define different group of informative latent variables. This will lead to representation of data with finer structure.
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L. Dinh, J. Sohl-Dickstein, and S. Bengio. Density estimation using Real NVP. arXiv:1605.08w8 088803 [cs, stat], Feb. 2017.
08803 arXiv: 1605.08803.
I. Goodfellow, J. Shlens, and C. Szegedy. Explaining and Harnessing Adversarial Examples. arXiv: 1412.6522 [cs, stat], Mar. 2015. 1412.6572.
6572 arXiv:
D. Hendrycks and K. Gimpel. A Baseline for Detecting Misclassified and Out-of-Distribution Ex- amples in Neural Networks. Nov. 2016. URL https: http//pppeevveeeeeee net. formm?id Hkg4T19x11
I. Higgins, D. Amos, D. Pfau, S. Racaniere, L. Matthey, D. Rezende, and A. Lerchner. Towards Definition of Disentangled Representations. 18Xiv:1812.081 22.02220 [cs, stat], Dec. 2018. URL http:://axii: org/ abs/ 02230
arXiv: 1812.02230.
A. Hyvarinen and H. Morioka. Unsupervised feature extraction by time-contrastive learning and nonlinear ica. Advances in Neural Information Processing Systems, pages 3765-3773, 2016.
J. Jacobsen, A. Smeulders, and Oyallon. i-revnet: Deep invertible networks. arXiv preprint wwXiv:1022.07080 2018
I. Khemakhem, D. Kingma, Monti, and A. Hyvarinen. Variational autoencoders and nonlinear ica: A unifying framework. In International Conference on Artificial Intelligence and Statistics, pages 2207-2211 2020.
Y. Li, J. Bradshaw, and Y. Sharma. Are Generative Classifiers More Robust to Adversarial Attacks? In International Conference Machine Learning, pages -814. PMLR, May 2019. URL http::/procceddnng: mlr. prwss http:///////111// wtt.l/..ree/////111111
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Sorrenson, C. Rother, and Köthe. Disentanglement by nonlinear ica with general incompressible-font networks (gin). In International Conference on Learning Representations, 2020. URL Stthhtps::.. httpp:::/ppenreview.... wtwp:e/wwwwww..e//orunne.t
http:///wwweeieeeeeeeeeeereeeaar -tw.:///ffruu./ddgggggggg rygeHggFDi
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A NONLINEAR ICA
In this section, we provide the formal statements about the identifiability of Nonlinear ICA in Sor- renson et (020) give context of discussions the paper.
In the problem setting of Nonlinear ICA, the data x E I is assumed to be generated by latent variable E R" through nonlinear function f with noise variable e. There are two different formulations about the generation process, where both formulations would normally allow n I d. By assuming S d, the framework is able to model data with intrinsic/informative support on lower dimensional manifold.
In Khemakhem et al. 20200> 8 setting, the nonlinear function f' assumed to be an injection from _ to I and another noise variable E IR is applied after f. i...
fz) +
In Sorrenson et al. (2020)* ' setting, the nonlinear function / is assumed to be a bijection from i to IR and another noise variable Rd- is concatenated to before i.e.,
x f(z,e).
In this work we only discuss the second setting.
Apart from f being injective in the first setting and bijective the second, normally do not put more assumptions on the generating function f. As indicated by the name of ICA, we would require independence among entries in the latent variable z. Although for linear ICA problems one usually assumes the mutual independence entries in z, one would usually introduce an auxiliary variable u and assume the conditional independence 2./u 1 2j ju for any i * due to obtain identifiability.
The identifiability of the latent variables is at the core of independent component analysis theory. It worth noting that simple transforms, e.g., scaling, shifting and permutation, of the underlying true latent variables do not hurt the indepedence/conditionn identifiability actually considered under the equivalence relationship induced by these transforms.
independence among them. As result, the
Next review the results from Sorrenson et al. (2020). We refer to the relationship between the ground truth latent variables and data as the generation process and that between the estimated latent variables and data the estimation process.
Notation 1.
(Generation) Assume and we do not have e for d.
1. Denote by x E Rd the observable data.
2. Denote by the auxiliary variable which is also observable. u can be either scalar or vector.
3. Denote by z E R = 2, x ... x Zn the informative latent variable
4. Denote by E [2d- the noise variable.
5. Denote by f bijection from Rd to Rdd
Assumption 1. (Generation)
1. Assume the observable data x is generated as x = f(z, c).
2. Assume the informative latent z conditioned on the auxiliary variable u admity an expo- nential family distribution
Qx(2,) (1)
k ETs(ts)
p(zlu)
exp
where Tıj 's the sufficient statistics. Ar 's the coefficients and Z Z are the normal- ising constants. This assumption implies the conditional independence between entries in
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3. Assume the distribution of must not dependent This assumption indicates that has zero mutual information with u by definition.
Notation (Estimation)
1. Denote by pdd the estimated latent variable.
2. Denote by ge IRd E2 bijection from a parametric family with parameter 0 E O We may omit the parameter 0 for notation simplicity.
Assumption 2. (Estimation)
1. Assume there exists a such that x 9et (w).
2. Assume the estimated latent variable w conditioned on u also admits an exponential family distribution
(ww. (u)
p(w/u) u)) = II
exp
(10)
i=1
j=1
where Ti, 's are the sufficient statistics. X. 's are the coefficients and Z are the normal- ising constants.
Theorem 1. [Sorrenson er al. 2020 addition Assumption 2 Hand we further assume:
1. The sufficient statistics (z)(2)ss are differentiable almost everywhere and their derivatives dTaa(=) are nonzero almost surely for all z E Z; and all S n and S S k.
dz
2. There exist nk + different values of u, i.e., uo..... Unk such that the matrix
L = [={(ui)> - A(wo), , Auunk) - X(uo)l where A(u) = (A),1(u), )1,((1), ... >,,,((u))
(11)
is invertible.
Then the sufficient statistics of the generating latent space are related to those of the estimated latent space by the following equation:
T(z) AT' + c,
(12)
where is a constant, fill row rank nk dk matrix and с is constant vector.
From Theorem I can see that the estimated latent variables connected to the ground truth latent variables by the unique linear relationship. Without further knowledge about the matrix A, entries of are still entangled in in general. However, for two-parameter exponential family distributions, each row in the mixing matrix only has one non-zero element (Sorrenson et al,, 2020 Here recall the results for Gaussian distributions and refer interested readers Sorrenson et al. 2020) for more results.
(Sorressnn et al. 2020 Assume both p(zlu) and p(wlu) u) are Gaussian distributions, then each row in A only has non-zero entry and each column has most non-zero entry. In other words, for each 2; in z [21,.... 2n} there exist unique W. Og + 0 and b;, such that
Theorem 2.
2, a,w, + b;-
(13)
From Theorem transform.
that for each 2,, there is only my estimating it up affine
B GENERAL INCOMPRESSIIBLE-FL.ON NETWORK
General Incompressible. flow Network (GIN) is a flow-based model integrating the features of both RealNVP (Dinh al. 2017) and NICE (Dinh et al. 2015).
Flow based models, a.k.a. normalising flows, describe discrete dynamics for random variables and thus induce transports of probabilistic densities via change of variable. In the continuous time limit,
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neural networks can also be used to model the time derivatives and thus define neural ODEs (Chen et al. 2018a). The absolute value of the determinant of the Jacobian matrix in the formula of change of variable usually requires highly complex computation. The design of flows calls for the balance between sufficient capacity of the model family for flexible modelling and sophisticated structure of the models for efficient computation. RealNVP approaches the problem by stacking coupling layers with form
Y1 Y2 = X2 O exp(s(xx)) + ((11);
(14)
where the input x and output y are respectively divided into two parts (x1,x2) and W2). Due the clever construction, evaluation of the Jacobian's determinant is reduced the summation of s(x1). The computation of the determinant can be completely eliminated by constructing s(.) to maintain zero sum. In the coupling structure of NICE, there is no elementwise multiplication on which de facto defines a zero sum s(-). The GIN paper proposed to output the last entry of s(-)
as the negative sum of previous entries. As summing over huge amount of dimensions might cause numerical issues in extreme cases, we replace it with a mean subtraction mechanism
s(x)) = s(x1) - 1 -
Σ s(x1),
(15)
where s(xi); is the i-th entry of the output vector of s(xi) and p is dimension of s(x1). ). A later release of the official GIN implementation also employs this mechanism independently. Note that by using cutro-uuuu--umm network, we could not only save the computation of the Jacobian's 's deter- minant but also preserve the volume of the object passing the flow.
C TRAINABLE STATISTICS VERSUS EMPIRICAL STATISTICS
As we mentioned Section there are two different objectives proposed by GIN: trainable statistics versus batch evaluations. To compare the performance of these two approaches, also train the batch evaluation version of GIN on synthetic data for two different variance choices of e: Oe, and oe; 10-2, The results are shown in Figure and Figure [1 and they are very similar the results of trainable statistics in Figure 3 and Figure
Infommate Lebee
Becwery VAR
Varsince
Reanery MI
Mirnal Information
25
04-
3
3.5
-50
-00
(a)
(b)
(c)
(d)
(e)
Figure 10: An example of GIN trained with empirical distribution parameters with high noise level. a) Ground truth informative latent dims. 2 dimensions with largest variance. e) Standard deviation decay, d) 2 dimensions with largest mutual information. e) Mutual information decay.
D EXPERIMENTS CONFIGURATIONS
Since the main scope of our paper to compare the mutual information criterion with the vari- ance criterion for informative latent variable identification and to show the benefit for downstream tasks, we adopt the same configurations of network structure and training methods with the orig- inal GIN paper. The networks are composed of serial coupling layers and dimension permutation layers. While the random permutation could allow different segmentation of the variable x and add additional flexibility coupling layers, it will not affect the Jacobian determinant's evaluation. The permutations are applied on the length axis for synthetic data and the chamnel axis for the EMNIST data. For the EMNIST data, the coupling layers mainly based convolutional layers, while
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75
16
181
04-
00
255
00
(a)
(b)
(c)
(d)
Figure 11: An example of GIN trained with empirical distribution parameters with low noise level. a) Ground truth informative latent dims. b) 2 dimensions with largest variance. c) Standard deviation decay. d) 2 dimensions with largest mutual information. e) Mutual information decay.
for the synthetic data, the coupling layers are based on fully connected layers. Besides, for EM- NIST data, the model also adopts invertible downsampling layers proposed InDacobsen et al. (2018) and fully connected couplings after flattening layer. More details can be found in Appendix D of Sorrenson et al. (2020).
Here provide more information about the data used in our experiments.
For the synthetic data, the parameters of Gaussian mixtures are randomly generated from uniform distributions. As with the GIN paper, we use interval [-5, 5] to sample the expectations and [0.5, 3] to sample the standard deviation of ž1, 22. For the one shown in Section 3, the statistics are
F1
2.9240
4.1936
T
1.7167
1.7170
H2
-3.9942
-3.7325
2.5244
2.7930
μ43
-4.1159
4.1676
and
43
0.8198
2.8419
(16)
4.3792
2.9258
2.3446
1.5909
4.4143
-4.9151
0.9997
0.9298.
And the distributions of OOD samples come from two Gaussian distributions with statistics
A [-20,, 20]
δ1 [1.34, 1.35] and
[20, 0]
62 [1.21, 0.96).
(17)
Figure 12 and 13 respectively show all the possible 2D projections of the ground truth latent variable (z, E) and observation x adopted in the results represented in Section 3. Figure 14 shows the esti- mated representation w by the trained GIN model shown in Section 3. While the correct informative latent dimensions are reconstructed within w, the mutual information criterion can correctly identify them but the variance criterion cannot.
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Prosser Ravie
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35
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Figure 12: All 45 possible 2--projections of the 10 dimensional latent (z, E) used in Section 3.
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Prosdter 10
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Figure 13: All 45 possible 2D-projections of the 10 dimensional observation f(z, E) used in Section
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Proseder d0
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Figure 14: All 45 possible 2D-projections of the 10 dimensional estimated representation w used in
Section 3.
17


Reviews:
[{"review_id": "odthAeSvw_", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper considers the problem of disentangled representation learning with an existing normalizing-flow-based approach, called general incompressible-flow networks (GIN). In the original approach, informative latent variables were separated from noise by considering the variances of learned latent variables and selecting ones with high variances. The current paper shows empirically that this approach can fail even when the underlying data generating process satisfies all assumptions needed for identifiability. The reasoning is simple: Latent variables can have different scales and their variances may not be indicative of informativeness. Instead, the current paper proposes to use the Shannon mutual information between the auxiliary observed variable (conditioned on which the true underlying factors of variation are conditionally independent and belong to the exponential family) and latent variables to separate informative latent variables from noise. With experiments on synthetic data and on EMNIST, the paper successfully shows that the proposed mutual-information-based approach works better than the original variance-based approach (in terms of recovering the ground truth latent factors, downstream classification accuracy, and out-of-distribution detection).\n\nThe paper achieves a good improvement in informative latent variables selection in GIN, but is not a significant contribution for ICLR publication. To improve the paper, the authors may consider proving that the mutual-information-based approach recovers the ground truth factors in the setting of equation (1) or (2). Additionally, the proposed approach of selecting informative latent variables can be considered more broadly in the context of other disentangled representation learning approaches.\n\n# Update\nThanks for the rebuttal. I have read the changes the authors did. The updated manuscript is more readable and more self-contained now. I am still of the opinion that the presented method should be put in a broader context (i.e., considering it in broader settings and/or for other methods) or be better analyzed theoretically. This way it will be much more useful for the community. For this reason, I keep the score the same.", "extra_scores": {}, "extra_text": {}}, {"review_id": "ecxvwBCOjSA", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "## Summary\n\nThe authors propose an alternative method for finding informative latent variables in a model called General Incompressible-flow Networks (GIN). While previous work relied on the variance of the variables to assess informativeness, the authors argue that this is problematic when the scale of noise epsilon is large. They propose instead to use the mutual information between the variables and u instead.\n\nThe authors evaluate their alternative identification method on two tasks: a toy example of a mixture of gaussians and EMNIST. They show that when the variance of epsilon is large, then their method outperforms the variance based method (figure 3). However when the variance of epsilon is small, then VAR performs similarly to MI (Figure 8).\n\nOn EMNIST, it's clear that MI is able to identify the most important variables and outperforms the best VAR setting, however when more variables are included the MI method deteriorates quicker than VAR. \n\n## Review\n\nIn general I find the paper difficult to follow. Many paragraphs have an unclear structure and sentences are not linked. There are also many unsubstantiated, or unclear/imprecise claims. The derivations and equations seem correct and the figures are well made and understandable. Especially figures 1 and 2 are informative and clear.\n\nI think the paper would be a lot stronger if it was positioned as \"This is a failure case of using the VAR method with GIN, we propose a rigorous solution\". Currently the authors make statements such as \"Though Sorrenson et al. successfully establish the identifiability, their interpretation on the meaning of this result and the method they propose are incorrect\", \"it is unplausible to use the variances of the learned representation\" and \"while the original VAR criterion is not competent at all\" - despite clearly showing that the VAR method works well in a number of situations. I think it's worth being more nuanced about alternative methods and explicitly showing when the other method does not behave as described/expected (which you also do!).\n\nCould the authors comment on how the MI was computed and in particular the computational trade offs between using MI and VAR to select variables? I'm also curious why the MI method degrades so quickly in figure 6, how did you set up the classification task?\n\nHow did you \"apply defence\" in the adversarial defence experiment? You mention that VAR300 is only fooled by 56% of the cases without defence, but this cannot be compared because its \"clean accuracy\" is lower, could you expand on that? It seems to me that higher accuracy is always better? It seems in contradiction with the claim at the end of the paragraph that \"all results indicate that ... features selected by MI criterion are more reliable.\"\n\nIn general, a lot of the experiments seemed to be based on Sorrenson et al, but are not properly described in the paper. I would like to see at least a basic description.\n\nIn summary, I think the authors have identified an interesting improved that is worth publishing. However I think that the paper in its current form is not ready: it can benefit from significant rewriting for clarity (both in structure, as well as spelling) and further analysis of the behaviour in figure 6.\n\n## Notes:\n\n\"Incompressive\" --> incompressible  \n\"in nonlinear independent analysis theory\" - missing \"component\"?  \nW_{d+1:n} should be W_{n+1:d} (below equation 4)?  \n\"unplausible\" -> \"Implausible\"?  \n\"interpretation on\" -> \"interpretation of\"\n\nThere's a number of other, non-existent, words in the paper that will be caught by a spellcheck.\n\n## After updating paper\n\nAfter reading the updates, I think the authors have considerably improved their paper and I have increased my score.\n\nWhile the core contribution, using MI over VAR, is clear. The evaluation is not strong enough, I still don't understand why adversarial defense is a reasonable way to evaluate the different metrics. Further it's now clear that the author's use the sklearn implementation of MI estimation, which depends on several entropy estimators which have high variance in practice. While the authors comment that it's \"negligible\" I am not convinced that this is actually easy or reliable.\n\nIn summary I think the paper is not ready for publication.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "Y4BVCW8ivWl", "reviewer": "AnonReviewer5", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper builds on top of the paper “Disentanglement by nonlinear ICA with General Incompressible-Flow Networks (GIN)” (Sorrenson, 2020) and argues that that paper’s method of identifying informative latent variables was wrong and instead suggests that informative latent variables can be identified by thresholding their mutual information with the auxiliary variable of the conditional generative model.\n\nOverall, I score this paper as a reject. Without judging the content as reason, the exposition of the paper is overly difficult to understand. It also makes very specific references to Sorrenson (2020) and Khemakhem (2020) and their proofs without providing context (Section 2). Similarly, it introduces nonlinear ICA without defining the model well (Section 1). This reviewer had to read the mentioned papers side-by-side to make sense of the paper under review. \n\n### Comments about the content\n\nGIN/Nonlinear ICA wants to recover the true generative latent variables $\\mathbf{z}$ for data $\\mathbf{x}$ given the conditioning variable $\\mathbf{u}$, which acts as the cause for the true latent. In particular, it would be important to mention that both data $\\mathbf{x}$ and auxiliary $\\mathbf{u}$ are observed. An example is given in Sorrenson (2020) for (E)MNIST: $\\mathbf{u}$ is the digit to draw, $\\mathbf{z}$ are the parameters that guide the drawing, $\\mathbf{x}$ is the output. The mentioned papers show that the true latent variables can be recovered up to affine transformations, and additional latent variables are noise.\n\nThis paper sets out to examine the question on how to separate informative latent variables from latent variables that encode noise (in the recovered latent) and claims that Sorrenson (2020) chooses the wrong metric for thresholding (they use the std deviation of the different latent variables). This reviewer could not find any particular mentions of this question in Sorrenson (2020), except for implicit usage in the experiment section.\n\nHowever, this paper claims that mutual information is a better metric and Sorrenson (2020) made a mistake in their proof. This reviewer did not retrace the proofs. Moreover, this reviewer could not understand how eq. (5) in this paper, which uses the data processing quality with the joint of the latent, motivates using the mutual information of the individual latent variables to identify informative ones. \n\nMoreover, intuitively, the MI with the auxiliary variable will tell us about the dependence of the two, which will identify latent variables that encode local information for the conditioning variable (eg digit) and which change a lot in their distribution depending on the conditioning variable (using the language from Sorrenson (2020)).\n\nThis leads to the question about “global” parameters, which do not change in distribution. For example, if MNIST was not grayscale but had digits with different colors (equally likely), the color could be a global parameter, which would be uncorrelated with the conditioning variable and thus have zero mutual information. This would mean that this paper would treat it as a noise variable, which it is not.\n\nAs such, this reviewer would ask the authors for clarification.\n\n### Additional comments\n\n1. The figures lack axis labels.\n2. FGSM is a rather weak adversarial attack.\n3. Section 5 and Figure 9 mentions that the learnt representation $\\mathbf{w}$ is very close to a permutation of the real latent variables. This is explained in Sorrenson (2020), Section 3.2: “when both the generating and estimated latent spaces follow a Gaussian distribution, the latent space variables are recovered up to a trivial translation and scaling.” This is the case here: there is no rotation, which leads to a permutation in the correlation matrix.\n\n---\nThis reviewer wants to thank the authors for their detailed reply and for updating the paper to make it more self-contained. It reads much better now and is much clearer. The review score has consequently been updated from 4 to 6. Stronger adversarial experiments would be encouraged. Could the authors also include a definition of the VAR criterion (even though it is trivial, just to avoid any ambiguities) and maybe include a paragraph for the future applications of this? For this reviewer, it is not entirely clear yet what the value of this finding is: is it going to help with downstream tasks for ICA? Is this another argument in favour of preferring Mutual Information as a metric over variances in general?", "extra_scores": {}, "extra_text": {}}, {"review_id": "DiCRXvfcoFd", "reviewer": "AnonReviewer1", "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper considers the previously established Generative Incompressible Flow (GIN) model to perform disentangled representation learning and argues that the original method of identifying explanatory latent variables via their variance magnitude is flawed. Identification is necessary because flow models require the same number of latent variables as the data dimensionality, thus many latent variables are expected to be uninformative, due to the general low-dimensional data manifold hypothesis. This paper proposes the mutual information between a latent variable and an auxiliary variable (needed because otherwise, the disentanglement objective is unsolvable in general) as an identification criterion and shows that this outperforms the variance criterion, especially in cases where the noise magnitudes are of the same order as the magnitudes of the explanatory variables.\n\nI think this paper makes solid arguments for why the original variance criterion is flawed under certain conditions and why the mutual information criterion is better and I lean towards accepting.\n\nHere are a couple of comments, weak points, and questions:\n- Labeled axes would make the plots more readable without having to re-read the caption multiple times\n- At some point, you claim that the importance of any latent variable can be dependent on which auxiliary variable is used. The claim seems to be that the same data could have a set of latent variables that would have high MI with u1, and another set that would have high MI with u2. I see that this is feasible (and even probable), but you never show this, not even in synthetic data. In fact, in the synthetic data, your 8 noise variables are very idealistic. I would therefore either take out the claims or include experiments that show these effects, both in synthetic and real data.\n- In Figure 6, the performance decreases for MI as you include more variables. You discuss this in the text, but it is not satisfactory to me. Could you formulate more clearly what you mean here? Is the degradation an artifact of not being able to estimate these variables well? Or is your core assumption of a low-data manifold violated? Or could it be that the MI is a better, but still not the correct criterion?\n- How do you estimate the MI, in detail?", "extra_scores": {}, "extra_text": {}}, {"review_id": "1tP-042Vix6", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper proposes a method to select informative latent variables for representation learning. The idea is to find variables that maximize mutual information with respect to observed auxiliary variables. The paper points out that in Sorrenson et al. the latent variable selection criteria does not lead to good guarantees, and empirically the new selection criteria has superior performance. \n\nPro: \n\nThe paper points out the problem that since the scale of the latent variable cannot be recovered, the scale cannot be used as a criteria for variable selection. This seems true and an important point to note. \n\nThe proposed alternative variable selection criteria seem reasonable, but the algorithm and its theoretical properties need to be rigorously stated.\n\nCon: \n\n\nThe writing is quite difficult to follow. One big issue is that this paper is not self contained, i.e. it inherits the notation in prior work (especially Sorrenson et al) without explaining what the symbols mean. Many notations are also confusing. For example in Eq.(5) what does w(z,eps) mean? I can kind of get you are trying to represent the dependence of w on z and eps, but this expression does not type check. \n\nHow is the mutual information criteria implemented in practice? I’m assuming you want to find a subset of variables that maximize mutual information w.r.t. u, but this is a problem known to be difficult (i.e. a naive solution has to enumerate every subset of variables). I inferred that you are finding **individual** variables that maximize mutual information with respect to u; However there is a big gap between Eq.(5) and the claim that you can maximize mutual information of each individual variable w.r.t. u. How is this justified, or did I misunderstand? I think you need to explicitly say exactly what algorithm you are using, and what property it should have. \n\nThe experimental settings are also not clearly explained. As a few confusing sentences (among many others), \"each latent variable is labeled by auxiliary variable\" or \"the signal provided by the informative latent variables may not be able to outstand\". I don't think I can understand what is going on exactly here with so much ambiguity. \n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a method to identify informative latent variables by thresholding based on the conditional generative model. While the exposition of the paper has substantially improved during the discussion period, some major concerns remain after the discussion among the reviewers. In particular, the problem considered in the paper has a very limited scope. Moreover, the evaluation of the methods needs to be improved. The paper could benefit from discussing how it situates in the broader context.
Decision: Reject