SpreadNUTS -- Moderate Dynamic Extension of Paths for No-U-Turn Sampling & Partitioning Visited Regions

Markov chain Monte Carlo (MCMC) methods have existed for a long time and the field is well-explored. The purpose of MCMC methods is to approximate a distribution through repeated sampling; most MCMC algorithms exhibit asymptotically optimal behavior in that they converge to the true distribution at the limit. However, what differentiates these algorithms are their practical convergence guarantees and efficiency. While a sampler may eventually approximate a distribution well, because it is used in the real world it is necessary that the point at which the sampler yields a good estimate of the distribution is reachable in a reasonable amount of time. Similarly, if it is computationally difficult or intractable to produce good samples from a distribution for use in estimation, then there is no real-world utility afforded by the sampler. Thus, most MCMC methods these days focus on improving efficiency and speeding up convergence. However, many MCMC algorithms suffer from random walk behavior and often only mitigate such behavior as outright erasing random walks is difficult. Hamiltonian Monte Carlo (HMC) is a class of MCMC methods that theoretically exhibit no random walk behavior because of properties related to Hamiltonian dynamics. This paper introduces modifications to a specific HMC algorithm known as the no-U-turn sampler (NUTS) that aims to explore the sample space faster than NUTS, yielding a sampler that has faster convergence to the true distribution than NUTS.

PartitionVAE -- a human-interpretable VAE

VAEs, or variational autoencoders, are autoencoders that explicitly learn the distribution of the input image space rather than assuming no prior information about the distribution. This allows it to classify similar samples close to each other in the latent space's distribution. VAEs classically assume the latent space is normally distributed, though many distribution priors work, and they encode this assumption through a K-L divergence term in the loss function. While VAEs learn the distribution of the latent space and naturally make each dimension in the latent space as disjoint from the others as possible, they do not group together similar features -- the image space feature represented by one unit of the representation layer does not necessarily have high correlation with the feature represented by a neighboring unit of the representation layer. This makes it difficult to interpret VAEs since the representation layer is not structured in a way that is easy for humans to parse. We aim to make a more interpretable VAE by partitioning the representation layer into disjoint sets of units. Partitioning the representation layer into disjoint sets of interconnected units yields a prior that features of the input space to this new VAE, which we call a partition VAE or PVAE, are grouped together by correlation -- for example, if our image space were the space of all ping ping game images (a somewhat complex image space we use to test our architecture) then we would hope the partitions in the representation layer each learned some large feature of the image like the characteristics of the ping pong table or the characteristics and position of the players or the ball. We also add to the PVAE a cost-saving measure: subresolution. Because we do not have access to GPU training environments for long periods of time and Google Colab Pro costs money, we attempt to decrease the complexity of the PVAE by outputting an image with dimensions scaled down from the input image by a constant factor, thus forcing the model to output a smaller version of the image. We then increase the resolution to calculate loss and train by interpolating through neighboring pixels. We train a tuned PVAE on MNIST and Sports10 to test its effectiveness.