Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

Abstract

Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C d , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.

On The Radon--Nikodym Spectral Approach With Optimal Clustering

“…The (5) solution is n pairs (λ [i] , ψ [i] (x)). For positively defined G jk = x j | x k the solution exists and is unique.…”

“…Here f = f / 1 . The difference between (16) (5) invariance the variation expansion (16) will be the same for arbitrary non-degenerated linear transform of x components: x j = n−1 k=0 T jk x k . In the basis of the Lebesgue quadrature Radon-Nikodym derivative expression (13) is the eigenvalues weighted with (20) weights.…”

“…Obtained optimal clustering solution assumes there is a scalar function f , that can be put to (5) to obtain ψ [i] , then to construct the · L measure and to obtain optimal clusters (27). For unsupervised learning the function f does not exist and the best what we can do is to put the Christoffel function as f (x) = K(x):…”

“…• Take original basis, perform basis regularization of Appendix A, obtain "product" attributes (43) from X k , then solve (5) of N (n, D) dimension. Obtain the Lebesgue quadrature (15).…”

“…Find the Coverage(x). The Christoffel function is often used as a proxy for the coverage [3][4][5], however a genuine Coverage(x) is a very important characteristics in ML.…”