  We consider the multivariate max-linear regression problem where the model
parameters
$\boldsymbol{\beta}_{1},\dotsc,\boldsymbol{\beta}_{k}\in\mathbb{R}^{p}$ need to
be 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 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 $k^{2}p$ 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.
