Thus, fMRI BOLD activity reflects an indirect, noisy measure ofneural activity,mismatched toneural timescales byseveral orders ofmagnitude.

For example, while prior work has suggested that theglobally optimal VAEsolution canlearn thecorrect manifold dimension, anecessary (butnotsufficient)condition forproducing samplesfrom the true data distribution, this has never been rigorously proven. Moreover, it remains unclear how such considerations would change when various types of conditioning variablesare introduced, or when the data support is extended to a union of manifolds (e.g., as is likely the case for MNIST digits and related). In this work, we address these points by first proving that VAE global minima are indeed capable of recovering the correct manifold dimension.

The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable invertible transformations.

We note that we modified the reference implementation provided with that work from in order to have consistent 4 4 kernels as shown in the architecture in Table 1.

Hence, extensiveresearch has been proposed to improvethisbound through richer distributions [55,53,36,13].

Autoencoders (AEs) [1, 2] and its modern variants like the widely used variational autoencoders (VAEs) [3], are a powerful paradigm for self-supervised representation learning for generative modeling [4], compression [5], anomaly detection [6] or natural language processing [7]. Since autoencoders can learn low dimensional representations without requiring labeled data, they are particularly useful forcomputer vision taskswhere samples canbeveryhighdimensional making processing, transmitting, and search prohibitively expensive.

As currently described, we can only apply our method to block design, fully observed settings where alldatapoints andtasks areobservedatthesame time.