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Max-affine regression refers to a model where the unknown regression functionis modeled as a maximum of $k$ unknown affine functions for a fixed $k \geq 1$.This generalizes linear regression and (real) phase retrieval, and is closelyrelated to convex regression. Working within a non-asymptotic framework, westudy this problem in the high-dimensional setting assuming that $k$ is a fixedconstant, and focus on estimation of the unknown coefficients of the affinefunctions underlying the model. We analyze a natural alternating minimization(AM) algorithm for the non-convex least squares objective when the design israndom. We show that the AM algorithm, when initialized suitably, convergeswith high probability and at a geometric rate to a small ball around theoptimal coefficients. In order to initialize the algorithm, we propose andanalyze a combination of a spectral method and a random search scheme in alow-dimensional space, which may be of independent interest. The final ratethat we obtain is near-parametric and minimax optimal (up to a poly-logarithmicfactor) as a function of the dimension, sample size, and noise variance. Inthat sense, our approach should be viewed as a direct and implementable methodof enforcing regularization to alleviate the curse of dimensionality inproblems of the convex regression type. As a by-product of our analysis, wealso obtain guarantees on a classical algorithm for the phase retrieval problemunder considerably weaker assumptions on the design distribution than waspreviously known. Numerical experiments illustrate the sharpness of our boundsin the various problem parameters.

Max-Affine Regression: Provable, Tractable, and Near-Optimal Statistical Estimation