New M-estimator of the leading principal component

Joni Virta; Una Radojicic; Marko Voutilainen

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arXiv · 2510.02799

New M-estimator of the leading principal component

Joni Virta·Una Radojicic·Marko Voutilainen

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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.

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Joni Virta, Una Radojicic, Marko Voutilainen. 2025-10-03. New M-estimator of the leading principal component. https://arxiv.org/abs/2510.02799

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