Shape-Constrained Regression Using Sum of Squares Polynomials

The authors present a hierarchy of semidefinite programs (SDPs) for the problem of fitting a shape-constrained (multivariate) polynomial to noisy evaluations of an unknown shape-constrained function. These shape constraints include convexity or monotonicity over a box. The authors show that polynomial functions that are optimal to any fixed level of their hierarchy form a consistent estimator of the underlying shape-constrained function. As a by-product of the proof, the authors establish that sum of squares-convex polynomials are dense in the set of polynomials that are convex over an arbitrary box. A similar sum-of-squares-type density result is established for monotone polynomials. In addition, the authors classify the complexity of convex and monotone polynomial regression as a function of the degree of the polynomial regressor. Whereas the authors' results show NP-hardness of these problems for degree three or larger, they can check numerically that their SDP-based regressors often achieve a similar training error at low levels of the hierarchy. Finally, on the computational side, the authors present an empirical comparison of their SDP-based convex regressors with the convex least squares estimator introduced in Hildreth [Hildreth C (1954) Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49(267):598–619] and Holloway [Holloway CA (1979) On the estimation of convex functions. Oper. Res. 27(2):401–407] and show that their regressor is valuable in settings in which the number of data points is large and the dimension is relatively small. The authors demonstrate the performance of their 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.