Fast Approximation of Rotations and Hessians matrices
This work addresses the problem of speeding up inference in Gaussian models and Hessian estimation in optimization for machine learning practitioners, though it appears incremental as it builds on existing approximation techniques.
The paper introduces a method for approximating rotation matrices with linearithmic complexity, enabling efficient representation of symmetric matrices like covariance and Hessian matrices, which was validated through experiments on synthetic and real-world data.
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.