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Shape-Constrained Regression Using Sum of Squares Polynomials

Mihaela Curmei¹; Georgina Hall²

¹ Department of Electrical Engineering and Computer Science, University of California, Berkeley, California 94720 · ² Decision Sciences, INSEAD, Fontainebleau 77300, France;

Shape-Constrained Regression Using Algebraic Techniques How can one fit a multivariate polynomial to data points in such a way that this polynomial is guaranteed to have certain shape constraints, such as convexity or monotonicity in some of its variables? In “Shape-constrained regression using sum-of-squares polynomials,” M. Curmei and G. Hall propose a hierarchy of semidefinite programs to address this problem. They show that polynomial functions that are optimal to any fixed level of our hierarchy form a consistent estimator of the underlying shape-constrained function, which generates the data points. As a by-product of the proof, they establish that sum-of-squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. They further demonstrate the performance of the regressor for the problem of computing optimal transport maps in a color transfer task and that of estimating the optimal value function of a conic program. A real-time application of the latter problem to inventory management contract negotiation is presented.

DOI

10.1287/opre.2021.0383

Volume

73 (1)

Pages

543-559

Language

en

Export

BibTeX

Sources

crossref openalex