黎曼流形上使用图拉普拉斯近似的张量 Chernoff 界
Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximation
AI总结：
本文提出一种基于图拉普拉斯近似的张量 Chernoff 界方法，用于分析黎曼流形上随机游走的尾概率，通过谱相似性确保近似精度，扩展了高维随机对象的尾界理论。
摘要：
本文致力于推进概率尾界分析，这是一种评估随机变量偏离其期望值的大偏差概率的关键统计工具。传统的尾界，如马尔可夫界、切比雪夫界和 Chernoff 界，已在众多科学和工程领域中被证明是有价值的。然而，随着数据复杂性的增长，迫切需要将尾界估计从标量变量扩展到高维随机对象。现有研究通常依赖于高维随机对象之间的独立性假设，而这一假设并不总是成立的。在 Garg 等人和 Chang 等研究者使用随机游走来建模高维集合的工作基础上，本研究通过探索流形上的随机游走，引入了一种更通用的方法。为了解决为流形构建合适的底层图的挑战，我们提出了一种新方法，该方法增强了在近似流形的图上的随机游走。这种方法确保了原始流形与近似图之间的谱相似性，包括匹配的特征值、特征向量和特征函数。利用 Burago 等人提出的流形图近似技术，我们推导出了张量 Chernoff 界，并根据底层流形的谱特征确定了其在黎曼流形上随机游走的适用范围。
英文摘要：
This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's, Chebyshev's, and Chernoff bounds, have proven valuable across numerous scientific and engineering fields. However, as data complexity grows, there is a pressing need to extend tail bound estimation from scalar variables to high-dimensional random objects. Existing studies often rely on the assumption of independence among high-dimensional random objects, an assumption that may not always be valid. Building on the work of researchers like Garg et al. and Chang, who employed random walks to model high-dimensional ensembles, this study introduces a more generalized approach by exploring random walks over manifolds. To address the challenges of constructing an appropriate underlying graph for a manifold, we propose a novel method that enhances random walks on graphs approximating the manifold. This approach ensures spectral similarity between the original manifold and the approximated graph, including matching eigenvalues, eigenvectors, and eigenfunctions. Leveraging graph approximation technique proposed by Burago et al. for manifolds, we derive the tensor Chernoff bound and establish its range for random walks on a Riemannian manifold according to the underlying manifold's spectral characteristics.
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