Comptes Rendus
Statistics
A disintegration of the Christoffel function
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1071-1079.

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of “the CF of the disintegration is the disintegration of the CFs”. In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.

Nous montrons que la fonction de Christoffel (CF) se factorise en le produit de deux fonctions de Christoffel dont une est celle de la marginale et l’autre est liée à la probabilité conditionnelle. La démonstration utilise une propriété apparemment ignorée (mais intéressante en soi), qui stipule que tout polynôme qui est somme de carrés est aussi la fonction de Christoffel d’une forme linéaire (représentée par une mesure dans le cas univarié). Il en va de même pour le cône convexe des polynômes positifs sur un ensemble basique semi-algébrique. Cette interprétation de la fonction de Christoffel fournit un pont supplémentaire entre l’optimisation polynomiale et les polynômes orthogonaux.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.380
Classification: 42C05, 47B32
Jean B. Lasserre 1

1 LAAS-CNRS and Institute of Mathematics, BP 54200, 7 Avenue du Colonel Roche, 31031 Toulouse cédex 4, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jean B. Lasserre. A disintegration of the Christoffel function. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1071-1079. doi : 10.5802/crmath.380. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.380/

[1] Michael Grant; Stephen Boyd CVX: Matlab Software for Disciplined Convex Programming, version 2.1, 2014 (http://cvxr.com/cvx)

[2] J. William Helton; Jean B. Lasserre; Mihai Putinar Measures with zeros in the inverse of their moment matrix, Ann. Probab., Volume 36 (2008) no. 4, pp. 1453-1471 | MR | Zbl

[3] Andras Kroó; Doron S. Lubinsky Christoffel functions and universality in the bulk for multivariate orthogonal polynomials, Can. J. Math., Volume 65 (2013) no. 3, pp. 600-620 | DOI | MR | Zbl

[4] Jean B. Lasserre Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2015 | DOI

[5] Jean B. Lasserre; Edouard Pauwels Sorting out typicality via the inverse moment matrix SOS polynomial, Advances in Neural Information Processing Systems (2016), pp. 190-198

[6] Jean B. Lasserre; Edouard Pauwels The empirical Christoffel function with applications in data analysis, Adv. Comput. Math., Volume 45 (2019) no. 3, pp. 1439-1468 | DOI | MR | Zbl

[7] Jean B. Lasserre; Edouard Pauwels; Mihai Putinar The Christoffel-darboux Kernel for Data Analysis, Cambridge Monographs on Applied and Computational Mathematics, 38, Cambridge University Press, 2022 | DOI | Zbl

[8] Yurii Nesterov Squared functional systems and optimization problems, High Performance Optimization (Applied Optimization), Volume 33, Springer, 2000, pp. 405-440 | DOI | MR | Zbl

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