New method finds lower-dimensional shape behind complex data sets
Recovering Lower-Dimensional Semialgebraic Support of a Measure from its Moments
Machine Learning
Summary
Sometimes, people want to understand the shape or boundary where certain invisible things, called measures, live. This problem gets tricky when the shape has lower dimensions, like a flat surface inside a 3D space. The authors came up with a new way to approximate these hidden shapes using math tools called moment matrices and Christoffel--Darboux kernels. They tested their method with examples where the shape isn’t just points but has more complex structures, showing their approach can work beyond previous simpler cases.
What this means in practice
- •For optimization engineers: Identify the shape of constraints’ solution sets when global minimizers are non-unique using moments data.
- •For statistical modelers: Approximate the support of probability distributions from computed moments when supports have lower-dimensional boundaries.
Authors
Ruben Karapetyan, Shenyuan Ma, Ales Wodecki, Jakub Marecek
Abstract
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.