Sum-of-Squares Polynomial Flow

TL;DR

This paper introduces a new Sum-of-Squares (SOS) flow method for high-dimensional density estimation, unifying existing autoregressive and flow-based approaches, and demonstrating its effectiveness through synthetic and real-world experiments.

Contribution

It proposes a novel SOS flow framework that is interpretable, universal, and easy to train, unifying and extending existing density estimation methods.

Findings

SOS flows achieve competitive results on various datasets.

The framework reveals commonalities and differences among existing methods.

Synthetic experiments demonstrate the benefits and limitations of the approach.

Abstract

Triangular map is a recent construct in probability theory that allows one to transform any source probability density function to any target density function. Based on triangular maps, we propose a general framework for high-dimensional density estimation, by specifying one-dimensional transformations (equivalently conditional densities) and appropriate conditioner networks. This framework (a) reveals the commonalities and differences of existing autoregressive and flow based methods, (b) allows a unified understanding of the limitations and representation power of these recent approaches and, (c) motivates us to uncover a new Sum-of-Squares (SOS) flow that is interpretable, universal, and easy to train. We perform several synthetic experiments on various density geometries to demonstrate the benefits (and short-comings) of such transformations. SOS flows achieve competitive results in simulations and several real-world datasets.