# On the Upper Bounds for the Matrix Spectral Norm

arXiv: https://arxiv.org/abs/2506.15660 · HTML: https://mlsanity.com/paper/2506.15660/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 18 Jun 2025.

## Builds on (most cited first, 3 of 3)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Hutch++: Optimal Stochastic Trace Estimation](https://mlsanity.com/paper/2010.09649/) (2020)

## Built on by (most cited first, 1 of 1)

- [Matrix-Free Two-to-Infinity and One-to-Two Norms Estimation](https://mlsanity.com/paper/2508.04444/) (2025)

## Abstract (arXiv)

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.