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

TL;DR

AQuaMaM is a neural network model that efficiently learns and computes exact likelihoods of complex 3D rotation distributions on SO(3) in a single forward pass, outperforming previous methods in speed and accuracy.

Contribution

It introduces AQuaMaM, a novel autoregressive quaternion manifold model that enables rapid, exact likelihood estimation of SO(3) distributions with fewer parameters and faster inference.

Findings

AQuaMaM closely matches true data distributions in experiments.

It achieves 14% higher log-likelihood than IPDF on a die rotation dataset.

AQuaMaM is 52 times faster than IPDF on a GPU.

Abstract

Accurately modeling complex, multimodal distributions 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 effective approach for learning arbitrary distributions on SO(3) up to a given precision. However, inference with IPDF requires $N$ forward passes through the network's final multilayer perceptron (where $N$ places an upper bound on the likelihood that can be calculated by the model), which is prohibitively slow for those without the computational resources necessary to parallelize the queries. In this paper, I introduce AQuaMaM, a neural network capable of both learning complex distributions on the rotation manifold and calculating exact likelihoods for query rotations in a single forward pass. Specifically, AQuaMaM autoregressively models the projected components of unit quaternions as mixtures of uniform distributions that partition their geometrically-restricted domain of values. When trained on an "infinite" toy dataset with ambiguous viewpoints, AQuaMaM rapidly converges to a sampling distribution closely matching the true data distribution. In contrast, the sampling distribution for IPDF dramatically diverges from the true data distribution, despite IPDF approaching its theoretical minimum evaluation loss during training. When trained on a constructed dataset of 500,000 renders of a die in different rotations, AQuaMaM reaches a test log-likelihood 14% higher than IPDF. Further, compared to IPDF, AQuaMaM uses 24% fewer parameters, has a prediction throughput 52$\times$ faster on a single GPU, and converges in a similar amount of time during training.