Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent

2024-06-11

arXiv.org

AI Summary

This paper addresses the robust recovery of Hankel matrices under sparse outliers and massive missing entries. To tackle this ill-posed low-rank structured recovery problem, we propose the Hankel-Structured Newton-like Descent (HSNLD) algorithm: it models robust loss directly on the low-rank Hankel manifold and employs a structured Newton approximation for efficient optimization. Theoretically, HSNLD achieves linear convergence with a rate independent of the Hankel matrix’s condition number—overcoming a key limitation of conventional methods that suffer from ill-conditioning. Moreover, we establish lightweight nonconvex optimization convergence guarantees. Experiments on both synthetic and real-world data demonstrate that HSNLD significantly outperforms state-of-the-art approaches, delivering both faster convergence and higher recovery accuracy.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper studies the robust Hankel recovery problem, which simultaneously removes the sparse outliers and fulfills missing entries from the partial observation. We propose a novel non-convex algorithm, coined Hankel Structured Newton-Like Descent (HSNLD), to tackle the robust Hankel recovery problem. HSNLD is highly efficient with linear convergence, and its convergence rate is independent of the condition number of the underlying Hankel matrix. The recovery guarantee has been established under some mild conditions. Numerical experiments on both synthetic and real datasets show the superior performance of HSNLD against state-of-the-art algorithms.

Problem

• Recovering Hankel matrices with missing entries and outliers
• Developing efficient non-convex algorithm for robust recovery
• Achieving linear convergence independent of matrix condition

Innovation

• Non-convex Hankel Structured Newton-Like Descent algorithm
• Linear convergence independent of condition number
• Robust recovery with sparse outlier removal