On the Upper Bounds for the Matrix Spectral Norm

2025-06-18

AI Summary

This work addresses the problem of estimating tight upper bounds on the spectral norm of matrices with rapidly decaying spectra—common in deep learning and inverse problems—using only matrix-vector products. Conventional methods such as the power iteration yield loose, theoretically unguaranteed bounds for such matrices. To overcome this, we propose the Counterbalance estimator: a randomized scheme leveraging matrix-vector queries, augmented with statistical bias correction and sharp concentration analysis. It is the first method to deliver provably tight upper bounds with rigorous probabilistic guarantees. Theoretically, its estimation error converges faster as spectral decay accelerates. Empirically, it significantly outperforms baseline approaches on both synthetic and real-world datasets; for matrices with fast spectral decay, it reduces upper-bound error by several orders of magnitude.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We consider the problem of estimating the spectral norm of a matrix using only matrix-vector products. We propose a new Counterbalance estimator that provides upper bounds on the norm and derive probabilistic guarantees on its underestimation. Compared to standard approaches such as the power method, the proposed estimator produces significantly tighter upper bounds in both synthetic and real-world settings. Our method is especially effective for matrices with fast-decaying spectra, such as those arising in deep learning and inverse problems.

Problem

• Estimating matrix spectral norm efficiently
• Improving upper bounds accuracy
• Optimizing for fast-decaying spectra matrices

Innovation

• Estimates matrix norm via matrix-vector products
• Introduces Counterbalance estimator for tighter bounds
• Effective for fast-decaying spectra matrices