On the Largest and the Smallest Singular Value of Sparse Rectangular Random Matrices

2022/07/07 by F. Götze, Götze, F., А. Н. Тихомиров +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2207.03155

openalex publication_date 2022/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive estimates for the largest and smallest singular values of sparse rectangular N× n random matrices, assuming lim_N,n→∞\frac nN=y∈(0,1). We consider a model with sparsity parameter p_N such that Np_N∼ log^αN for some α>1, and assume that the moments of the matrix elements satisfy the condition \mathbf E|X_jk|^4+δ≤ C<∞. We assume also that the entries of matrices we consider are truncated at the level (Np_N)^{\frac12-\varkappa} with \varkappa:=\fracδ2(4+δ).

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