Timezone: »
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.
Author Information
Edouard Pauwels (IRIT)
Jean Lasserre (LAAS-CNRS)
More from the Same Authors
-
2022 Poster: Automatic differentiation of nonsmooth iterative algorithms »
Jerome Bolte · Edouard Pauwels · Samuel Vaiter -
2021 Poster: Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification »
Tong Chen · Jean Lasserre · Victor Magron · Edouard Pauwels -
2021 Poster: Nonsmooth Implicit Differentiation for Machine-Learning and Optimization »
Jérôme Bolte · Tam Le · Edouard Pauwels · Tony Silveti-Falls -
2021 Poster: Numerical influence of ReLU’(0) on backpropagation »
David Bertoin · Jérôme Bolte · Sébastien Gerchinovitz · Edouard Pauwels -
2020 Poster: Semialgebraic Optimization for Lipschitz Constants of ReLU Networks »
Tong Chen · Jean Lasserre · Victor Magron · Edouard Pauwels -
2016 : The moment-LP and moment-SOS approaches in optimization and some related applications »
Jean Lasserre -
2014 Workshop: OPT2014: Optimization for Machine Learning »
Zaid Harchaoui · Suvrit Sra · Alekh Agarwal · Martin Jaggi · Miro Dudik · Aaditya Ramdas · Jean Lasserre · Yoshua Bengio · Amir Beck