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Accurately modeling complex, multimodal distributions is necessary foroptimal decision-making, but doing so for rotations in three-dimensions, i.e.,the SO(3) group, is challenging due to the curvature of the rotation manifold.The recently described implicit-PDF (IPDF) is a simple, elegant, and effectiveapproach for learning arbitrary distributions on SO(3) up to a given precision.However, inference with IPDF requires $N$ forward passes through the network'sfinal multilayer perceptron (where $N$ places an upper bound on the likelihoodthat can be calculated by the model), which is prohibitively slow for thosewithout the computational resources necessary to parallelize the queries. Inthis paper, I introduce AQuaMaM, a neural network capable of both learningcomplex distributions on the rotation manifold and calculating exactlikelihoods for query rotations in a single forward pass. Specifically, AQuaMaMautoregressively models the projected components of unit quaternions asmixtures of uniform distributions that partition their geometrically-restricteddomain of values. When trained on an "infinite" toy dataset with ambiguousviewpoints, AQuaMaM rapidly converges to a sampling distribution closelymatching the true data distribution. In contrast, the sampling distribution forIPDF dramatically diverges from the true data distribution, despite IPDFapproaching its theoretical minimum evaluation loss during training. Whentrained on a constructed dataset of 500,000 renders of a die in differentrotations, AQuaMaM reaches a test log-likelihood 14% higher than IPDF. Further,compared to IPDF, AQuaMaM uses 24% fewer parameters, has a predictionthroughput 52$\times$ faster on a single GPU, and converges in a similar amountof time during training.

AQuaMaM: An Autoregressive, Quaternion Manifold Model for Rapidly Estimating Complex SO(3) Distributions