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Context: Taken together, this implies that the \beta-vae’s ability to encourage disentanglement is predominantly through direct control over the level of overlap.It places no other direct constraint on the latents to disentangle (although in some cases, the annealed prior may inadvertently encourage better disentanglement), but instead helps avoid the pitfalls of inappropriate overlap.Amongst other things, this explains why large \beta is not universally beneficial for disentanglement, as the level of overlap can be increased too far. This shows that the \beta-vae objective does not directly encourage latent variables to take on meaningful representations when using the standard choice of an isotropic Gaussian prior.In fact, on its own, it encourages latent representations which match the true generative factors no more than it encourages any arbitrary rotation of these factors, with such rotations capable of exhibiting strong correlations between latents.This view is further supported by our empirical results (see Figure 2), where we did not observe any gains in disentanglement (using the metric from Kim and Mnih [25]) from increasing \beta>0 with an isotropic Gaussian prior trained on the 2D Shapes dataset [38].It may also go some way to explaining the extremely high levels of variation we found in the disentanglement-metric scores between different random seeds at train time. Given the characterisation set out above, we now develop an objective that incorporates the effect of both factors (a) and (b).Our analysis of the \beta-vae tells us that its objective allows direct control over the level of overlap, i.e. factor Item a.To incorporate direct control over the regularisation Item b between the marginal posterior and the prior, we add a divergence term \mathbb{D}(q_{{\phi}}\left(z\right),p(\bm{z})), yielding\displaystyle\begin{split}\mathcal{L}_{\alpha,\beta}&(\bm{x})=\operatorname{{}\mathbb{E}}_{q_{{\phi}}\left(\bm{z}\mid\bm{x}\right)}\left[\log p_{{\theta}}\left(\bm{x}\mid\bm{z}\right)\right]\\&-\beta~{}\operatorname{\scalebox{0.95}{\text{KL}}}\left(q_{{\phi}}\left(\bm{z}\mid\bm{x}\right)\,\|\;p(\bm{z})\right)-\alpha~{}\mathbb{D}(q_{{\phi}}\left(\bm{z}\right),p(\bm{z}))\end{split}(7)allowing control over how much factors (a) and (b) are enforced, through appropriate setting of \beta and \alpha respectively. Connecting prior work on disentanglement to this framework, we analysed the \beta-vae objective to show that its contribution to disentangling is primarily through direct control of the level of overlap between encodings of the data, expressed by maximising the entropy of the encoding distribution.In the commonly encountered case of assuming an isotropic Gaussian prior and an independent Gaussian posterior, we showed that control of overlap is the only effect of the \beta-vae.Motivated by this observation, we developed an alternate objective for the elbo that allows control of the two factors of decomposability through an additional regularisation term.We then conducted empirical evaluations using this objective, targeting alternate forms of decompositions such as clustering and sparsity, and observed the effect of varying the extent of regularisation to the prior on the quality of the resulting clustering and sparseness of the learnt embeddings.The results indicate that we were successful in attaining those decompositions. To connect our framework with existing approaches for encouraging disentanglement, we provide a theoretical analysis of the \beta-vae [17, 3, 2], and show that it typically only allows control of latent overlap, the first decomposition factor.We show that it can be interpreted, up to a constant offset, as the standard vae objective with its prior annealed as p_{{\theta}}\left(\bm{z}\right)^{\beta} and an additional maximum entropy regularization of the encoder that increases the stochasticity of the encodings.Specialising this result for the typical choice of a Gaussian encoder and isotropic Gaussian prior indicates that the \beta-vae, up to a scaling of the latent space, is equivalent to the vae plus a regulariser encouraging higher encoder variance.Moreover, this objective is invariant to rotations of the learned latent representation, meaning that it does not, on its own, encourage the latent variables to take on meaningful representations any more than an arbitrary rotation of them.

Question: Does β-VAE's objective control overlap ?
The \beta-vae objective to show that its contribution to disentangling is primarily through direct control of the level of overlap between encodings of the data, expressed by maximising the entropy of the encoding distribution. \beta-vae’s ability to encourage disentanglement is predominantly through direct control over the level of overlap. Hence, β-VAE's objective control overlap.