Sorting out typicality with the inverse moment matrix SOS polynomial

TL;DR

This paper reveals that a specific sum-of-squares polynomial derived from the inverse empirical moment matrix effectively captures the shape of data clouds, with applications demonstrated in network intrusion detection.

Contribution

It introduces a novel polynomial based on the inverse moment matrix that accurately represents data shape and connects it to orthogonal polynomials and the Christoffel function.

Findings

The polynomial's sublevel sets closely match data cloud shapes.

The method performs comparably to existing intrusion detection techniques.

Provides a mathematical understanding of the polynomial's extremality properties.

Abstract

We study a surprising phenomenon related to the representation of a cloud of data points using polynomials. We start with the previously unnoticed empirical observation that, given a collection (a cloud) of data points, the sublevel sets of a certain distinguished polynomial capture the shape of the cloud very accurately. This distinguished polynomial is a sum-of-squares (SOS) derived in a simple manner from the inverse of the empirical moment matrix. In fact, this SOS polynomial is directly related to orthogonal polynomials and the Christoffel function. This allows to generalize and interpret extremality properties of orthogonal polynomials and to provide a mathematical rationale for the observed phenomenon. Among diverse potential applications, we illustrate the relevance of our results on a network intrusion detection task for which we obtain performances similar to existing dedicated methods reported in the literature.