A Christoffel-like function for high-dimensional support inference in graphical models

TL;DR

This paper introduces a computationally efficient modification of Christoffel polynomials for high-dimensional support inference in graphical models, leveraging sparsity and treewidth to improve scalability.

Contribution

It proposes a new Christoffel-like function that reduces computational cost while maintaining key properties, utilizing graphical model sparsity and lower-dimensional factorizations.

Findings

The modified Christoffel function exhibits support dichotomy.

It is a rational function with factorizations into lower-dimensional polynomials.

Computational complexity depends on the graphical model's treewidth.

Abstract

Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $\mu$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a moment matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. In particular, it (1) exhibits a so-called support dichotomy and (2) it is a rational function, whose numerator and denominator factor into `lower-dimensional' Christoffel polynomials whose coefficients can be computed by inverting potentially much smaller moment matrices. Our approach relies on sparsity of the underlying measure $\mu$, described by a graphical model. The complexity of our modification depends on the treewidth of this model.