From abstract items to latent spaces to observed data and back: Compositional Variational Auto-Encoder

The architecture proposed for CompVAE is depicted as a graphical model on Fig. 1. As said, the $i$-th part belongs to category $\ell_{\_}i$ and is associated with a latent variable $w_{\_}i$ (different parts with same category are associated with different latent variables). The ensemble of the $w_{\_}i$s is aggregated into an intermediate latent variable $\widetilde{w}$. A key requirement is for $\widetilde{w}$ to be invariant w.r.t. the order of elements in $x$. In the following $\widetilde{w}$ is set to the sum of the $w_{\_}i$, ${\mbox{$\widetilde{w}$}}=\sum_{\_}iw_{\_}i$. Considering other order-invariant aggregations is left for further work.
The intermediate latent variable $\widetilde{w}$ is used to condition the $z$ latent variable; both $\widetilde{w}$ and $z$ condition the observed datapoint $x$. This scheme corresponds to the following factorization of the generative model $p_{\_}\theta$:

In summary, the distribution of $x$ is conditioned on the ensemble $\{\ell_{\_}i\}$ as follows: The $i$-th part of $x$ is associated with a latent variable $w_{\_}i$ modeling the generic distribution of the underlying category $\ell_{\_}i$ together with its specifics. Variable $\widetilde{w}$ is deterministically computed to model the aggregation of the $w_{\_}i$, and finally $z$ models the specifics of the aggregation.

Notably, each $w_{\_}i$ is linked to a single $\ell_{\_}i$ element, while $z$ is global, being conditioned from the global auxiliary $\widetilde{w}$. The rationale for introducing $z$ is to enable a more complex though still learnable distribution at the $x$ level $-$ compared with the alternative of conditioning $x$ only on $\widetilde{w}$. It is conjectured that an information-effective distribution would store in $w_{\_}i$ (respectively in $z$) the local information related to the $i$-th part (resp. the global information describing the interdependencies between all parts, e.g. the fact that the households face the same weather, vacation schedules, and so on). Along this line, it is conjectured that the extra information stored in $z$ is limited compared to that stored in the $w_{\_}i$s; we shall return to this point in section 4.1.

The property of invariance of the distribution w.r.t. the order of the $\ell_{\_}i$ is satisfied by design. A second desirable property regards the robustness of the distribution w.r.t. the varying number of parts in $x$. More precisely, two requirements are defined. The former one, referred to as size-flexibility property, is that the number $K$ of parts of an $x$ is neither constant, nor bounded a priori. The latter one, referred to as size-generality property is the generative model $p_{\_}\theta$ to accommodate larger numbers of parts than those seen in the training set.

Letting $p_{\_}D(x|\{\ell_{\_}i\})$ denote the empirical data distribution, the learning criterion to optimize is the data likelihood according to the sought generative model $p_{\_}\theta$: $\operatorname*{\mathbb{E}}_{\_}{p_{\_}D}\log p_{\_}\theta(x|\{\ell_{\_}i\})$.
The (intractable) posterior inference of the model is approximated using the Evidence Lower Bound (ELBO) [14], following the Variational AutoEncoder approach [17, 27]. Accordingly, we proceed by optimizing a lower bound of the log-likelihood of the data given $p_{\_}\theta$, which is equivalent to minimizing an upper bound of the Kullback-Leibler divergence between the two distributions :

The learning criterion is, with $q_{\_}\phi(z,\{w_{\_}i\}|x,\{\ell_{\_}i\})$ the inference distribution:

The inference distribution is further factorized as $q_{\_}\phi(\{w_{\_}i\}|z,x,\{l_{\_}i\})q_{\_}\phi(z|x)$, yielding the final training loss:

The training of the generative and encoder model distributions is described in Alg. 1.

In CompVAE, the sought distributions are structured as a Bayesian graph (see $p_{\_}\theta$ in Fig. 1), where each node is associated with a neural network and a probability distribution family, like for VAEs. This neural network takes as input the parent variables in the Bayesian graph, and outputs the parameters of a distribution in the chosen family, e.g., the mean and variance of a Gaussian distribution. The reparametrization trick [17] is used to back-propagate gradients through the sampling.

A concern regards the training of latent variables when considering Gaussian distributions. A potential source of instability in CompVAE comes from the fact that the Kullback-Leibler divergence between $q_{\_}\phi$ and $p_{\_}\theta$ (Eq. (4)) becomes very large when the variance of some variables in $p_{\_}\theta$ becomes very small¹¹1Single-latent variable VAEs do not face such problems as the prior distribution $p_{\_}\theta(z)$ is fixed, it is not learned.. To limit this risk, some care is exercised in parameterizing the variances of the normal distributions in $p_{\_}\theta$ to making them lower-bounded.

As said, CompVAE is meant to achieve a programmable generative model. From a set of latent values $w_{\_}i$, either derived from $p_{\_}\theta(w_{\_}i|\ell_{\_}i)$ in a generative context, or recovered from some data $x$, it should be able to generate values $\hat{x}$ matching any chosen subset of the $w_{\_}i$. This property is what we name the ”ensemblist disentanglement” capacity, and the first goal of these experiments is to investigate whether CompVAE does have this capacity.

A second goal of these experiments is to examine whether the desired properties (section 3.1) hold. The order-invariant property is enforced by design. The size-flexibility property will be assessed by inspecting the sensitivity of the extraction and generative processes to the variability of the number of parts. The size-generality property will be assessed by inspecting the quality of the generative model when the number of parts increases significantly beyond the size range used during training.

A last goal is to understand how CompVAE manages to store the information of the model in respectively the $w_{\_}i$s and $z$. The conjecture done (section 3.1) was that the latent $w_{\_}i$s would take in charge the information of the parts, while the latent $z$ would model the interactions among the parts. The use of synthetic problems where the quantity of information required to encode the parts can be quantitatively assessed will permit to test this conjecture. A related question is whether the generative model is able to capture the fact that the whole is not the sum of its parts. This question is investigated using non-linear perturbations, possibly operating at the whole and at the parts levels, and comparing the whole perturbed $x$ obtained from the $\ell_{\_}i$s, and the aggregation of the perturbed $x_{\_}i$s generated from the $\ell_{\_}i$ parts. The existence of a difference, if any, will establish the value of the CompVAE generative model compared to a simple Monte-Carlo simulator, independently sampling parts and thereafter aggregating them.

Two synthetic problems have been considered to empirically answer the above questions.²²2These problems are publicly available at https://github.com/vberger/compvae .

CompVAE is trained as a mainstream VAE, except for an additional factor of difficulty: the varying number of latent variables (reflecting the varying number of parts) results in a potentially large number of latent variables. This large size and the model noise in the early training phase can adversely affect the training procedure, and lead it to diverge. The training divergence is prevented using a batch size set to 256. The neural training hyperparameters are dynamically tuned using the Adam optimizer [15] with $\alpha=10^{-4}$, $\beta_{\_}1=0.5$ and $\beta_{\_}2=0.9$, which empirically provide a good compromise between training speed, network stability and good convergence. On the top of Adam, the annealing of the learning rate $\alpha$ is achieved, dividing its value by 2 every 20,000 iterations, until it reaches $10^{-6}$.

For both problems, the data is generated on the fly during the training, ³³3The data generator is given in supplementary material, section LABEL:app:data-generation. preventing the risk of overfitting. The overall number of iterations (batches) is up to⁴⁴4Experimentally, networks most often converge much earlier. 500,000. The computational time on a GPU GTX1080 is 1 day for the 1D problem, and 2 days for the 2D problem.

Empirically, the training is facilitated by gradually increasing the number $K$ of parts in the datapoints. Specifically, the number of parts is uniformly sampled in $[[1,K]]$ at each iteration, with $K=2$ at the initialization and $K$ incremented by 1 every 3,000 iterations, up to 16 parts in the 1D problem and 8 in the 2D problem.

Fig. 4 displays in log-scale the losses of the $w_{\_}i$s and $z$ latent variables along time, together with the reconstruction loss and the overall ELBO loss summing the other three (Eq. (4)). The division of labor between the $w_{\_}i$s and the $z$ is seen as the quantity of information stored by the $w_{\_}i$s increases to reach a plateau at circa 100 bits, while the quantity of information stored by $z$ steadily decreases to around 10 bits. As conjectured (section 3.1), $z$ carries little information.

Note that the $x$ reconstruction loss remains high, with a high ELBO even at convergence time, although the generated curves ”look good”. This fact is explained from the high entropy of the data: on the top of the specifics of each part (its amplitude and phase), $x$ is described as a $T$-length sequence: the temporal discretization of the signal increases the variance of $x$ and thus causes a high entropy, which is itself a lower bound for the ELBO. Note that a large fraction of this entropy is accurately captured by CompVAE through the variance of the generative model $p_{\_}\theta(x|z,{\mbox{$\widetilde{w}$}})$.

The ability of ”ensemblist disentanglement” is visually demonstrated on Fig. 6: considering a set of $\ell_{\_}i$, the individual parts $w_{\_}i$ are generated (Fig. 6, left) and gradually integrated to form a whole $x$ (Fig. 6, right) in a coherent manner.

The size-generality property is satisfactorily assessed as the model could be effectively used with a number of parts $K$ ranging up to 30 (as opposed to 16 during the training) without requiring any re-training or other modification of the model (results omitted for brevity).

As shown in Fig. 7, the 2D problem is more complex. On the one hand, a 2D part only has a local impact on $x$ (affecting a subset of pixels) while a 1D part has a global impact on the whole $x$ sequence. On the other hand, the number of parts has a global impact on the range of $x$ in the 1D problem, whereas each pixel value ranges in the same interval in the 2D problem. Finally and most importantly, $x$ is of dimension 200 in the 1D problem, compared to dimension $3,072$ ($3\times 32\times 32$) in the 2D problem. For these reasons, the latent variables here need to store more information, and the separation between the $w_{\_}i$ (converging toward circa 200-300 bits of information) and $z$ (circa 40-60 bits) is less clear.

Likewise, $x$ reconstruction loss remains high, although the generated images ”look good”, due to the fact that the loss precisely captures the discrepancies in the pixel values that the eye does not perceive.

Finally, the ability of ”ensemblist disentanglement” is inspected by incrementally generating the whole $x$ from a set of colored sites (Fig. 8). The top row displays the colors of $\ell_{\_}1\ldots\ell_{\_}5$ from left to right. On the second row, the $i$-th square shows an image composed from $\ell_{\_}1\ldots\ell_{\_}i$ by the ground truth generator, and rows 3 to 6 show images generated by the model from the same $\ell_{\_}1\ldots\ell_{\_}i$. While the generated $x$ generally reflects the associated set of parts, some advents of black and white glitches are also observed (for instance on the third column, rows 3 and 5). These glitches are blamed on the saturation of the network (as black and white respectively are represented as $(0,0,0)$ and $(1,1,1)$ in RGB), since non linear combinations of colors are used for a good visual rendering⁵⁵5Color blending in the data generation is done taking into account gamma-correction..