Algebraic Geometric Comparison of Probability Distributions

TL;DR

This paper introduces an algebraic geometric framework for comparing probability distributions via their cumulants, enabling more efficient and accurate solutions to subspace learning problems.

Contribution

It develops a novel algebraic approach to probability distributions using cumulants, applying algebraic geometry for improved analysis and solution methods.

Findings

Direct algebraic solution reduces computational cost.

Higher accuracy in subspace identification.

Provides a new theoretical identifiability criterion.

Abstract

We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of Algebraic Geometry, which we demonstrate in a compact proof for an identifiability criterion.