正则化稀疏随机矩阵的集中性：基于非回溯算子的谱边界的界
Concentration of Regularized Sparse Random Matrices: Spectral Edge Bounds via Nonbacktracking Operators
- University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文针对正则化稀疏随机矩阵，首次给出依赖方差和截断的极端奇异值与特征值界，并推广非回溯算子谱半径方法至依赖情形，在更弱条件下获得更优误差。
中文摘要 AI 辅助
在稀疏随机矩阵中，谱离群值（位于谱主体之外的特征值和奇异值）因度波动而出现：高度数会增大算子范数，而低列度数会减小最小奇异值。正如Feige和Ofek（2005）以及Le、Levina和Vershynin（2017）所证明的，度正则化使得矩阵在期望范数尺度上集中。然而，包含截断的精确界仍未得到探索且具有挑战性，因为正则化在条目之间引入了依赖性。在文献中首次，我们为正则化非齐次随机矩阵的极端奇异值和特征值提供了依赖于方差和截断的界。在没有正则化的情况下，我们对最小奇异值的下界与Brailovskaya和van Handel（2024）获得的前导常数相同。此外，在较温和的条件$d/\log N\to\infty$下，我们的误差项消失，而他们的更强要求是$d/(\log N)^4\to\infty$。一个关键要素是将非回溯矩阵的谱半径界扩展到依赖情形。我们基于Benaych-Georges、Bordenave和Knowles（2020）以及Dumitriu和Zhu（2024）建立的独立情形的方法，并仔细处理仅被遍历一次的边。我们的证明框架将确定性谱比较与概率估计分开：一旦建立了Loewner不等式和逐列方差控制，剩余的概率分析归结为验证本文中提出的图矩条件。我们希望该框架能够扩展到处理具有更复杂依赖性的通用随机矩阵。
英文摘要
In sparse random matrices, spectral outliers (eigenvalues and singular values located away from the bulk) emerge due to degree fluctuations: high degrees inflate the operator norm, while low column degrees reduce the least singular value. As proved by Feige and Ofek (2005) and Le, Levina, and Vershynin (2017), degree regularization enforces concentration at the expected norm scale. However, precise bounds incorporating the cutoffs remain unexplored and challenging since regularization introduces dependencies among entries. For the first time in the literature, we provide variance- and cutoff-dependent bounds for extreme singular values and eigenvalues of regularized inhomogeneous random matrices. In the absence of regularization, our lower bound for the least singular value matches the same leading constant obtained by Brailovskaya and van Handel (2024). Moreover, our error term vanishes under the milder condition $d/\log N\to\infty$, as opposed to their stronger requirement $d/(\log N)^4\to\infty$. A key ingredient is to extend spectral radius bounds for nonbacktracking matrices to the dependent setting. We build on approaches for independent cases established by Benaych-Georges, Bordenave, and Knowles (2020), as well as Dumitriu and Zhu (2024), and carefully handle edges traversed only once. Our proof framework separates deterministic spectral comparisons from probabilistic estimates: once Loewner inequalities and columnwise variance controls are established, the remaining probabilistic analysis boils down to verifying the graph moment conditions formulated in this paper. We hope this framework can be extended to handle general random matrices with more complex dependencies.
