# Sorting out typicality with the inverse moment matrix SOS polynomial

arXiv: https://arxiv.org/abs/1606.03858 · HTML: https://mlsanity.com/paper/1606.03858/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2016.

## Built on by (most cited first, 6 of 6)

- [RCoNet: Deformable Mutual Information Maximization and High-order Uncertainty-aware Learning for Robust COVID-19 Detection](https://mlsanity.com/paper/2102.11099/) (2021)
- [Data-Driven Reachability analysis and Support set Estimation with Christoffel Functions](https://mlsanity.com/paper/2112.09995/) (2021)
- [Scalable anomaly detection via a univariate Christoffel function](https://mlsanity.com/paper/2606.12483/) (2026)
- [CLOE: Christoffel Loss Autoencoder for Anomaly Detection](https://mlsanity.com/paper/2607.20530/) (2026)
- [Cost Trade-offs in Matrix Inversion Updates for Streaming Outlier Detection](https://mlsanity.com/paper/2603.16697/) (2026)
- [Sum-of-Squares Degree Barriers for the Reweighted-Hinge Method in Robust Halfspace Learning: A Christoffel-Function Characterization](https://mlsanity.com/paper/2606.17215/) (2026)

## Abstract (arXiv)

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.