Ian Välimaa | Searcharxiv

arXiv2026

Sub-Poisson distributions: Concentration inequalities, optimal variance proxies, and closure properties

We introduce a nonasymptotic framework for sub-Poisson distributions with moment generating function dominated by that of a Poisson distribution. At its core is a new notion of optimal sub-Poisson variance proxy, analogous to the variance parameter in the sub-Gaussian setting. This framework allows us to derive a Bennett-type concentration inequality without boundedness assumptions and to show that the sub-Poisson property is closed under key operations including independent sums and convex combinations, but not under all linear operations such as scalar multiplication. We derive bounds relating the sub-Poisson variance proxy to sub-Gaussian and sub-exponential Orlicz norms. Taken together, these results unify the treatment of Bernoulli and Poisson random variables and their signed versions in their natural tail regime.

math.PR↗

arXiv2025

Consistent spectral clustering in sparse tensor block models

High-order clustering aims to classify objects in multiway datasets that are prevalent in various fields such as bioinformatics, recommendation systems, and social network analysis. Such data are often sparse and high-dimensional, posing significant statistical and computational challenges. This paper introduces a tensor block model specifically designed for sparse integer-valued data tensors. We propose a simple spectral clustering algorithm augmented with a trimming step to mitigate noise fluctuations, and identify a density threshold that ensures the algorithm's consistency. Our approach models sparsity using a sub-Poisson noise concentration framework, accommodating heavier than sub-Gaussian tails. Remarkably, this natural class of tensor block models is closed under aggregation across arbitrary modes. Consequently, we obtain a comprehensive framework for evaluating the tradeoff between signal loss and noise reduction incurred by aggregating data. The analysis is based on a novel concentration bound for sparse random Gram matrices. The theoretical findings are illustrated through numerical experiments.

math.ST↗