Exploring the Latent Space of Autoencoders with Interventional Assays
Authors: Felix Leeb
Stefan Bauer
Michel Besserve
Bernhard Schölkopf
Venue: NeurIPS 2022 Accept (2022)
Keywords: Autoencoders
Interventions
Structured Representation
Manifold Learning

Abstract: Autoencoders exhibit impressive abilities to embed the data manifold into a low-dimensional latent space, making them a staple of representation learning methods. However, without explicit supervision, which is often unavailable, the representation is usually uninterpretable, making analysis and principled progress challenging. We propose a framework, called latent responses, which exploits the locally contractive behavior exhibited by variational autoencoders to explore the learned manifold. More specifically, we develop tools to probe the representation using interventions in the latent space to quantify the relationships between latent variables. We extend the notion of disentanglement to take the learned generative process into account and consequently avoid the limitations of existing metrics that may rely on spurious correlations. Our analyses underscore the importance of studying the causal structure of the representation to improve performance on downstream tasks such as generation, interpolation, and inference of the factors of variation.

Exploring the Latent Space of Autoencoders with Interventional Assays
Felix Leeb
Stefan Bauer
Michel Besserve
Bernhard Schölkopf
Abstract
Autoencoders exhibit impressive abilities to embed the data manifold into a low- dimensional latent space, making them a staple of representation learning methods. However, without explicit supervision, which is often unavailable, the representa- tion is usually uninterpretable, making analysis and principled progress challenging. We propose framework, called latent responses, which exploits the locally contrac- tive behavior exhibited by variational autoencoders to explore the learned manifold. More specifically, develop tools to probe the representation using interventions in the latent space to quantify the relationships between latent variables. We extend the notion of disentanglement take the learned generative process into account and consequently avoid the limitations of existing metrics that may rely on spuri- correlations. Our analyses underscore the importance of studying the causal structure of the representation to improve performance on downstream tasks such as generation, interpolation, and inference of the factors of variation.
Introduction
Autoencoders (AEs) [2] and its modern variants like the widely used variational autoencoders (VAEs) [3, are a powerful paradigm for sulf-supervised representation learning for generative modeling 4 compression [5), anomaly detection (6) natural language processing [7]. Since autoencoders can learn low dimensional representations without requiring labeled data, they are particularly useful for computer vision tasks where samples be very high dimensional making processing, transmitting, and search prohibitively expensive. Here VAEs have shown impressive results, often achieving state-of-the-ard results compared to other paradigms (8] 10
The striking performance coupled with the relatively flexible approach has prompted an explosion of variants to learn representation with some structure that is particularly conducive
given
task [12 13 14]. In addition to meaningful
dineeniimnsii representations of our world [[3], the focus may be to improve generalization [16] I8) increase interpretability by disentangling the underlying mechanisms 19
20
21
22 or even to enable causal reasoning 23 24
In designing ever more intricate training objectives to learn more specialized structures part of complicated model pipelines, it becomes increasingly important to gain a better understanding of what the representation actually looks like more quickly identify and resolve any weaknesses of a proposed method. Here the manifold learning community provides principled formulation to analyze and control the geometry of the data manifold learned by the representation 25
26 27 28
29 30 31
32 33 Developing tools to better understand the structure of the representation is not only useful as diagnostic to identify avenues for improving modelling and sampling [3 35 but also has crucial importance for fairness [6 37 and safety [8 38
Our focus here is on taking advantage of common properties of autoencoders to gain deeper understanding of the structure of the representation. We summarize our contributions as follows:
Email: flesb@tuebingen.c wwwpenggn..n..........mm.. de
Max Planck Institute for Intelligent Systems, Tübingen, Germany
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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We propose framework, called latent responses, which exploits the locally contractive behavior of autoencoders distinguish the informative components from the noise in the latent space and to identify the relationships between latent variables.
We develop tools to analyze how the data manifold is embedded in the latent space by estimating the extrinsic curvature which also enables semantically meaningful interpolations.
Where true labels are available, we use conditioned latent responses to assess how each true factor of variation encoded in the representation and introduce the Causal Disentanglement Score to quantify how disentangled the learned generative process is.
We release our code at [tttps//ggthb...c/////aaaoaaaaaa
Background
Representation learning begins with set of N observation samples X' C IR which originate from some unknown stochastic generative process with distribution x ~ p(X) and support X. This data manifold X' embedded into low-dimensional (d << D) latent space R2, and is modelled by the support Z an encoder f: X + Z with the decoder learning the inverse mapping Z + X where after training & = X.
Variational Autoencoders (VAEs) 3 40 are framework for optimizing a latent variable model p(X) s PX Z; ()))(ZZ) with parameters 0, typically with a fixed prior p(2) = (Z; 0, I),
using amortized stochastic variational inference. variational distribution q X; Φ) with parame- ters @ approximates the intractable posterior p(2 XX). The encoder and decoder are parameterized such that q(2
(z)), and E [p(X Z = 2; 0)] = g²(z) where f*(x and 8 (x). and g (2) are neural networks which are jointly optimized using the reparameteriza- tion trick (3 to maximize the ELBO (Evidence Lower BOund) which is a lower bound to the log likelihood:
[ X = d) = N fz o
log (X: Ey(2 (x,) [log((X] 2;0) DRL. (x; ())))((()
L0.0
(X):
(1)
Note that practice, p(X) is unknown, and we only have access to samples (1x f so p(X) approximated by x(X = x) = n Σ 8(x - (0)) The first term in the objective corresponds to reconstruction loss, while the second can be interpreted as regularization term to encourage the posterior to match the prior.
2.1 Related Work
Representation geometry A closely related approach is manifold learning which aims exploit the geometry of the data manifold, usually by regularizing the geometry of the representation by
estimating the intrinsic curvature of the data manifold [26] 27 28
30 or by improving sampling and interpolation using the the Riemmanian metric B1 32 25 33 In comparison, our response
29
maps estimate the extrinsic curvature, which focuses the specific embedding, rather than being an intrinsic property of the dataset (see appendix A.4).
Interpretability and Disentanglement Another general approach is to gain a better understanding of the representation by making it more interpretable, by, for example, disentangling the true factors While our analysis is more similar approaches that improve the representation by focusing on the structure of the representation, for example by learning extra post-hoc models in the latent space 45 46 47), we develop a metric to evaluate how disentangled the learned generator is, rather than just the encoder.
of variation [19 22 41
Autoencoder Consistency VAEs have been investigated from an information theoretic viewpoint and with respect to training problems like posterior collapse [50] or the holes problem 51 to better understand common failure modes. Similarly, mismatches between the encoder and
48
49
decoder [52 53 have spurred research into increasing the self-consistens
of autoencoders [54] 58 Our latent response framework relies on a very similar approach and formulation, but crucially, thus far these methods focus on regularizing the training of the representation to impose certain desired structure, while we focus on analyzing the structure that is learned rather than modifying the training objective, making our tools directly applicable to virtually all VAEs.
55
56
57

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One implication of separating the deterministic and stochastic part of the encoder is a new perspective on the training signal for the decoder from the VAE objective. Substituting our definitions for S and U into the reconstruction loss term (shown equation 5 reveals the decoder trained map all 3 Latent Responses
On a high level, to explore the structure of how the data manifold is embedded in the latent space, it is necessary to separate the semantic information from the noise in the latent space. So our goal is to decompose the latent variables Z into an endogenous S and exogenous U component, which, for simplicity, we choose to relate to one another as shown in equation 2
Z S=
Conceptually, S should capture the semantic information necessary to reconstruct the sample X, while U is local noise model in the latent space for given observation X which does not mean- ingfully affect the semantics. In the context of VAEs with a Gaussian prior, we propose equations 3
and 4 which recover the familiar posterior for VAEs g(Z X = z; 6) [(S|x;))g(( (x
( (z)
( = X- = s (z) s)
(3)
q(U = xx) -U0
(4)
samples from the posterior match the same observation sample 2z(() As result, the decoder learns to filter out any exogenous component u from z around 8, making the latent space locally contractive around S to the extent of U. Although contractive behavior is even observed in unregularized autoencoders to some extent [59] suggesting this may be more fundamental feature of the inductive biases in deep autoencoders.
observe this byproduct of the VAE objective, this
[u [=z [logp(X = x0() | u; 0)
(5)
where ((() = (z((0) is the (deterministic) latent code corresponding to the observation 2((i)
Starting from some latent sample = ~ p(Z), we would like separate the constituent exogenous u and endogenous 8 components. If we had the matching & such that z 4(2 = x; Φ). then the separation would be trivial since, by definition s = (x) and u = 2 - s. However, since the VAE is optimized to reconstruct observations from the latent space. we approximate the missing with è = (z). Subsequently, we encode the generated sample to infer an approximation of 8, = (t). Now by expanding g around and f around æ to the first order, we are left with three terms shown in equation 6 (neglecting higher order terms, full derivation in appendix A.2).
$ =
+J (z) - x)) (z) (s)u [c]] + [u²]
(6)
where c = 2 - x is the reconstruction error, J (x: is the jacobian of f evaluated at x and J (s) is the jacobian of g evaluated at
Term aligns with our conceptual interpretation of the semantic information should remain invariant to any noise information also contained in z. Meanwhile the second and third terms correspond two different sources of error to this interpretation we must potentially take into account. The term 2 derives from the encoder struggling to encode samples that were not seen during training, since 7 (X) A p(X Z: ())).. However, provided the reconstructions sufficiently faithful across latent space (i.e. (s))
is small), this term may be ignored. This error can be further mitigated by training the encoder to be more robust with respect to the input (diminishing J (x)) with mild additive noise on the input observations or additional regularization terms (54 52 58
Finally, term 3 in equation 6 originates from the decoder having to filter out the stochastic exogenous information from 2 when decoding. As discussed above in equation [5] the VAE objective already directly minimizes this term by training the decoder output to be invariant to noise samples u, thus diminishing (s).
The bottom line is, as long as the decoder can filter out the exogenous information u from latent sam- ples and the encoder can recognize the resulting generated samples &, the endogenous information
3

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is preserved. We call this process of decoding and reencoding latent samples, h = ff o g" the latent response function. Crucially, the latent response function allows us to extract the semantic information from the latent space without knowing how the information is encoded, so although we can identify the structure of the representation, the representation is not necessarily interpretable without ground truth label information. A statistical treatment of this phenomenon is discussed in the appendix, however provided the error terms are sufficiently small the corresponding response distribution r [ Z: 0, Φ) is shown in equation 7
r( 0) = E | ())
One subtlety of our interpretation remains to be addressed: how ambiguities due to overlapping pos- terior distributions resolved by latent responses. To match the prior, there will be overlap between posterior distributions which correspond to semantically different samples. Statistically speaking, these ambiguities are captured by the higher moments of r 2 Z: f, Φ). Despite 5=(212:0.0) [$] = S, the variance can be interpreted to relate to the uncertainty of the encoder in the inference of the endogenous variable from the generated sample, suggesting a potential signal for anomaly detec- tion 555 From another perspective, a strong deviation between å and S implies there is some inconsistency between the encoder and decoder, which identifies the "holes" in the latent space 51 where the encoder has trouble recognizing the samples generated by the decoder.
3.1 Interventions
intervention response
To control the learned generative process, need to know how to manipulate specific se- mantic information in the representation. We conceptualize these manipulations as interven- tions where a chosen latent variable is modified while all others remain unchanged. For exam- ple, given latent sample 2, A (z))=)))) refers to the interventional sample where the jth latent variable is resampled from the marginal of the aggregate posterior žj
Figure I: Encoded data points are shown as black points with the surrounding latent manifold shaded blue. An intervention depicted as the orange ar- row might replace the semantic information along the horizontal latent dimension with that of the sample with a yellow border. In this case, the in- tervention results in the latent sample leaving the latent manifold. Now the latent response function approximately filters out the exogenous noise to ef- fective project the sample back onto the manifold. Note that this projection (green arrow) changes both the horizontal and vertical dimensions, from which we infer there is non-trivial relationship between the horizontal and vertical.
(z)
When intervening on latent variables Z, there are two possible outcomes: either the resulting generated sample is affected significantly, that is to say, the semantic information S' is affected by the intervention, or there is no significant change in which case only the noise was affected. In the first case, we identify a specific kind of in- tervention manipulate the learned generative process.
Latent Response Matrix This brings us to the first practical tool we introduce based on latent responses, which aims to describe to what extent the latent variables causally affect one another with respect to the learned generative process. Each element in the matrix M E RR quantifies the
degree to which an intervention latent variable j causes a response in latent variable & as seen in
equation g where f refers to the kth variable of the latent response (see figure
M3,
1
2 B==)(()), [ ())))=))))))) - n (2)
(8)
Along the diagonal, M,j can be interpreted as quantifying the extent which intervention along the jth latent variable is detectable at all. As the value approaches O, changes in the latent variable do not affect the generated sample in any way detectable by the encoder. For VAEs, this is frequently due to posterior collapse [60, 61, 50 62] On the other hand, if the latent variable is maximally informative and the intervention ž, is fully recoverable it implies negligible exogenous noise, so
then 00 (2) R 2 and h (2) = 2j. and Mjj approaches 1, since both 2, and 2j are sampled independently from standard normal, and the elements are normalized with a factor of 1.

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Perhaps even more interesting. the off- off-diagonal elements of Mi show to what extent an intervention on variable j affects variable k. Consequently, the latent response matrix can be interpreted weighted adjacency matrix of directed graph of the learned structural causal graph. Note that there may be cycles in this graph if there is some non-trivial relationship between latent variables. For example, the model could embed a periodic variable into two latent dimensions, such J1 and j2, to keep the representation continuous. In that case, we would expect an intervention along dimension to elicit similar response in j2 the response of ji from intervention on j2. Consequently, Mi Mizii > O suggests ji and j2 should be treated jointly as a single latent variable. Thus, the latent response matrix not only describes the causal structure of the learned generative process, but it can also identify more complex relationships between latent variables for further analysis.
Conditioned Response Matrix While the latent response matrix M, lets us quantitatively compare how much each latent dimension affects another, without manual inspection or label information, the correspondence between the learned causal variables the true causal factors is, in general. unknown [99.
However, when we have access the true generative process, at least the ground truth labels y(t y Rd' corresponding to the observation samples z01), then there is variant of the latent response matrix, termed the conditioned response matrix, to quantify how well the learned variables match with the true ones, which is closely related to the disentanglement of the representation.
The key is to carefully select our interventions such that they only affect one true factor at a time, and then evaluate to what extent these interventions for each of the latent variables are still detectable. Intuitively, if latent variable Zj only captures information pertaining to factor Yz, then if we select interventions that only change factor Ye' where c' o c, then interventions Zj do not produce response, so M₁ R O.
To condition the set of interventions on a specific factor Y.. we choose a subset of observations which are all semantically identical except for a single factor Y,, x ~ p(X Ye, (-)) where p(X (Y) refers to the true generative process given label Y and Y-c refers to all true causal variables except Yz- Then the latent response matrix is computed with interven- tions exclusively sampled from the resulting aggregate posterior of this subset g(Z ( +) [ X; Φ)p(X X (Y, Y (Y))d(Y)
Mr 2 1E 2==) (-s)))((-)))) [ (A(e+
h2 (:)
(9)
In the context of controllable generation, the conditioned response matrix quantifies how much an intervention on each latent variable can affect each of the true factors of variation. Ideally, each true causal factor would only be manipulable by disjoint subsets of the latent variables, which is commonly referred to as "disentangling" the factors of variation. From the conditioned response matrix, can identify not only which latent variables contain information pertaining each of the true factors of variation, but also which factors are affected by an intervention in each of the latent variables.
Causal Disentanglement Score To more easily compare representations, we can aggregate the information in the conditioned response matrix into a single value to measure the degree to which the representation is able to disentangle the true factors of variation. The causal disentanglement score (CDS) equation T0 which allows each latent variable to causally affect single factor, but penalizes any additional responses. As written, the score is between de and 1, but re-scale it to [0, 1].
ma MC
CDS Ees M
(10)
(To our knowledge) none of the existing disentanglement metrics take the learned generative process into account all. If the task only to infer the true labels from the observations (as common), then the decoder is admittedly superfluous. which is why disentanglement methods generally focus on the encoder. However, in tasks such as controllable generation, where disentanglement is obviously valuable, the behavior of the decoder is critical. Here, the main risk in evaluating disentanglement from the encoder alone is that some latent variables may correlated with true factors of variation without them having any causal effect those factors when generating new samples. These spurious
5

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correlations also potentially decrease the resulting disentanglement score, which may falsely penalize larger representations.
The conditioned response matrix and associated CDS mirrors the responsibility matrix and disentan- However, erucially, the responsibility matrix identifies how well each latent variable correlates with each true factor of variation.
glement score introduced by 42
Response Maps The last type of analysis we propose focuses on gaining qualitative understanding of how the data manifold is embedded in the latent space. Specifically, we exploit the ability to probe the latent manifold using the response function to map out the manifolds extent, including estimating its extrinsic curvature. From the definition of s = (xx) where s E Z. x E X and fo is a smooth projection of the data manifold X into the latent space.
deterministic function, may interpret
Usually, our analysis of Z is limited to the observation samples of x we have access, from which man- ifold learning methods often estimate the intrinsic curvature of the data manifold X' However, using latent responses, we can use any latent sample 2 ~ p(Z) to probe as long as á =z s. Rearranging the terms gives us u(2) in equation 11 whose magnitude can be interpretted as the unsigned distance function to the latent manifold, which is evocative of the neural implicit functions 663 suggesting a variety of further tools we leave for future work.
64
u(=)
2 R 1 00 (z)
(11)
Treating [u[2]] as an approximate distance function to the manifold, we compute the mean curvature H of the latent manifold (see appendix for further discussion). In practice, we estimate the necessary gradients by finite differencing across 2D grid in the latent space, we call the resulting map the response map, visualized similarly as in [65].
Qualitatively, with our sign convention, high curvature corresponds to regions where u=z) is small and locally convergent, and consequently where we find the data manifold. Empirically, we also find the divergence of u(2), which is closely related to the mean curvature, and is useful for identifying the regions of the latent space where u diverges, which may be interpreted as possible "holes" the latent space.
When interpolating between latent samples z, ideally we would follow the geodesic of the data manifold. This can be done by estimating the Riemannian metric from the data samples and integrating an expensive ODE to find the geodesic connecting samples [25). The Riemmanian metric relates to the intrinsic curvature of the data manifold, which is independent of the embedding. consequently guaranteeing we find the geodesic independent of the representation. However, in our case, we use the latent responses to estimate the extrinsic curvature which does depend on the embedding, so the resulting path may not be optimal with respect to the underlying data manifold. Nevertheless, we optimize the path along the response map to stay in high curvature regions, thereby effectively are finding a path in the latent space which stays near the data manifold.
4 Toy Example: The Double Helix
To illustrate how the latent response framework can be used to study the representation learned by VAE, we show the process when learning a 2D representation for samples from double helix embedded in R Disregarding the additive noise, the data manifold has two degrees of freedom (data manifold formally defined in the appendix). This analysis is largely independent of the precise neural network architecture, provided the model has sufficient capacity to learn a satisfactory representation (hyperparameters/ in the appendix).
Figure 2 provides an example for how the response maps can be used to trace the latent representation through the mean curvature and divergence. Note that the magnitude of the response [u(2)] is not sufficient to identify the latent manifold since both the regions with maximal and minimal curvature have minimal response. The divergence shows most of the latent space has a slightly negative divergence, implying most of the latent space converges, rather than diverges, which is consistent with expectations. The mean curvature shows what regions of the latent space the map converges to in yellow, from which we can recognize the structure of the learned latent manifold. Finally, the right most plot shows the aggregate posterior g(S x;) of the 1024 training samples.
This toy example also motivates the value of meaningful interpolations as seen in figure 3 The path in red shows the shortest euclidean path between the two samples in orange, but note that the

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Mogritude of Ressonse
Divergence
Moan Curvettue
Posteror
14
05
003
08
600
5.4
0.44
-20
0.45
Figure 2: Depicted are three quantities derived using the latent response function compared to the aggregate posterior shown on the far right for the representation learned by a VAE for the double helix toy dataset. Note that almost all the density of the posterior is in regions that have positive curvature, corresponding to the data manifold.
corresponding path in the observation space jumps from one strand to another twice. Meanwhile, the path that maximizes the estimated mean curvature shown in green and produces much more reasonable interooaation.
15
1.0
1.-
0.5
-0.5
-1.0
-55
1.0 -1.0
-0.5
0.5
1.0
-20
-2
Figure 3: The left plot shows the 2D latent space including the aggregate posterior density in black, and possible interpolations between the two pink points. Meanwhile, the plot the right shows the ambient space with the black points being the observed data samples, with the blue points showing the reconstructed samples, and the paths in the ambient space corresponding the ones in the latent space. Note how the path in green follows the learned manifold and consequently much more consistent in the ambient space compared the shortest euclidean path in red.
5 Experiments & Results
Experimental Setup We apply our new tools on a small selection of common benchmark datasets, including 3D-Shapes [66], MNIST [67], and MANIS [68] Our methods directly applicable to any VAE-based model, and can readily be extended to any autoencoders. Nevertheless, focus our empirical evaluation vanilla VAES and some 3- B-VASs (denoted by substituting the 3, so 4- VAE refers to 3-VAE where B = 4). Specifically, here we mostly analyze a 4- VAE model referred
with d 24 latent space trained on 3D-Shapes (except for table [
Model A, and include results on range of other models in the appendix. All our models use four convolution and two fully-connected layers in the encoder and decoder. The models are trained using Adam 69 with a learning rate of 0.001 for 100k steps (see appendix for details).
Qualitative Understanding The direct way get better understanding of the manifold structure is the visualization of the response maps, in particular with the mean curvature (see figure Unfortunately, since the curvature is estimated numerically using a grid of samples, the maps do scale well to the whole latent space. Here the latent response matrices help identify pairs related latent dimensions, which can then be analyzed more closely with response map. Furthermore. currently, all these response maps are aligned to the axes of the latent space. Although VAEs do align information along the axes somewhat [70), any off-axis structure is missed since the off-axes responses are completely ignored. Since, an implicit requirement of disentangled representations that the information axis-aligned [19, the response maps present the most striking results for disentangled representations (see appendix for more examples).
All provided with MIT. Apache or Creative Commons License

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400
(a) Divergence Map and Aggregate Pos- terior
(c) Reconstructions over
(b) Mean Curvature Map
the same space
Figure 4: A projection along dimensions 16 (horizontally) and 22 for Model A (4-VAE) shows the computed divergence of the response field in blue and red while the green points are samples from the aggregate posterior. [4b] shows the mean curvature, which identifies 10 points where the curvature spikes and the boundaries between the regions corresponding to different clusters in the posterior. Finally, from the corresponding reconstructions in 4c (with all other latent variables fixed) becomes clear that each the clusters in the posterior corresponds to a different floor hue. Note that, although the aggregate posterior is highly concentrated at few points, the negative divergence almost everywhere suggests the extent of X the decoder can handle extends well beyond the posterior (as confirmed by the reconstructions).
(c) Shortest Path Reconstructions
(b) Interpolation between latent samples the manifold.
(a) Mean Curvature Map.
(d) Best Path Reconstructions
Using Model A (4- VAE) in similar setting to figure we now search for an interpolation between two latent samples from the posterior along the latent manifold, visualized in 15/5a Figure 5b compares this best path (in green) to the shortest path in euclidean distance (in red), Finally, latent samples along each of the paths even intervals are decoded showing that the shortest path results in blurry, unrealistic shadows in the middle of 5c compared to 5d
Figure 5:
that the manifold is particularly nonlinear along these two latent dimensions (13 and 14). Consequently, there can be a dramatic difference between the geodesic (here approximated using the mean curvature), compared the shortest path in cuclidean space as seen in figure 5
From the mean curvature response map in figure
Causal Disentanglement Table unsurprisingly, shows the disentanglement increasing with increasing 8. Our proposed CDS score correlates strongly with the other disentanglement metrics. Perhaps noteworthy that even though the CDS and DCI-D scores are computed similar ways (the vital difference being whether responsibility rests a causal link statistical correlation), the DCI-D scores are consistently lower than the CDS scores. This may be explained by the DCI-D metric
Name
CDS
DCI-D
IRS
MIG
Table Comparing disentanglement met- rics for B-VAEs trained on 3D-shapes with varying 3. For the models the first four rows d = 12, while d 24 for the remain- ing four. While the CDS generally correlates well with other metrics, notably, the DCI-D score is consistently slightly lower which may be due to spurious correlations between
I- I-VAE
0.44
0.3
0.44
0.07
2-VAE
0.49
0.36
0.46
0.09
4-VAE
0.58
0.46
0.51
0.16
8-VAE
0.71
0.66
0.63
0.21
1-VAE
0.52
0.49
0.48
0.09
2-VAR
0.61
0.58
0.52
0.15
VAE
0.72
0.67
0.57
0.17
8-VAE
0.78
0.73
0.64
0.2
latent variables and the true factors.
8

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8:00
000
0.34
GOC
9:03
0.03
9/0
5/02 0.09
1000
0.90
503
9.64
6/2
0.09
0.52
502 6.00
4-033
0:07
232
5:03
6.22
0.02
0.03
0.05
5/02 0.00
38-
13-0.02
2.00
202
0:33
0.02
0.02
5.02 0.00
14 0000
0.05
5.00
5.44
0:02
0.00
0.02
(b) DCI Responsibility Matrix
0000
5.0
1.25
6:16
26-0006
0.05
5.04
5.82
2.52
0.53
0.50
0.00
07
6.48
5,02
Paa
21 02 0.05
5.0
5.0 3:53 0.02
0.01
0.00
0.75
505
5.03
042 0.00
8.0
6.0
0.00 0:03
2.04
2.04
3:05
0.28
8.03 0.00
0.04
0:0
1.52
000
6.0
A04
oc0
Resporse
ateraibs
6 2:99
(d) Graph conrespond- ing to the Response Matrix
(a) Latent Response Matrix
(c) Conditioned Response Matrix
Figure 6: These are the (6a) latent response matrix, (6b) DCI Responsibility matrix 42). (6c conditioned response matrix, and the (6d) derived from the latent response matrix for Model A (4-VAE) (d = 24) trained on 3D-Shapes. The responsibility matrix shows the predictability of each latent dimension (column) for each factor of variation (row), while the conditioned response matrix measures the effect an intervention in each latent dimension (column) has provided that the intervention can changes a specific factor of variation (row). Note that from the conditional response matrix see only dimensions 16 and 22 are causally linked to the floor hue, for which the structure is further visualized in figure 4
taking additional spurious correlations between latent variables into account (as seen figure 6). while the CDS metric focuses on the causal links, so the DCI-D has an undeservedly low score.
Causal Structure Closer comparison between the conditioned response matrices and the responsi- bility matrices reveal more how the DCI-D metric and CDS differ. Figure/osshove the responsibility matrix matches the conditioned response matrix for the most part. The only exception being latent variables 4 and 20. Since the DCI framework identifies which latent variables are most predictive for the true factors, cannot distinguish between correlation and a causal link. In this case, the DCT metric recognized a correlation between dimension 4 and the "scale" factor. However, from the latent response matrix (and the graph (d) we see interventions on dim 20 have a significant effect on dim 4, but not vice versa. Consequently. we identify dim 20 is parent of dim 4 in the learned causal graph. Since dim 20 is closely related with the "scale" " factor the causal link to dim 4 results in dim being correlated with "scale". The conditioned response matrix correctly identifies that it is dimension 20 which primarily affects the scale, but indirectly also affects the shape through dimension 4. This is prime example of how causal reasoning can avoid misatrributing responsibility due spurious correlations.
6 Conclusion
In this work, we have introduced and motivated the latent response framework including variety of tools to better visualize and understand the representations learned by variational autoencoders. Given an intervention on a sample in the latent space, the latent response quantifies the degree to which that intervention affects the semantic information in the sample. Therefore, we can think of this analysis as leveraging the interventional consistency of representation to study the geometric and causal structure therein. Notably, the current analysis relies on a certain degree of axis-aligned structure in the latent space, which makes these tools especially useful for understanding the structure of disentangled representation. Another limitation is that computing the latent response maps to, for example, improve interpolations, does not scale well for the large representations of high fidelity generative models [8. Consequently, our experiments thus far have focused on synthetic datasets designed for evaluating disentanglement methods. However, latent responses do not require any ground truth label information, which is particularly promising for better understanding representations of real datasets and consequently speeding up development of not just better performing representation learning techniques, but also more interpretable and trustworthy (71) models.

---

Acknowledgements
This work was supported by the German Federal Ministry of Education and Research (BMBF); Tübingen AI Center, FKZ: 011S18039B, and by the Machine Learning Cluster of Excellence, EXC number 2064/1 Project number 390727645. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Felix Leeb.
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13


Reviews:
[{"review_id": "ztyBhsuJ8v", "reviewer": "Reviewer_g1w9", "summary": "This paper describes an approach to understanding the structure in VAE models. At the core of this approach is what the authors term the latent response map, the function obtained by first decoding and the re-encoding a point in the latent space. This is used to build a latent response matrix, which measures the degree to which modifying each latent variable affects the values of each other latent variable. This can then be used to understand the underlying causal structure in the latent space underpinning a VAE. \n\nIn cases where the true generative model is known an extension of this, which the authors term the conditioned response matrix, follows a similar approach to measure the degree to which changes in a true generating factor influences each latent factor. This can then be used to derive a new disentanglement metric. \n\nFinally, the latent response function can also be used to build what the authors term a response map, which uses the latent response function to estimate the mean curvature in the latent manifold. This provides insight into the structure of the latent space and can be used to efficiently compute approximate geodesics between latent samples (if I understand correctly)\n\nThe method is first evaluated on simple toy data: a 3D double-helix generated from 2D latent space (samples from a uniform and Bernoulli r.v.). This is used to illustrate how the method gives insight into the learned latent manifold and can be used to meaningfully interpolate between points in the latent space\n\nThe method is then evaluated on the 3D-shapes dataset, a synthetic data set that is often used for benchmarking methods for learning disentangled representations. This is used to demonstrate:\n(i) the mean curvature map captures meaningful structure in the latent space - the clusters implied by the curvature map for a pair of latent variables coincide with different floor colours in the generated data;\n(ii) the response map can be used to meaningfully interpolate between points in the latent space; and\n(iii) the disentanglement score inferred from the conditioned response matrix aligns well with other metrics of disentanglement\n\n\n", "questions": "None", "limitations": "n/a", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n- The method described in this paper seems novel and tackles important problems in representation learning\n\nWeaknesses\n\n- I found the writing in the paper difficult to follow at times. For example, in the explanation of the conditioned latent response matrix it took me a while to understand \"subset of observations which are all fully identical except for the factor Yc\" referred to a subset of observations which agree on all other true factors are fixed and which take different values for Yc. I initially thought \"fully identical\" referred to agreeing on X also.\n\n- Empirical evaluation was limited to toy / synthetic data. For a paper like this, I feel this is just about sufficient but the paper would certainly be strengthened by the application of the method to real data", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "P0EUMBEE_QR", "reviewer": "Reviewer_4nGX", "summary": "This paper concentrates on studying the learnt latent space of VAE. Unlike many existing works which view VAE as simply two parts, reconstruction and alignment between posterior and prior, this paper decomposes the latent variables $Z$ to endogenous and exdogenous components, $S$ and $U$, respectively. This separation of deterministic and stochastic parts of the encoder is indeed quite novel. The paper further proposes a decoding-then-reencoding process to test if the encoder can recognize the resulting sample and if the decoder can filter out exogenous information. This is named as \"latent response\". Latent responses can help extract the semantic information from the latent space without the need of true label information. Besides, after obtaining the aggregated posterior, each individual dimension's marginal can help build the latent response matrix which reflects the surrounding latent manifold for the input $x$. While this matrix is not informative enough to indicate the relations with semantic meanings, once the label information is available, one can build the conditional response matrix to draw the correspondence. The element scale in the latent response matrix also implies the inter-dimension connections. The resulting analysis tool is applied to a toy example, the double helix, and several common benchmark datasets, including 3D shapes, MNIST, and fashion-MNIST. ", "questions": "1. Typically, when we say aggregated posterior, it refers to $q(z)=E_x[q(z|x)]$. Could you specify if the subscript $j$ from on page 4? is the index of dimensions? Is there any consideration when you choose square $|\\cdot|^2$?\n2. On line 181, it is sometimes non-trivial to say if $M_{j_1, j_2}$ approximates $M_{j_2, j_1}$. Do you set a fixed threshold?\n3. Why use CDS rather than a more entropy-base metric to evaluate the disentanglement? The current CDS might go better or worse due to some individual $j$'s.\n4. How do you determine whether $\\hat{s}=s$ for boundaries in response map?\n5. If I know the labels, how would you compare your analysis tool with some supervision-guided methods where the separate latent variables are learnt for different semantic variations?\n6. This analysis method targets on each individual dims and then re-group some of them. Some other works [1,2] tend to directly learn the factors for different groups. Given that group factor learning is becoming another trending direction, how would you compare your method and these methods?\n\n[1] Li, Y. and Mandt, S., 2018. Disentangled sequential autoencoder. arXiv preprint arXiv:1803.02991.\n[2] Bai, J., Wang, W. and Gomes, C.P., 2021. Contrastively disentangled sequential variational autoencoder. Advances in Neural Information Processing Systems, 34, pp.10105-10118.", "limitations": "1. It would be nicer if the authors can use the double helix example as the running example to show how to derive the divergence and mean curvature. \n2. Since disentanglement and the understanding of the latent space is mostly for the generation tasks, training an additional discriminator to distinguish different factors could be a fun validation experiment. ", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pros:\n1. This is a very interesting idea. Re-encoding the decoded sample is a novel way to analyze the latent space. Building a correlation matrix for the latent dimensions without any supervision but only interventions is also potentially very helpful for VAE studies.\n2. Grouping similar or semantically correlated latent dimensions is always a goal in VAE research. I think this work provides a very useful tool to this end. The proposed analysis tool also facilitates the post-process when some semantic hints become available after a VAE is self-supervisedly trained.\n\nCons:\n1. The latent space is continuous, sometimes it is hard to say if there is any edge between the posterior distributions. For example, when mnist digit \"2\" transits to \"1\", it might first become \"7\". But one cannot say \"7\" is the boundary. \n2. The choice of the square in build the (conditional) response matrix can be elaborated.\n3. While the construction of the response matrix $M$ is neat and elegant, I think it should be more clear how the scale in the latent space would affect this matrix. Note that there is no normalized scaling process during VAE encoding. But the euclidean distance can be greatly affected by the scales. \n4. It would be great if the analysis tool can be applied to a larger-scale dataset to show its effectiveness. \n5. There are many typos and grammar issues in the paper. e.g. \"can changes\" --> \"can change\"", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "823PLP9Fh0", "reviewer": "Reviewer_dJu3", "summary": "The paper introduces the latent response framework for evaluating the disentanglement of a representation that is learned by Variational AEs. The framework is based on the so-called latent response function, which takes in input a latent sample, decodes it, and then re-encodes it in order to get a version of the initial latent sample \"pushed\" into the learned manifold. By performing intervention on individual dimensions of the latent sample, and then applying the latent response function to them, it is possible to quantify how the latent dimensions are causally related to each other and to the true factors of variations (when there are any). \n", "questions": "METHOD:\n\n1. The authors should state in a more explicit way what do they mean for intrinsic and extrinsic curvature of the latent manifold, since they are not standard concepts in the machine learning community.\n\n2. (Line 62) the second term of the ELBO does not encourage the aggregate posterior to match the prior. If that was the case, the term would be  KL(q(Z)|| p(Z)), as q(Z) is the aggregate posterior. The second term it actually encourage the conditional posterior q(Z|X) to match the prior.\n\n3. (Line 132) \"the posterior distributions must overlap to, in aggregate, match the prior\". As before, the ELBO actually encourages each conditional posterior to match the prior. This sentences can be misleading. You still have the overlapping, as each posterior is pushed towards the standard normal distribution.\n\n4. I would appreciate a complete and explicit derivation of equation 6 to be included in the appendix, since it is one of the main contribution of this work.\n\n5. The method seems to work for standard normal prior, but can it still be used also for other types of priors?\n\n6. The CDS score assumes that a disentangled representation encodes each generative factor into a single separate latent axis. This is, in general a restrictive assumption. Since the latent response matrix can catch the case where a generative factor is connected to more than one latent dimension, I think that the definition of CDS metric should take into account that case. \n\nEXPERIMENTS:\n\n7. The two interpolations of figure 5.c and 5.d do not seem visually differente: I would appreciate to see more examples of latent interpolations on different datasets in the appendix. Also, the caption of figure 5c does not specify which latent dimensions are considered.\n\n8. The matrices of figure 6 show only a subset of latent dimensions. It would seems that, as \\beta increases, most of the latent dimensions do not encode any information about the data at all. A deeper discussion on the phenomenon should be added. \n\n9. (lines 305-306) \"This may be explained by the DCI-D metric taking additional spurious correlations\". The author should provide some evidences of this claim (either via formal proofs or empirically).\n\n10. For the sake of self-containment, the authors should provide a brief definition of responsibility matrix, discussing the differences between it and the conditional response matrix of this work, since judging by Figure 6 the results seems to be quite similar.\n\nMINORS AND TYPOS:\n- Line 18: \"representations\" instead of \"representation\"\n- 32: the reference [31] is cited twice.\n- 54: (last word of the line) \"are\" instead of \"and\"\n- 57: a reference to the reparameterization trick should be included.\n- 103-104: \"then we could the separation would be trivial\"\n- 132: \"the\" instead of \"The\" after the colon.\n- 134: which corresponding.\n- 136: missing \".\" before \"Despite\".\n- 136: \"interpretted\". \n- Overall the sentence from line 132 to line 135 is a bit convoluted and difficult to understand. It should be restated more clearly.\n- 138: \"from a another\" instead of \"from an another\".\n- 191: to measure how quantify how well\n- 226: \"identifies to how\"\n- Equation 6: g(s) in the second term does not have the \\theta superscript.", "limitations": "The final discussion only hints at a (methodological) limitation of their work, that is the requirement for axis-aligned disentanglement. A larger discussion of such a limitation could have been interesting to have.\nNo discussio of societal impact of the work is provided: maybe the Authors could consider how mixed disentanglement-and-causality methodologies can be supportive of Trustworthy AI.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "ORIGINALITY\n\nThe paper fits into a quite active and consolidated research topic (studying disentanglement in VAE models) with a twist on more recent and heavily building research area concerning causal learning and causal explanations of deep models. The result is a work which has some value and novelty, contributing both on disentanglement and causality.  What I did not particularly like in the presentation of the work is a certain tendency to underlook relationship with early works on contractive autoencoders and manifold learning in AEs. These are sometimes references in the paper (e.g. [60]) but the relationship is not discussed. Referencing without discussion of the relationship with earlier work is not \"useful\" for the reader (it only builds up citation counts, which is not the objective of bibliography). \n\nQUALITY and CLARITY\n\nThe organization of the content is adequate and so is, mostly, the level of technical depth in the main text. However, there are aspects in the derivation of the main results which can be better clarified (sometimes it suffices to have those in the supplementary materials): please see the Questions section for detailed aspects on the matter. The writing style is not always clear and this is also due to diffused misprints and errors, which could have easily been spotted by a proper proofreading of the work. The authors should spend a little more effort to bring the paper to speed with the expected linguistic and presentation quality of the venue. \n\nSIGNIFICANCE\n\nAs stated the key catch underlyign the paper is interesting, though there are some techical questions that needs clarification to convince the reader that everything adds up (again please refer to the Questions section). One open question that is important to address pertains to the generality of the approach: the paper shows how the method can be applied to both vanilla and \\beta-VAEs, but it is largely unclear whether the same approach would work with priors different from a single Gaussian.  The experimental setting seems sound, although all the experiments are based on synthetic toy datasets. The experimental results are not always compelling, in particular those on the qualitative side, where it is quite difficult to visually spot the adde value of the proposed approach.  \n\nOverall, the work has some merits but there are also open issues that need addressing and clarification as part of the rebuttal discussion. Please refer to questions for such points of discussions and suggestions (also some minors).\n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept