On a least absolute deviations estimator of a multivariate convex function

When estimating a performance measure f* of a complex system from noisy data, the underlying function f* is often known to be convex. In this case, one often uses convexity to better estimate f* by fitting a convex function to data. The traditional way of fitting a convex function to data, which is done by computing a convex function minimizing the sum of squares, takes too long to compute. It also runs into an “out of memory” issue for large-scale datasets. In this paper, we propose a computationally efficient way of fitting a convex function by computing the best fit minimizing the sum of absolute deviations. The proposed least absolute deviations estimator can be computed more efficiently via a linear program than the traditional least squares estimator. We illustrate the efficiency of the proposed estimator through several examples.