Gauss quadrature for matrix inverse forms with applications

TL;DR

This paper introduces a Gauss quadrature-based framework to efficiently compute bilinear inverse forms for large matrices, significantly accelerating machine learning algorithms like determinantal point processes and submodular optimization.

Contribution

It is the first to demonstrate the convergence and bounding properties of Gauss-type quadrature for inverse forms, enabling faster large-scale computations in machine learning.

Findings

Achieves linear convergence of bounds on inverse forms

Enables significant speedups in machine learning tasks

Scales efficiently to large, sparse matrices

Abstract

We present a framework for accelerating a spectrum of machine learning algorithms that require computation of bilinear inverse forms $u^\top A^{-1}u$, where $A$ is a positive definite matrix and $u$ a given vector. Our framework is built on Gauss-type quadrature and easily scales to large, sparse matrices. Further, it allows retrospective computation of lower and upper bounds on $u^\top A^{-1}u$, which in turn accelerates several algorithms. We prove that these bounds tighten iteratively and converge at a linear (geometric) rate. To our knowledge, ours is the first work to demonstrate these key properties of Gauss-type quadrature, which is a classical and deeply studied topic. We illustrate empirical consequences of our results by using quadrature to accelerate machine learning tasks involving determinantal point processes and submodular optimization, and observe tremendous speedups in several instances.