New M-estimator of the leading principal component

We study the minimization of the non-convex and non-differentiable objective function $v\mapsto \mathrm{E}(\mathrm{\parallel }X-v\mathrm{\parallel }\mathrm{\parallel }X+v\mathrm{\parallel }-\mathrm{\parallel }X{\mathrm{\parallel }}^2)$ 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.