On a least absolute deviations estimator of a multivariate convex function

Abstract

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.

On Consistency of Absolute Deviations Estimators of Convex Functions

“…We start with Proposition 1, provided in Lim & Luo (2014), that reveals howĝ n can be computed numerically.…”

“…Numerical results presented in Lim & Luo (2014) suggests that the least squares estimatorĝ n is computed faster and for a larger data sets than the least squares estimatorg n .…”

“…Recently, there have been extensive studies on how to compute the best fit of an unknown convex function more efficiently. Lim & Luo (2014) suggest computingĝ n : [0, 1] d → R that minimizes the sum of absolute deviations…”