Polynomial Learning of Distribution Families

TL;DR

This paper proves that Gaussian mixture distributions in high dimensions with a fixed number of components can be learned efficiently using polynomial time and samples, by analyzing polynomial families and applying algebraic geometry tools.

Contribution

It introduces a polynomial-time learning algorithm for high-dimensional Gaussian mixtures with fixed components, utilizing polynomial families and dimensionality reduction techniques.

Findings

Gaussian mixtures are polynomially learnable in high dimensions with fixed components.

Distribution parameters can be estimated efficiently using algebraic geometry methods.

A deterministic dimensionality reduction algorithm simplifies high-dimensional learning tasks.

Abstract

The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a high-dimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions.