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Context: We characterise the decomposition of latent spaces in vaes to be the fulfilment of two factors (as shown in Figure 1):a.An “appropriate” level of overlap in the latent space—ensuring that the range of latent values capable of encoding a particular datapoint is neither too small, nor too large.This is, in general, dictated by the level of stochasticity in the encoder: the noisier the encoding process is, the higher the number of datapoints which can plausibly give rise to a particular encoding.b.The aggregate encoding q_{{\phi}}\left(\bm{z}\right) matching the prior p\left(\bm{z}\right), where the latter expresses the desired dependency structure between latents. We posit a generalisation of disentanglement in vaes—decomposing their latent representations—that can help avoid such pitfalls.We characterise decomposition in vaes as the fulfilment of two factors:a) the latent encodings of data having an appropriate level of overlap, andb) the aggregate encoding of data conforming to a desired structure, represented through the prior.We emphasize that neither of these factors is sufficient in isolation: withoutan appropriate level of overlap, encodings can degrade to a lookup tablewhere the latents convey little information about data, and without the aggregate encoding of data following a desired structure, the encodings do not decompose as desired.

Question: What are the factors of fullfilement for the decomposition of latent spaces in VAEs ?
The decomposition in vaes as the fulfilment of two factors: a) the latent encodings of data having an appropriate level of overlap—ensuring that the range of latent values capable of encoding a particular datapoint is neither too small, nor too large.This is, in general, dictated by the level of stochasticity in the encoder: the noisier the encoding process is, the higher the number of datapoints which can plausibly give rise to a particular encoding, and b) the aggregate encoding of data conforming to a desired structure, represented through the prior.