New M-estimator of the leading principal component

2025/10/03 by Joni Virta, Virta, Joni, Una Radojičić +3
Computer Science · Mathematics · #Advanced Statistical Methods and Models #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2510.02799

openalex publication_date 2025/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the minimization of the non-convex and non-differentiable objective function v ↦ E ( ‖ X - v ‖ ‖ X + v ‖ - ‖ X ‖² ) in 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.

Citations

• Unsupervised linear discrimination using skewness
• On the behavior of extreme d-dimensional spatial quantiles under minimal assumptions
• On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

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