A general error analysis for randomized low-rank approximation with application to data assimilation

2024-05-08

arXiv.org

AI Summary

Existing error analyses for randomized low-rank approximation rely on restrictive structural assumptions about the covariance matrix, limiting their applicability. Method: We develop the first general error analysis framework for centered nonstandard Gaussian random matrices, yielding tight upper bounds—both in expectation and with high probability—in the Frobenius norm, under only a minimal covariance condition. Contribution/Results: Our framework dispenses with conventional structural assumptions, unifying and strictly improving upon classical results while significantly tightening the bounds. It theoretically characterizes how covariance design governs approximation efficiency, offering interpretable guidance for applications such as data assimilation. Numerical experiments confirm that problem-adapted covariance matrices substantially enhance approximation accuracy, aligning closely with theoretical predictions.

Technology Category

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

Application Category

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

Abstract

Randomized algorithms have proven to perform well on a large class of numerical linear algebra problems. Their theoretical analysis is critical to provide guarantees on their behaviour, and in this sense, the stochastic analysis of the randomized low-rank approximation error plays a central role. Indeed, several randomized methods for the approximation of dominant eigen- or singular modes can be rewritten as low-rank approximation methods. However, despite the large variety of algorithms, the existing theoretical frameworks for their analysis rely on a specific structure for the covariance matrix that is not adapted to all the algorithms. We propose a general framework for the stochastic analysis of the low-rank approximation error in Frobenius norm for centered and non-standard Gaussian matrices. Under minimal assumptions on the covariance matrix, we derive accurate bounds both in expectation and probability. Our bounds have clear interpretations that enable us to derive properties and motivate practical choices for the covariance matrix resulting in efficient low-rank approximation algorithms. The most commonly used bounds in the literature have been demonstrated as a specific instance of the bounds proposed here, with the additional contribution of being tighter. Numerical experiments related to data assimilation further illustrate that exploiting the problem structure to select the covariance matrix improves the performance as suggested by our bounds.

Problem

• Unified error analysis for randomized low-rank approximation algorithms
• Deriving probabilistic bounds under minimal covariance matrix assumptions
• Improving data assimilation performance through structured covariance selection

Innovation

• Unified stochastic analysis framework for low-rank approximation
• Minimal covariance assumptions enable tighter error bounds
• Structure-aware covariance selection improves algorithm performance