Fast Approximation of Rotations and Hessians matrices

TL;DR

This paper introduces a fast, learnable method for approximating rotation matrices and Hessians with linearithmic complexity, enabling efficient covariance and inverse Hessian estimation in machine learning tasks.

Contribution

The paper presents a novel FFT-like approach to approximate rotation matrices and Hessians using gradient descent, improving computational efficiency in high-dimensional settings.

Findings

Effective approximation of synthetic matrices

Accurate covariance matrix estimation for real data

Reliable inverse Hessian tracking in ML optimization

Abstract

A new method to represent and approximate rotation matrices is introduced. The method represents approximations of a rotation matrix $Q$ with linearithmic complexity, i.e. with $\frac{1}{2}n\lg(n)$ rotations over pairs of coordinates, arranged in an FFT-like fashion. The approximation is "learned" using gradient descent. It allows to represent symmetric matrices $H$ as $QDQ^T$ where $D$ is a diagonal matrix. It can be used to approximate covariance matrix of Gaussian models in order to speed up inference, or to estimate and track the inverse Hessian of an objective function by relating changes in parameters to changes in gradient along the trajectory followed by the optimization procedure. Experiments were conducted to approximate synthetic matrices, covariance matrices of real data, and Hessian matrices of objective functions involved in machine learning problems.