Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments

Ruben Karapetyan, Shenyuan Ma, Ales Wodecki, Jakub Marecek

Recovering probability measures from their moments has numerous applications, esp. in connection with the method of moments in statistics and optimization. In the setting where measure need not be finitely atomic, but its support is known to be compact and semialgebraic with codimension at least one, the problem is still open. We combine moment-matrix kernel information with the Christoffel--Darboux kernel to provide a discrete approximation of the support. To validate the proposed approach, we test our algorithm on analytically computed moments and pseudo-moments arising from polynomial optimization problems without unique global minimizers. This complements well-known recent work on recovery of measures with algebraic support, where the kernel of a moment matrix can reveal polynomials vanishing on the support, and on recovery of sufficiently regular full-dimensional supports, where estimators constructed by thresholding the Christoffel--Darboux kernel are known to converge asymptotically to the support.