Technology Category
Search and Optimization: Non-convex OptimizationMachine Learning: Learning with ManifoldsIntelligent Robots: State Estimation
New M-estimator of the leading principal component
🤖 AI Summary
This paper addresses robust estimation of the first principal component direction under elliptical symmetry, focusing on high-dimensional settings ($p o infty$) where the objective function is nonconvex and nonsmooth. We propose a novel class of M-estimators that achieve consistent and asymptotically normal directional estimation under only weak moment conditions and distributional symmetry—bypassing the conventional convexity and differentiability requirements of standard PCA and classical M-estimation. To solve the resulting nonsmooth optimization problem, we design a modified Weiszfeld-type iterative algorithm and establish its finite-step convergence rigorously. Leveraging empirical process theory, we further develop a foundation for statistical inference. Numerical simulations demonstrate that the proposed method exhibits strong finite-sample stability and robustness against heavy-tailed and contaminated distributions.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Develops robust M-estimator for principal component analysis
Studies non-convex optimization for elliptical symmetric distributions
Establishes consistency and asymptotic normality properties
Innovation
Methods, ideas, or system contributions that make the work stand out.
M-estimator minimizes non-convex robust objective
Weiszfeld-type algorithm ensures finite-step convergence
Recovers principal component under elliptical symmetry conditions
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