Max-Linear Regression by Scalable and Guaranteed Convex Programming
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We consider the multivariate max-linear regression problem where the model parameters β_1,…,β_k∈ℝ^p need to be estimated from n independent samples of the (noisy) observations y = max_1≤ j ≤ kβ_j^𝖳x + noise. The max-linear model vastly generalizes the conventional linear model, and it can approximate any convex function to an arbitrary accuracy when the number of linear models k is large enough. However, the inherent nonlinearity of the max-linear model renders the estimation of the regression parameters computationally challenging. Particularly, no estimator based on convex programming is known in the literature. We formulate and analyze a scalable convex program as the estimator for the max-linear regression problem. Under the standard Gaussian observation setting, we present a non-asymptotic performance guarantee showing that the convex program recovers the parameters with high probability. When the k linear components are equally likely to achieve the maximum, our result shows that a sufficient number of observations scales as k^2p up to a logarithmic factor. This significantly improves on the analogous prior result based on alternating minimization (Ghosh et al., 2019). Finally, through a set of Monte Carlo simulations, we illustrate that our theoretical result is consistent with empirical behavior, and the convex estimator for max-linear regression is as competitive as the alternating minimization algorithm in practice.
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