A modified Christoffel function and its asymptotic properties

We introduce a certain variant (or regularization) $\tilde{\Lambda}^\mu_n$ of the standard Christoffel function $\Lambda^\mu_n$ associated with a measure $\mu$ on a compact set $\Omega\subset \mathbb{R}^d$. Its reciprocal is now a sum-of-squares polynomial in the variables $(x,\varepsilon)$, $\varepsilon>0$. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with $n$ of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed $\varepsilon>0$, and under weak assumptions, $\lim_{n\to\infty} \varepsilon^{-d}\tilde{\Lambda}^\mu_n(\xi,\varepsilon)=f(\zeta_\varepsilon)$ where $f$ (assumed to be continuous) is the unknown density of $\mu$ w.r.t. Lebesgue measure on $\Omega$, and $\zeta_\varepsilon\in\mathbf{B}_\infty(\xi,\varepsilon)$ (and so $f(\zeta_\varepsilon)\approx f(\xi)$ when $\varepsilon>0$ is small). This is in contrast with the standard Christoffel function where if $\lim_{n\to\infty} n^d\Lambda^\mu_n(\xi)$ exists, it is of the form $f(\xi)/\omega_E(\xi)$ where $\omega_E$ is the density of the equilibrium measure of $\Omega$, usually unknown. At last but not least, the additional computational burden (when compared to computing $\Lambda^\mu_n$) is just integrating symbolically the monomial basis $(x^{\alpha})_{\alpha\in\mathbb{N}^d_n}$ on the box $\{x: \Vert x-\xi\Vert_\infty<\varepsilon/2\}$, so that $1/\tilde{\Lambda}^\mu_n$ is obtained as an explicit polynomial of $(\xi,\varepsilon)$.Comment: Rapport LAAS n 2300

A bounded degree SOS hierarchy for polynomial optimization

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine's positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) In contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems. Finally (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging

Sorting out typicality with the inverse moment matrix SOS polynomial

International audienceWe 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