Semi-algebraic Approximation Using Christoffel–Darboux Kernel

Abstract

We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to non-linear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with non-smoothness implicitly so that a single scheme can be used to treat smooth or non-smooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular we observe empirically that our method does not suffer from the the Gibbs phenomenon when approximating discontinuous functions.

Graph Recovery from Incomplete Moment Information

“…The recent work [15] describes an algorithm based on the Christoffel-Darboux kernel, to recover the graph of a function from knowledge of its moments. Its novelty (and distinguishing feature) is to approximate the graph of the function (rather than the function itself) with a semialgebraic function (namely a minimizer of a sum of squares of polynomials) with L 1 and pointwise convergence guarantees for an increasing number of input moments.…”

“…with discontinuities). In particular, this distinguishing feature allows to sometimes avoid a typical Gibbs phenomenon (as well as oscillations) encountered when approximating a discontinuous function by polynomials; for more details and illustrative examples the interested reader is referred to [15].…”

Rate of convergence for geometric inference based on the empirical Christoffel function

“…The idea of this estimator S d,n comes from Marx et al (2019), Section 4.1. The difference is that we let 0 < < 1 arbitrarily small for a better rate of convergence (instead of setting = 1/2 like in Marx et al (2019)).…”

“…The idea of this estimator S d,n comes from Marx et al (2019), Section 4.1. The difference is that we let 0 < < 1 arbitrarily small for a better rate of convergence (instead of setting = 1/2 like in Marx et al (2019)). Moreover, by choosing carefully the threshold, we obtain an estimator S d,n such that not only S d,n is contained in a small enlargement of S (which has been shown in Marx et al (2019)), but we also have a small enlargement of S d,n that contains S. The explicit result is as follows.…”

“…Indeed, Lasserre and Pauwels (2019) provides a thresholding scheme using the Christoffel function which approximates the compact support S of the measure µ with strong asymptotic guarantees. More precisely, Lasserre and Pauwels (2019) considers a family of polynomial sublevel sets S k = {x ∈ R p : Λ µ,d k (x) ≥ γ k } with k ∈ N, where the degree d k increases with k and where the threshold γ k is well-chosen between a lower bound of the Christoffel function inside S and an upper bound outside S. Another thresholding scheme can be found in Marx et al (2019), which provides useful results on the relation between S and its estimator S k . The topic of set estimation based on the population Christoffel function is thus currently a subject of active interest with a large range of applications in machine learning (see Pauwels and Lasserre (2016); Lasserre and Pauwels (2019)).…”