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We consider the multivariate max-linear regression problem where the modelparameters$\boldsymbol{\beta}_{1},\dotsc,\boldsymbol{\beta}_{k}\in\mathbb{R}^{p}$ need tobe estimated from $n$ independent samples of the (noisy) observations $y =\max_{1\leq j \leq k} \boldsymbol{\beta}_{j}^{\mathsf{T}} \boldsymbol{x} +\mathrm{noise}$. The max-linear model vastly generalizes the conventionallinear model, and it can approximate any convex function to an arbitraryaccuracy when the number of linear models $k$ is large enough. However, theinherent nonlinearity of the max-linear model renders the estimation of theregression parameters computationally challenging. Particularly, no estimatorbased on convex programming is known in the literature. We formulate andanalyze a scalable convex program as the estimator for the max-linearregression problem. Under the standard Gaussian observation setting, we presenta non-asymptotic performance guarantee showing that the convex program recoversthe parameters with high probability. When the $k$ linear components areequally likely to achieve the maximum, our result shows that a sufficientnumber of observations scales as $k^{2}p$ up to a logarithmic factor. Thissignificantly improves on the analogous prior result based on alternatingminimization (Ghosh et al., 2019). Finally, through a set of Monte Carlosimulations, we illustrate that our theoretical result is consistent withempirical behavior, and the convex estimator for max-linear regression is ascompetitive as the alternating minimization algorithm in practice.

Max-Linear Regression by Scalable and Guaranteed Convex Programming