New M-estimator of the leading principal component

Abstract: We study the minimization of the non-convex and non-differentiable objective function $v \mapsto \mathrm{E} ( | X - v | | X + v | - | X |^2 )$ in $\mathbb{R}^p$. In particular, we show that its minimizers recover the first principal component direction of elliptically symmetric $X$ under specific conditions. The stringency of these conditions is studied in various scenarios, including a diverging number of variables $p$. We establish the consistency and asymptotic normality of the sample minimizer. We propose a Weiszfeld-type algorithm for optimizing the objective and show that it is guaranteed to converge in a finite number of steps. The results are illustrated with two simulations.