On Disentangled Representations Learned From Correlated Data
Authors: Frederik Träuble
Elliot Creager
Niki Kilbertus
Anirudh Goyal
Francesco Locatello
Bernhard Schölkopf
Stefan Bauer
Venue:  (2021)
Keywords: representation learning
disentanglement

Abstract: Despite impressive progress in the last decade, it still remains an open challenge to build models that generalize well across multiple tasks and datasets. One path to achieve this is to learn meaningful and compact representations, in which different semantic aspects of data are structurally disentangled. The focus of disentanglement approaches has been on separating independent factors of variation despite the fact that real-world observations are often not structured into meaningful independent causal variables. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of most prominent methods and disentanglement scores on correlated data in a large scale empirical study (including 4260 models). We show that systematically induced correlations in the dataset are being learned and reflected in the latent representations, while widely used disentanglement scores fall short of capturing these latent correlations. Finally, we demonstrate how to disentangle these latent correlations using weak supervision, even if we constrain this supervision to be causally plausible. Our results thus support the argument to learn independent mechanisms rather than independent factors of variations.

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ON DISENTANGLED REPRESENTATIONS LEARNED FROM CORRELATED DATA
Anonymous authors Paper under double-blind review
ABSTRACT
Despite impressive progress the last decade, it still remains an open challenge to build models that generalize well across multiple tasks and datasets. One path achieve this is to learn meaningful and compact representations, in which different semantic aspects of data are structurally disentangled. The focus of disentanglement approaches has been on separating independent factors of variation despite the fact that real-world observations are often not structured into meaningful independent causal variables. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of most prominent methods and disentanglement scores correlated data in large scale empirical study (including 4260 models). We show that systematically induced correlations the dataset are being learned and reflected in the latent representations, while widely used disentanglement scores fall short of capturing these latent correlations. Finally, we demonstrate how to disentangle these latent correlations using weak supervision, even if we constrain this supervision to be causally plausible. Our results thus support the argument to learn independent mechanisms rather than independent factors of variations.
INTRODUCTION
Due to the induced structure, disentangled representations promise generalization unseen scenarios (Higgins et al. (2017b). increased interpretability (Adel et al. 2018 Higgins et al. 2018 and faster learn- ing on downstream tasks (van Steenkiste al. 2019 Locatello et al. 2019a). While the advantages of disentangled representations have been well established, they generally assume the existence of natural factors that vary independently within the given dataset, which is rarely the case in real-world settings. As an example, consider a scene with table and some chairs (see Fig. [ The higher-level factors of this representation are in fact correlated and what we actually want to infer independent (causal) mechanisms (Peters al. 2017 Parascandolo
Figure I: While in princi- ple we consider the pres- ence of the objects (coffee cup, table, chair) to be inde- pendent mechanisms, they tend to appear together in observed data.
et al. 2018 Suter et al. 2019 Goyal et al. 2019).
complex generative model can be thought of as the composition of independent mechanisms or "causal" modules, which generate high- dimensional observations (such images or videos). In the causality community, this is often considered prerequisite to achieve represen- tations which are robust to interventions upon variables determined by such models (Peters et al., 2017). One particular instantiation of this idea in the machine learning community is the notion of disentangled representations (Bengio et al.) (013). The goal of disentan- glement learning is to find representation of the data which captures all the ground-truth factors of variation (FoV) independently.
Despite the recent growth of the field, the performance of state-of-the-art disentanglement learners remains unknown for more realistic settings where FoV are correlated during training. Given the potential societal impact in the medical domain (Chartsias et al.] 2018) fair decision making (Locatello et al., 2019a Madras et al.) 20188) Creager et al.] 2019). the evaluation of the usefulness of disentangled representations trained correlated data of high importance.

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To go beyond the highly idealized settings considered thus far, we conducted large scale empirical study to systematically assess the effect of induced correlations between pairs of factors of variation in training data the learned representations. To provide qualitative and quantitative evaluation, we investigate multiple datasets with access to ground-truth labels. Moreover, we study the generalization abilities of the representations learned correlated data well their performance in particular for the downstream task of fair decision making.
Contributions. Our main contributions can be summarized follows:
We present the first large-scale empirical study (4260 models that examines how modern disen- tanglement learners perform when ground truth factors of the observational data are correlated. We find that factorization-based inductive biases are insufficient to learn disentangled representa- tions from observational data. Existing methods fail to disentangle correlated factors of variation, resulting in correlated latent space dimensions. Moreover, standard disentanglement metrics do not reveal these persisting correlations.
We investigate the usefulness cuni-supervised and weakly-supervised approaches to resolve latent entanglement. For the latter setting, we focus multiple observational and interventional distributions.
BACKGROUND AND RELATED WORK
Disentanglement. Current state-of-the- thr---rr-t disentanglement approaches use the framework of varia- tional auto-encoders (VAEs) (Kingma & Welling 2014) Rezende et al.] 2014). The (high-dimensional) observations modelled being generated from some latent features with chosen prior p( according to the probabilistic model ps (ac x pz). The generative model pa (x) x) as well as the proxy posterior go (z(2)at lar) can be parameterized by neural networks, which are optimized by maximizing the variational lower bound (ELBO) of log p(.),....... x.).
LV = Σ (allat)) ol (or - (x
The above objective does not enforce any structure on the latent space, except for similarity (in KL-divergence) to the prior p(z) (typically chosen as an isotropic Gaussian). However, the structure and semantic meaning latent representations can be relevant to study generation properties. Consequently, various proposals for structure-imposing regularization and commonly used evaluation
metrics measuring different notions of disentanglement of the learned representations have been made (Higgins et al.] 20 Kim & Mnih 2018 [Burgess et al. 2018 Kumar et al. 2018 Chen et al. 2018 Eastwood & Williams 2018) Mathieu et al. 2018). Recently, it has been shown that unsupervised disentangling by optimising marginal likelihood in generative model is impossible without further inductive bias (Locatello et al., 2019b). To address this theoretical limitation, methods have been proposed that do require explicitly labelled data but only some weak labeling information (Locatello et al. 2020 Shu et al.) 2019). Ideas related to disentangling the factors of variations date back the linerr ICA literature (Bach & Jordan 2002 Comon 1994] Jutten & Karhunen 2003; Hyvärinen & Pajunen 1999; Hyvarinen et al.) 2019; Hyvarinen & Morioaa |0166 Gresele al., 2019). Recent work combines non-linear ICA with disentanglement (Khemakhem et al. 20200 Sorrenson et al. 2020; Klindt et al.) 2020)
Correlations. A set of random variables XI=I,.., is not independent, if and only their joint distribution does not factorize
P(X1, X2 X2.......... X.,) + II P(X.).
In this case, we speak of dependence between the random variables, also commonly referred to as correlation Correlation between two variables can either stem from a direct causal relationship (one
'ach model has been trained for 300,000 iterations on Tesla V100 GPUs. Reproducing these experiments requires approximately 0.79 GPU years.
We use the term correlation here in a broad sense of any statistical association, not just linear dependencies.
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variable affects the other), but can also be due to different unobserved circumstances (confounders) affecting both. Real-world datasets display many of these ("spurious" and often a priori unknown) correlations (Geirhos et al.] (020). However, most work on learning disentangled representations assumes that there is an underlying set of independent ground truth variables that govern the generative process of observable data. These methods are hence predominantly evaluated on data that obey independence in the true factors of variation, which we then consider be the correct factorization. In the real world, the observation generating process is likely not always clearly "disentangled" and we do expect correlations in the collected datasets. It is thus an open question to what degree existing inductive biases from the encoder/decoder architecture, but more importantly the dataset biases, affect the learned representation. In our experiments, we introduce dataset correlations in controlled manner to understand to what degree state-of-the-art approaches can cope with such correlations. We believe these correlations to reflect a major feature of more realistic environments.
Other Related Work. Most popular datasets the disentanglement literature exhibit perfect independence in their FoV. At some level this is sensible as it reflects the underlying assumption in the inductive bias being studied. However, this assumption is unlikely to hold in practice as shown by [i et al. (2019), who propose methods based on a pairwise independence assumption instead. The literature so far has not thoroughly measured how popular inductive biases such as factorized priors behave when learning from correlated datasets, although several smaller experiments along these lines can be acknowledged. Chen et al.) 2018) studied correlated 3DFaces (Paysan et al. 2009) by fixing all except three factors in which the authors conclude that the 3-TC-VAE regularizer can help to disentangle imposed correlations. [Brekelmans et al. (2019) show that Echo noise results in superior disentanglement compared to standard Vet dSprites variant where randomly selected factor pairs are excluded. However, the latent structure was not studied in detail; our findings suggest that global disentanglement metrics are insufficient to diagnose issues when models learn from correlated data. Creager et al. (2019) based some of the evaluations of a proposed new autoencoder architecture the fairness context on a biased dSprites variant and Yang et al. (2200) study linear SCM in VAE architecure on datasets with dependent variables. However, their studies focused on representation learners that require strong supervision via VoV labels at train time.
small experiment on a downsampled
THE EFFECT OF CORRELATED DATA ON DISENTANGLEMES LEARNERS
In this section, we want to present the key findings from our empirical study of unsupervised disen- tanglement learning on a particular variant of correlated data. We start by outlining the experimental design of our study in Section 3.1 Based on this, analyze the latent spaces in Section B.2 and find that factorization-base inductive biases are insufficient to learn disentangled representations from observational data. Persisting pairwise correlations in the latent space are sufficiently revealed by standard disentanglement metrics that might be particularly relevant and problematic for fairness applications. Finally, in Section 33. we show extrapolation and generalization capabilities of the leamed representations towards unseen factor combinations due to the induced correlations.
3.1 EXPERIMENTAL DESIGN
For our first experiments we introduce correlations between single pairs of factors of variation on the following three datasets: Shapes3I with object size and azimuth (denoted "A"), dSprites with orientation and X- X-position ("B") and finally the real-world observations dataset MPI-3D with
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Figure 3: We show latent traversals (left) of the best DCI score model among all 180 trained models with strongest correlation (a 0.2) on Shapes3ID (A). The traversals in latent code dimensions 4 and (highlighted black), suggest that these dimensions encode a mixture of azimuth and object size, reflecting one major axis along the correlation line of the joint distribution and one smaller, locally orthogonal axis. This is supported by a heat map of the GBT feature importance matrix of this model (right) indicating entanglement of azimuth and object size encoded into both latent codes.
first and second degree of freedom ("C"). We focus on linear correlations with Gaussian noise between the two variables, which we denote cl, c2. The ground truth factors for cl, c2 take values Zel E - = and 2,2 E {O..... =22x} respectively. We then parameterize correlations by sampling the training dataset from the joint P x (= 0F 0)) where The strength of the correlations can be tuned by a. for which we choose 0.2, 0.4. 0.7 in normalized units with respect to the range of values in 2e1,c2. Lower indicates stronger correlation. See Fig. for an example of P(=c1, Rc2) for correlating azimuth and object size in Shapes3ID with a 0.2. Additionally, we study the uncorrelated limit in the literature. We train the same six VAE methods as discussed in Locatello et al. (2019b), including V-VAE, FactorVAE, AnnealedVAE, DIP-VAE-I, DIP- VAE-II and 3-TC-VAE, each with six hyperparameter settings. Each method has been trained using five different random seeds. All remaining factors variation are sampled uniformly at random. This first study sums up to a total of 2160 trained models, 180 models per dataset and correlation strength Appendix describes additional implementation details.
=c2 /=c1 m=a
oc), which amounts to the case typically studied
3.2 CAN UNSUPERVISED METHODS ACHIEVE DISENTANGLEMES OF CORRELATED DATA?
Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by single scalar measure. Perhaps surprisingly, Fig. 2 shows no clear trend among all implemented disentanglement scores w.r.t. correlation strength (see Fig. in the Appendix for the full result across all datasets and metrics). The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. However, as we will demonstrate along the following analysis, the latent spaces in this setting show some characteristic differences when trained common disentanglement metrics are limited when correlations are introduced into the training data. To conduct more careful analysis of the inductive data bias applied the learned representations we will instead evaluate pairwise metrics. Note that regarding BetaVAF and Factor VAE this observed trend is to some degree expected they would yield perfect disentanglement scores even if we would take the correlated ground truth factors or a linear transformation in the case of Bet as the representation.
strongly correlated pair of FoVs. We thus argue that
Latent structure and pairwise entanglement. We start by analysing latent traversals of some trained models on Shapes3D (A). For strong correlations = 0.2 and = 0.4), we typically observe trained models with two latent codes encoding the two correlated variables simultancously. In these cases, one the latent codes corresponds to data along the major axis of the correlation line whereas the other latent code dimension manifests in orthogonal change of the two variables along the minor axis. Still, a full traversal of the code corresponding to the minor axis often seems to cover only observations within the correlation line. Fig. (left) shows this effect for the latent space of a model trained on Shapes3D (A) with strongest correlation (a = 0.2).
Code for all experiments will be released after publication

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Figure 4: Left: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We see that stronger correlation leads to statistically more entangled latents compared the baseline score without correlation (blue). Right: The same behavior can be seen for the unfairness score between the correlated pair of factors.
To quantify this observation, we analyze the importance of individual latent codes in predicting the value of given ground truth FoV. An importance weight for each pair of (FoV, latent dimension computed by training gradient boosting trees (GBT) classifier predict the ground truth labels from the latents (10,000 samples). In the right panel of Fig. 3 we compute these importance weights for the model used to generate traversals in the left panel. The corresponding evaluation for a model trained on the same dataset with weak correlation does not reveal this feature visually (see Fig. 10 in Appendix).
To support this claim for a larger set of models, we calculate pairwise entanglement score that allows us to measure how difficult it is to separate two factors of variation from their latent codes. This computation involves grouping FoV into pairs based
ordering of their pairwise mutual information or GBT feature importance between latents and FoV; we defer to Appendix A for a detailed description of this procedure. Figure 4 (left) shows that across all datasets the pair of correlated FoV V has a higher score than the median of all other pairs, indicating that they are harder disentangle. This threshold decreases with weaker correlation and the pair becomes easier to disentangle for weaker correlations (a 2 0.7). These findings suggest that the models still manage to disentangle correlated factors if the correlation not too strong.
Finally, correlations between variables are of crucial importance in fairness applications motivating an additional investigation on ramifications of these entangled latent spaces. In this setting we are interested in the unfairness of predicting the second correlated variable while the first correlated variable is considered being protected or sensitive attribute. the following, we use variant of demographic parity (Dwork et al. 2012) that computes pairwise mutual information between latents and FoV (Locatello et al.j 2019a). In Fig. (right) we evaluate this score when correlations are present within the data in the case of Shapes3D (A). Unfairness tracks correlation strength in this scenario. These results suggest that we cannot expect disentangled representations learned unsupervisedly to help reduce unfaimesss beyond the benefits discussed in Locatello al. (2019a). More comprehensive results, that support the finding that the correlated pair is statistically more entangled in the latent representations across all unsupervised experiments and datasets is provided in Appendix [B.]] Summary. We find that existing methods fail to learn disentangled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient reveal these troublesome pairwise entanglements in the latent space.
3.3 GENERALIZATION PROPERTIES
In this Section, we aim to understand how the trained models perform on unseen training data far away from the correlation line; cuttof-diisribution (OOD) w.r.t. the train distribution. We analyse this capability for the model from Fig. 3 In this model, the remaining factors seem to be disentangled well enough to only focus the two latent dimensions encoding the entangled variables.
As first test, we sample observations from the FoV then set object size and azimuth to six distinct configurations of zero probability, see Fig- (left). The trained model is capable of reconstructing these observations despite never having encountered these configurations or neighbors thereof during
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Figure Generalization -fitt--diittibutionn (OOD) test data. Left: Reconstructions observations the model has never seen during training. Right: Latent space distribution of the two entangled dimensions. Circles without edges indicates encoded data from the (correlated) training distribution, while circles with edges indicate encoded OOD data where the correlation pattern is broken.
training. This suggests that the encoder maps representations to meaningful point in the latent space from which the decoder is equally capable generating expected observations. To test this hypothesis further, analysed latent traversals originating from these OOD points and observe that changes in the remaining factors reliably yield the expected reconstructions. Traversals with respect to the two entangled latent codes continue to encode object size and azimuth.
To fully understand this models' generalization properties we visualise the occupied latent space spanned by the two identified dimensions encoding both correlated factors in Fig. (right). We are particularly interested in where these points are located with respect to the ground truth value of each correlated variable, depicted via color. The two sets of depicted points (1) latent codes sampled from the correlated training data and (2) latent codes sampled with (object size, azimuth) configuration that has zero probability under the correlated training distribution. We observe that contours of equal color (ground truth) are not aligned with the latent axes. This indicates that the two latent dimensions encode both FoV at the same time. Likewise, we can understand the generalization capabilities of this model far away from the trained data. Extreme configurations such as small azimuth and large object size are encoded to space regions corresponding to the intersections of the manifolds with constant value for each correlated variable. This shows that all out-oo-distributtin points are encoded in this representation space in a way that obeys the natural ordering of each respective factor. This behaviour remains even in the default value ordering as stored in our ground truth table. For this we additionally trained 360 models two additional Shapes3I) variants each, where strongly correlated object color object size ("D") and object color azimuth ("E"") respectively (a = 0.2 and = 0.4). As the color values do not allow for a unique natural ordering, the trained models do often encode different color manifold ordering into the latent space. In Appendix B. we show some of their characteristic latent space visualizations with similar extrapolation and generalization capabilities. We conclude from these results that disentanglement methods can generalize towards unseen FoV configurations as long as each factor value contained in the training data within different configuration.
where the trained latent space does not mirror
4 FINDING THE RIGHT FACTORIZATION
The results from Section 3 illustrate the limitations of state-of-the-ar unsupervised methods on correlated data (and thus real-world observational data). We now investigate the usefulness several approaches for mitigating pairwise correlations in the latent code. We begin with post-hoc procedure Section 4.1 that uses limited label information on the ground truth factors and show that it achieves substantial correction of the pairwise latent correlation. We then consider a recently proposed approach leveraging recent advances in weakly supervised disentanglement learning that applics directly at train time. As will be seen in Section 4.2 this method results in substantially more disentangled representations, even when applied on correlated data from different sampling scenarios.

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Figure 6: Fast adaption with few labels:
4.1 POST-HOC ALIGNMENT CORRECTION WITH FEW LABELS
When limited number of FoV labels {y:} can be accessed, a reasonable option for resolving entan- gled dimensions of the latent code is by fast adaptation. To identify the two entangled dimensions (2,, 2,) we look at the maximum feature importance for given FoV from a GBT trained using these labels only. We then train a substitution function using supervised learning to replace these two dimensions with the predicted ground truth label. Crucially, both steps of this procedure rely on the same FoV labels, which should be as few as possible. In Fig. we show the pairwise entanglement score of the correlated FoVs under this fast adaptation with a linear regression as the substitution function, which succeeds with as few 100 labels, corresponding to less than 0.02% of all data points in Sha However, fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, or when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). To address this, a nonlinear substitution function such as MLP can reduce this pairwise entanglement to certain degree (see additional results in Appendix B.2
We find that the efficacy of fast adaptation depends on the level of disentanglement of the representa- tions with respect to all the other factors. This implies that if the representation is well disentangled at the start of the fast adaption procedure, it is possible to achieve perfectly disentangled model (according to our previous visual and quantitative evaluations). However, if all FoV are entangled at the beginning, the fast adaption method will have little effect. Finally, we note that model selection is impossible in purely unsupervised manner based on any of the used disentanglement metrics, as they all require labeled ground truth data. These shortcomings shall be resolved by the following method capable of disentangling the correlated factors of variation much more reliably.
4.2 ALIGNMENT DURING TRAINING USING WEAK SUPERVISION
Since the unsupervised disentangling by optimising marginal likelihood in generative model is impossible (Locatello et al.) 2019b Theorem 1). inductive biases like grouping information (Bouchacourt 2018) access to labels (Locatello et al. 2019c) is required. Changes in natural environments, which typically correspond to changes of only a few underlying factors of variation, provide weak supervision signal for representation learning algorithms (Goyal et al. 2019 Földiák
1991 Schmidt et al. 2007 Bengio et al., 2019). Without correlations it has been shown that this weak supervision helps in learning much more disentangled representations (Locatello al. 2020) Shu et al. 2019). Locatello et al. (020) showed access to observations which display differences in a known number of factors of variation (without knowing which ones specifically) is sufficient to learn disentangled representation. These additional weak assumptions render the generative model identifiable in contrast to unsupervised disentanglement. This kind of extra knowledge might be available in certain settings - e.g., in temporarily close frames from a video of moving robot arm where some factors remain unchanged. Hence, we want to investigate the usefulness of such a weakly supervised method applied in various scenarios when training data is correlated. Specifically we implement the Ada-GVAE variant Locatello et al. (2020) that was shown to allow for model
*On the other hand, in applications with fairness concerns it may be impossible to intervene on FoY representing sensitive and immutable attributes of individuals (gender, race, etc.); we refer to Madras et al. 2019) for more complete discussion.
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Figure 7: Weak supervision: Left: With weak supervision trained models on Shapes3D correlation object size and azimuth learn consistently improved, often perfect, disentangled representations across correlation strengths. Middle: Unfairness scores between correlated FoVs are much smaller (see scale). Right: Latent dimensions of best DCI model with strong correlation (a 0.2). Representations are axis-aligned with respect to both of the correlated variables ground truth values (right). Below: Traversals of model trained with strong correlation (a 0.2)
selection via the (unsupervised) reconstruction loss. The method requires a pair of observations that differs in known number of factors, without knowing which in particular.
Weak supervision mitigates pairwise latent entanglement. We trained the three correlated Shapes3D variants (A, D, E) with pairwise correlations between object color, object size and azimuth with the same correlation strength settings. Due to the definition of the regularizer we limit this study to the 3-VAE models with the 360 additional models. For the generation of pairs we study the case where the difference in the observation pairs is present in random FoV. Whenever of the correlated FoV, its respective value in each pair is drawn from the probability distribution conditioned on the other correlated FoV. This means the difference in this factor is typically very small and depends on the correlation strength. Note that this procedure assures that constructed pairs are consistent with the observational data such that the correlation never broken. Fig. 7 summarizes the weak supervision results when imposing correlations in object size and azimuth. We consistently observe much better disentangled models, often achieving perfect DCI score irrespective of correlations in the dataset. The latent spaces tend to strongly align their coordinates with the ground truth label axis. Finally, weak- wonnk--sprrssiion reduces unfairness relative to the unsupervised baseline, and occasionally achieves unfairness score.
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These results suggest that weak supervision can provide strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. As a prominent example, this is an issue in the fairness context where real-world datasets often display unknown discriminatory correlations. Additional results the other datasets can be found in Appendix B.2 including two additional scenarios where one has intervening capabilities to generate the pair. We consistently observe the same strong trends regarding disentangled correlations all of the above studies using weak supervision.

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5 CONCLUSION
We have presented the first large-scale empirical study examining how modern disentanglement learners cope with correlated observational data. We find that existing methods fail to learn disentan- gled representations of correlated factors of variation, and moreover that standard disentanglement metrics are insufficient to reveal these pairwise entanglements. We discussed practical implications for downstream tasks like fair decision making. Finally, we demonstrate how to correct for these latent correlations via various weakly supervised training scenarios. Our results thus support the importance and usefulness of learning independent mechanisms rather than independent factors of variations [Schölkopf] 2019; Parascandolo al. 2018)) [Suter et al. 2019) Goyal et al. 2019). Besides the simple correlations studied in this work, future work is needed to address the open question
whether these results extend more complex nonlinear correlations and settings where many more variables are correlated simultaneously.

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A IMPLEMENTATION DETAILS
Unsupervised Disentanglement methods. For reasons of comparison, the considered disen- tanglement methods in this work cover the full collection of state-of-the-art approaches in from Locatello et al. (2019b) based on representations learned by VAEs. The set contains six different methods that enforce disentanglement of the representation by equipping the loss with different regularizers that aim at enforcing the special structure of the posterior aggregate encoder distribution. A detailed description of the regularizer forms used in this work, specifically 3-VAE (Higgins et al., (017a), FactorVAE (Kim & Mnih, 2018). VnnealedVAE. (Burgess et al.) 2018), VIE-V VAE-II (Kumar et al. 2018) and 3-TC-VAE (Chen et al. 2018) is provided We use the same encoder architecture with 10 latent dimensions for every
in (Locatello et al. 2019b). model.
Evaluation metrics. To measure disentanglement of learned representation, various metrics have been proposed, each requiring access to the ground truth labels. The BetaVAE score is based upon the prediction of a fixed factor from the disentangled representation using linear classifier (Higgins) et al. 201 The FactorVAE score is intended to correct for some failures of the former by utilizing majority vote classifiers based on normalized variance of each latent dimension (Kim & Mnih 2018). The SAP score represents the mean distance between the classification errors of the two latent dimensions that are most predictable (Kumar et al. 2018). For the MIG score, one computes the mutual information between the latent representation and the ground truth factors and calculates the final score using a normalized gap between the two highest MI entries for each factor. Finally, disentanglement score proposed by [astwood & Williams (0118), often referred to as DCI score, calculated from dimension-wise entropy rellecting the usefulness of the dimension to predict single factor of variation.
Unfairness between a pair of Fo The scores reported are based on notion of demographic parity for predicting a target variable y given protected and sensitive variable s. Both y and s can be associated with factor of variation here. Rather than using the global total variation average as defined in Locatello et al. (2019a), we report the individual demographic parities for the correlated factors specifically.
Pairwise entanglement metric: In order to further analyze the differences between the GBT feature importance matrices between latent code and encoded factor of variation, we view them as weights on the edges of a bipartite graph encoding the statistical relation between each factor of variation and code. We can now delete all edges with weight smaller than some threshold and count (i) how many factors of variation are connected with at least latent code and (ii) the number of connected components with size larger than one. We can then compute which factors are merged at which threshold. Factors that merged at lower threshold statistically related to a shared latent dimension. This computation can be not only based on the GBT feature importances but likewise on weight matrices inferred from the Mutual Information.
more entangled in the sense that are more
Joint distributions of correlated factors in datasets: In Fig. 8 we show the joint probability distributions of the correlated pair of FoV for all datasets and correlation strengths considered in this study. Dataset A, B and C were designed with correlated factors of variation that are ordinal for natural visual interpretation of the traversals. contrast, datasets D and contain a correlated factor of variation that has such natural ordering.
B ADDITIONAL RESULTS
B. SECTION 3
Shortcomings of existing metrics. Following recent studies, we evaluate the trained models with the help of a broad range of disentanglement metrics that aim at quantifying overall success by the help of a single scalar measure. Perhaps surprisingly, as can be seen in Fig. 9 there is no clear trend among all implemented disentanglement scores w.r.t. correlation strength. The metrics have been evaluated by sampling from the correlated data distribution although they do not differ substantially when evaluated on the uncorrelated distribution. We thus argue that commonly used methods insufficient to provide insight into the latent space learned.
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Figure 10: We show latent traversals (left) of the best DCI score model among all 180 trained models with weak correlation (o = 0.7) in object size and azimuth. The traversals in latent code no and 7/8 (highlighted in black), suggest that these dimensions encode no mixture of azimuth and object size compared to the models with stronger correlation. This is supported by the GBT feature importance matrix of this model (right).
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Figure 11: Pairwise entanglement scores help to uncover still existent correlations in the latent representation. Left: Mean of the pairwise entanglement scores for the correlated pair (red) and the median of the uncorrelated pairs. We that stronger correlation leads to statistically more entanglement latents across all datasets studied compared to their baseline pairwise entanglemer where the data exhibits correlations (blue). Each pairwise score is the mean across 180 models for each dataset and correlation strength. Scores in the left table are based on GBT feature importance and scores presented in the right table are based Mutual Information.
Latent structure and pairwise entanglement. Our hypothesis that the latent representations are less correlated if the correlation strength is weaker is shown for model on Shapes3D (A) with weak correlation Fig. [0] Here the latent traversals do not mirror the major and minor axis of the correlated joint distribution. This conclusion is being backed by the thresholds of the pairwise entanglement metries for the correlated pair vs. the median of all other pairs across all datasets, either the mutual information. See Fig. [1 for these respective results. Another pairwise metric that tracks the correlation strength in our scenario is the unfairness score between the correlated pair of factors that is being shown for datasets A, B and C in Fig.
when computing them using the GBT feature importances
Generalization Properties In order to support our conclusion from these results that disentangle- ment methods can generalize towards unseen FoV configurations we show Fig. [3 latent traversals originating from these OOD point with smallest object size and largest azimuth. We observe that changes in the remaining factors reliably yield the expected reconstructions. Additionally, we are interested in where samples from correlated models are located with respect the ground truth value of each correlated variable. For this, we are visualizing the latent spaces with similar extrapolation and generalization capabilities of four models from the strongest correlation dataset variants of Shapes3d (D) and Shapes3d (E) in Fig. 14
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Figure 13: Generalization capabilities towards out-of-distribution test data. Latent traversals from an observations the model has never seen during training. The starting point corresponds to factor configuration in point I from Fig. | Shown are the results of the model with highest DCI score among all 180 trained models on Shapes3d (A) with very restricted correlation strength = 0.2 in object size and azimuth
B.2 SECTION
we see the axis alignment of the correlated latent space after fast adaptation using linear regression model trained Shapes3ID Fast adaptation with linear regression substitution fails in some settings: when no two latent dimensions encode the applied correlation isolated from the other latent codes, when the correlated variables do not have a unique natural ordering (e.g. color or categorical variables). Additionally, the functional form of the latent manifolds beyond the training distribution is unknown and in general expected to be nonlinear. We test the possibility of fast adaptation in this case using as substitution function a one-hidden layer MI.P classifier of size 100 on the correlated Shapes3D variants. Under this method, we sample the FoV from a uniform independent distribution. A small number of such samples could practically be labeled manually. Using only 1000 labeled data points for our fast adaptation method shows a significant reduction in searaannggemmeettteshhhlllle for the correlated pair (Fig. 16
Post-hoc alignment correction with few labels Fig. 13
(A). Fast
Alignment during training using weak supervision Using the studied weakly supervision Ada- GVAE method with l: 1 from Locatello et al. (2020), we showed that weak supervision can provide a strong inductive bias capable of finding the right factorization and resolving spurious correlations for datasets of unknown degree of correlation. Despite the results shown on Shapes3D (A) in main paper, results across all three correlation variants in Shapes3D (A, D, E) are shown in Fig. T7 as well as some representative latent space visualizations that show strong axis alignment Fig This study contains a total of 360 trained models.
In addition to the experiment from main paper where pairs are constructed solely from the correlated observational data, want to study two scenarios where have limited intervention capabilities on the FoV to generate training pairs. The resulting distribution of FoVs (still exhibiting correlations) in these pairs depends whether the correlation between two pairs is due to causal link or due to a common confounder.
Scenario I-1: We assume there is confounder causing spurious correlation between the factors, such that correlation is broken whenever we constraint the change to be one of the correlations.
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The value the changing variable then sampled uniformly in the second observation. Note that this still means that the vast majority of sampled pairs exhibit correlated Fo This is depicted in the substantially lifted disentanglement scores shown Fig. as well as some selected latent space visualizations that show strong axis alignment in Fig. 20 We consistently observe much better disentangled models, often achieving perfect DCI score irrespective
correlations in the data set. The latent spaces tend to strongly align their coordinates with the ground truth label axis. We chose 10 random seeds per configuration in this study, yielding 720 models in total.
Let us assume C₁ causes C2 in our examples, which manifests the studied linear correlations. Within this setting we cannot sample uniformly in C2 if we intervene (or "fix") all factors except for this causal factor. Intervening any value in C1 as is not causally affected by C2. To test the hypothesis that this constraint still allows for disentangling the correlation, trained on Shapes3D (A) and sample pairs consistent with this causal model. Besides observing visually disentangled factors in the latent traversals, we show summary of results in Fig. 21 with the same significant improvements regarding disentangling the correlated Fo Vs. Besides the above correlation strengths, we additionally trained the same models using a very strong correlation of a 0.1, yielding 300 models trained in this study.
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Figure 19: Standard disentanglement metrics evaluated on the correlated training sets for the weakly supervised scenario with intervening capabilities (I-1).
22

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Under review conference paper at ICLR 2021
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Figure 20: Left; For the weakly supervised scenario with intervening capabilities (I-1) trained models on Shapes3D (A), (D) and (E) correlating object color and azimuth learn consistently improved, often perfect, disentangled representation across all correlation strengths. Right: Latent dimensions of best DCI model with strong correlation (0.2). Representations are perfectly axis-aligned with respect both of the correlated variables ground truth values (right).
23

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Under review conference paper at ICLR 2021
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Figure 21: Disentanglement metrics (top), unfairness scores (middle) and latent spaces (bottom) show strong disentanglement using weak supervision with intervening capabilities (1-2) even under the stronger assumption that sampling of observation pairs follow its causal generative model. We show the learnt latent space encoding of the two correlated factors of variation for model with 0.1 and low reconstruction loss.
24


Reviews:
[{"review_id": "nteY3iilFno", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "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 addresses the important problem that most existing disentangled representation learning algorithms analyze statistical independence rather than causal independence. The paper conducts a large scale study to investigate whether statistical correlation prevents learning disentangled representations (according to human defined ground truth factors of variation). \n\nPro:\n\nThis is a timely contribution to clarify problems in evaluating disentangled representation learning algorithms. The paper argues that assuming statistical independence is unrealistic, in practice factors of variation are often causally independent but statistically correlated. This seems like an important missing piece in current empirical evaluation benchmarks. \n\nThere are several interesting observations: \n\n1.The existing metrics for disentanglement do not apply to situations with statistical correlation between the factors of variation. New metrics will be needed. \n\n2.For correlated factors of variation, even though disentanglement fails, the latent space still extrapolates to never seen combinations. \n\n3.Several semi-supervised or weakly supervised methods for disentanglement work well when the factors of variation are correlated. \n\nCon:\n\nSection 3 and 4 are a little hard to follow. There are many results with no clear logical connection to each other. Maybe it’s better to have some bullet points of the empirical findings, then point to specific paragraphs that explain the empirical methodology that produced these findings. \n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "QHmODdInJA", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "Summary: This paper systematically presents a large-scale empirical study on the disentangled representation learning when the underlying factors are possibly entangled. From the results of purely unsupervised settings, the authors have discovered the shortcomings of the existing metrics of disentanglement as well as the poor learned representations (in terms of disentanglement). However, with the help of small amount of factor labels or other weak supervision signals, recent approaches could learn fairly perfect representation.\n\nFirst of all, it is worthy to pay attention to the possible correlation between factors when you intend to learning a disentangled representation. And the whole empirical results are carried out by very large number of experimental batches, which to some extent could well support the conclusions displayed in the paper.\nHowever, there still exist several limitations in my opinion:\n(1)\tThe design of induced correlation is too simple. As you have mentioned in related work, there were some papers noticed the too ideal assumptions of traditional VAE-based models which may not be held in practice. IMO, the linear dependency between only two variables is far from reality as well. More complicated settings should be involved.\n(2)\tIn line with the former limitation, diagnostics of the potential entanglement should also not be limited to pairwise level, which cannot scale up to high dimensional latent factors.\n(3)\tThe novelty of Section 4 is somewhat limited as all the correction methods and even some conclusions were proposed by the previous work.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "5Z3qk4rmnzr", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper studies the behaviour of disentanglement methods and metrics on data where a couple of factors of variation (FoV) are correlated, a more realistic setup compared to the usual independent FoV setting in the literature. The paper shows how the correlation in the FoV is reflected in the representations learned by the models, and claims that the widely used disentanglement scores fail to capture these correlations. A couple of solutions that use weak supervision are suggested.\n\nStrengths:\n1. The paper attempts to address an important limitation of current unsupervised disentangling methods, in that they assume independent FoV. I agree that this is an unrealistic assumption, and only holds for the synthetic datasets used in the literature.\n2. The experimental evaluation of the paper is extensive\n3. The paper is presented clearly for the most part.\n\nWeaknesses:\n1. At the beginning of Section 3.2, the claim that the existing disentanglement metrics fail to capture correlations in the training data seems flawed. Figure 2, that the violin plots of metrics for 180 models (with different loss/hyperparam combinations/random seed) shows no clear trend, is used as evidence for the claim. However this argument ignores an important confounding factor, which is the behaviour of the models (trained with losses originally designed for data with independent FoV) when trained on a dataset with correlated FoV. Without understanding the behaviour of models after training, it is difficult to draw any conclusions from Figure 2. Moreover comparing violin plots of all 180 models does not seem sensible - the models with poor disentangling performance will likely be uninformative, creating noise for comparisons between different settings. Hence I think it makes more sense to compare results for the top models for a given metric (as is done in Fig3).\n2. I also have doubts regarding the conclusion itself, that all existing metrics fail to capture correlations in the training data. I think this only holds for some metrics and not for others. Suppose the representations that we are evaluating are the ground truth FoV values. Then the BetaVAE and FactorVAE metrics will give a perfect score regardless of the degree of correlation by design. But the MIG/SAP/DCI scores will be lower for higher correlation in the FoV, since the mutual information/prediction accuracy between the two correlated factors will be higher than between uncorrelated factors. \n3. The result presented by Figure 3, that training disentanglement models on strongly correlated data gives representations encoding mixtures of the correlated FoV, is unsurprising in my opinion, and adds little value to the pool of knowledge in the literature.\n4. In Section 2, I think that the statement “it has been shown that purely unsupervised learning of disentangled representations is impossible (Locatello et al., 2019b)” is a wrong interpretation of Locatello et al. They show that optimising marginal likelihood in a generative model (such as a VAE) cannot achieve disentangling without any inductive biases in the model. But the inductive biases in the models used by disentangling methods, along with the loss (that is a variant of the ELBO and not the marginal likelihood), are what allow disentangling in practice. There are also theoretical works such as [1] that explain this behaviour.\n5. In Section 3.2, the conclusion “These results suggest that we cannot expect disentangled representations learned unsupervisedly to help reduce unfairness beyond the benefits discussed in Locatello et al. (2019a)” seems problematic. It’s not the unsupervised learning that’s problematic, rather it’s the model assumption that the FoV are independent that is problematic in the scenario of correlated FoV, which is an unsurprising conclusion. \n6. The conclusion in Section 3.3, that disentanglement methods can generalize towards unseen FoV configurations, is old news that is already shown in the literature in disentanglement models trained on datasets with an incomplete set of FoV configurations e.g. 3D Faces, CelebA\n7. In Section 4.2, the claim “This kind of extra knowledge is often available at no cost , e.g., in temporarily close frames from a video of a moving robot arm where some factors remain unchanged” is unjustified. We still need to know the number of factors that have changed, which usually requires human labelling\n8. The weak supervision for the correlated FoV when applying Ada-GVAE relies on being able to generate data where the correlation is broken, which seems like an unrealistic assumption. The authors seem to address this by assuming a causal relation between the two factors. However even if there is a causal relation between the two factors (C1 causes C2), the correlation implies that some (C1,C2) pairs are very unlikely to appear in the data, so you cannot “sample any value in C1” given C2. Hence I don’t understand how the causal setting at the end of Section 4.2 helps address the problems of having to generate data where the correlation is broken.\n\nOverall the paper does address an important problem in the disentanglement literature, but the conclusions drawn from the extensive evaluation are either unsurprising or unjustified. Moreover, the proposed solution via weak supervision appears flawed because it requires generating data where the correlation is broken, a very unrealistic assumption.\n\nOther points:\n- Typo in Figure 2 caption: “lower \\sigma indicates less correlation” <- “higher \\sigma indicates less correlation”\n- Section 3.1: “P(z_c1, z_c2) ~ N(z_c2 - \\alpha z_c1, \\sigma)” How does the RHS define a joint distribution? The RHS shows a scalar normal distribution, whereas the LHS is a joint density. Do you want to replace “\\sim” with “ \\propto” ?\n- Why doesn’t entanglement decrease with #labels > 100 in Figure 6?\n- It might be helpful to also look at the disentanglement metrics for just the two correlated factors, to further highlight the differences between different models.\n\n[1] Rolinek, M., Zietlow, D. and Martius, G., Variational Autoencoders Pursue PCA Directions (by Accident). CVPR 2019.\n\n===========================\n\nScore raised to 5 following response to the rebuttal below, then to 6 following the re-rebuttal.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This submission considers the problem of learning disentangled representations from data in which there are correlations between underlying factors of variation (FoVs). Much of the work on learning disentangled representations has considered simulated datasets in which the FoVs are conditionally independent. The authors perform an extensive experimental evaluation (4000+ trained models). The main findings of this evaluation are that:

- Existing methods fail to disentangle when ground truth FoVs are correlated, in the sense that learned factors will reflect the correlations in the training data
- Metrics for disentanglement do not necessarily reveal correlations in underlying factors
- Semi-supervision and weak supervision can be used to induce learned factors that align with true FoVs

Reviewers expressed diverging opinions on this paper:

- R2 is a favor of acceptance, but does note that the paper is somewhat difficult to follow, owing to the fact that it presents a large number of results and does not quite arrive at a streamlined narrative.

- R4 was initially critical and posted detailed comments relating to framing, interpretation of existing work, and the conclusions that can be drawn from the presented experiments. The authors were able to address a number of these concerns in a detailed discussion with the reviewer. 

- R3 is critical of the experimental setup, which considers linear correlations between two underlying factors, and feels the semi-supervised and weakly supervised experiments have limited novelty. This reviewer did not respond to author feedback. 

The metareviewer has carefully read the reviews, author feedback, and subsequent discussion. Owing to the fact that reviews are diverging the meteareviewer also read the paper. 

As R4 notes, the results in this paper are in some sense unsurprising – we would expect correlations between underlying factors to lead to correlated learned factors. In fact one could even argue that dimensions in the latent space should reflect correlations in the training data. That said, the metareviewer feels that a paper need not present results that are surprising, as long as the experimental evaluation is rigorous and there are no major problems with framing and exposition. 

In this context, the metareviewer would like to express their appreciation to R4 for taking the time to follow up in detail with the authors, and for checking their revisions. The metareviewer feels that the fact that these revisions have a fairly large edit distance should not in itself not impede acceptance, as long as reviewers are agree that the edits improve the paper. 

The metareviewer is not entirely convinced by the criticisms presented by R3. The reviewer is of course correct that real-world datasets will not just have linear correlations between two factors. That said, an experimental evaluation of how correlations affect the degree of disentanglement has to start somewhere, and even these comparatively simple experiments represent a substantial effort on the part of the authors. 

Having read the submission, the metareviewer agrees with R2s assessment that the exposition is difficult to follow, even for readers who know the relevant literature. As the reviewer notes, the overall narrative could be clearer. Extracting a clear narrative is challenging when there are many experiments to report, but it is nonetheless something that the authors should spend additional time and thought on. Another factor that hurts clarity is that experiments are described in long paragraphs that often would benefit from an equation or two, for example to describe the substitution function in Section 4.1, or to more precisely describe the form of weak supervision that the authors employ in Section 4.2. 

As a final note, the metareviewer would suggest more explicit discussion on what authors think a VAE *should* do when factors are correlated. Arguably learning factors that reflect correlations is the "correct" when the training data exhibits such correlations. Currently the authors do not provide much of an arguments for *why* they think a VAE should learn a representation that ignores these correlations. A possible argument is that train-time correlations might not be representative of test-time correlations. Here, testing to what extent a learned representation generalizes to test-time data with a shift in correlation would also strengthen results.

On balance, the metareviewer's assessment is that this paper falls narrowly below the threshold for acceptance. While the experimental evaluation represents a substantial effort that in itself is above the bar, there are problems with narrative and the clarity of  writing that rise above the level of minor revisions that could be addressed by camera ready without additional review. Based on this, the metareviewer will recommend rejection. With a little bit more work on writing and exposition, this will make a great paper at a future conference. 
Decision: Reject