Sorting out typicality with the inverse moment matrix SOS polynomial
This provides a mathematical foundation for data shape analysis with potential applications in domains like cybersecurity, though it appears incremental as an extension of orthogonal polynomial theory.
The authors discovered that sublevel sets of a sum-of-squares polynomial derived from the inverse empirical moment matrix accurately capture the shape of data clouds, generalizing properties of orthogonal polynomials. They demonstrated its relevance by achieving network intrusion detection performance similar to existing dedicated methods.
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.