用矩阵-向量乘法估计矩阵谱范数，新方法更精准
On the Upper Bounds for the Matrix Spectral Norm
- 提出反向平衡估计算法，仅通过矩阵-向量乘法估算上界
- 在合成与真实数据中均显著优于幂法，尤其对快速衰减谱有效
- 适合深度学习与反问题中谱衰减快的矩阵分析
我们研究仅通过矩阵-向量乘法估算矩阵谱范数的问题。提出一种新的反向平衡估计算法，可提供谱范数的上界，并推导出其低估的概率保证。与标准方法（如幂法）相比，该算法在合成与真实场景下均产生更紧的上界。方法特别适用于谱快速衰减的矩阵，这类矩阵常见于深度学习与反问题中。
原文摘要 · Abstract (English)
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.
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