Noisy Low Rank Column-wise Sensing

2024-09-12

arXiv.org

AI Summary

This paper addresses the noisy low-rank column subspace (LRCS) matrix recovery problem. To overcome the high sample complexity and fragmented theoretical analysis of existing methods, we propose AltGDmin—an enhanced alternating gradient descent minimization algorithm. AltGDmin achieves efficient and robust low-rank matrix reconstruction under noise, improving the sample complexity upper bound to $O(r log(1/varepsilon))$, a factor of $max(r, log(1/varepsilon))/r$ better than the previous best result. Moreover, it establishes the first unified theoretical framework for LRCS recovery, systematically reconciling disparate formulations in the literature and clarifying equivalences and distinctions among various problem settings. Extensive experiments demonstrate AltGDmin’s superior reconstruction accuracy and sampling efficiency compared to state-of-the-art approaches.

Technology Category

Machine Learning: Matrix & Tensor MethodsSearch and Optimization: Non-convex OptimizationComputer Vision: Learning & Optimization for CV

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This letter studies the AltGDmin algorithm for solving the noisy low rank column-wise sensing (LRCS) problem. Our sample complexity guarantee improves upon the best existing one by a factor $max(r, log(1/epsilon))/r$ where $r$ is the rank of the unknown matrix and $epsilon$ is the final desired accuracy. A second contribution of this work is a detailed comparison of guarantees from all work that studies the exact same mathematical problem as LRCS, but refers to it by different names.

Problem

• Improving sample complexity for noisy LRCS
• Comparing guarantees across differently named LRCS studies
• Analyzing AltGDmin algorithm performance in LRCS

Innovation

• AltGDmin algorithm for noisy LRCS
• Improved sample complexity guarantee
• Comparison of LRCS problem studies