Positivity and Optimization: Beyond Polynomials

Abstract

We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function (CF)) in theory of approximation and orthogonal polynomials. We then (i) show a strong link between the CF and the SOS-based positive certificate at the core of the Moment-SOS hierarchy, and (ii) describe how the CDkernel provides a simple interpretation of the SOS-hierarchy of lower bounds as searching for some signed polynomial density (while the SOS-hierarchy of upper bounds is searching for a positive (SOS) density). This link between the CF and positive certificates, in turn allows us (i) to establish a disintegration property of the CF much like for measures, and (ii) for certain sets, to relate the CF of their equilibrium measure with a certificate of positivity on the set, for constant polynomials.

An Algebraic Perspective on Multivariate Tight Wavelet Frames

“…Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) from [33] are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semi-definite programming. The latter together with the results in [31,35] answer affirmatively the long standing open question of the existence of such tight wavelet frames in dimension d = 2; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension d = 3 showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames.…”

“…Namely, in subsection 3.3, we show how to reformulate Theorem 3.10 equivalently as a problem of semidefinite programming. This establishes a connection between constructions of tight wavelet frames and moment problems, see [24,30,31] for details.…”

“…Such a can be, for example, any scaling symbol of a 3-D orthonormal wavelet with 8 vanishing moments; in particular, the tensor product Daubechies symbol a(z) = m 8 (z 1 )m 8 (z 2 )m 8 (z 3 ) with m 8 in[15] satisfies conditions(31) andσ∈G a σ * a σ = 1. The properties of m and a imply that 1. p satisfies the sub-QMF condition on T 3 , since m is G-invariant and 0 ≤ 1 − c • m ≤ 1 on T p satisfies sum rules of order at least 6, 3. the Taylor expansion of 1 − p * p at z = 1, in local coordinates (y 1 , y 2 , y 3 ), has 2 • c • m as its homogeneous part of lowest degree.…”