Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

Jain, Vishesh

doi:10.1007/s11856-021-2144-y

Cited by 7 publications

(9 citation statements)

References 32 publications

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“…The methods we use in this paper can be further developed in various directions. In a recent work [8], the first two named authors utilized and extended some of the ideas introduced here in order to provide the best known upper bound for the well studied problem of estimating the singularity probability of random symmetric {±1}-valued matrices, and in upcoming work [11], the second named author uses some of the results in this paper to study the non-asymptotic behavior of the least singular value of different models of discrete random matrices. In another upcoming work of the second named author [10], it is shown how to extend the techniques introduced here and in [11] to study not-necessarily-discrete models of random matrices.…”

Section: Further Directions and Related Workmentioning

confidence: 99%

“…In a recent work [8], the first two named authors utilized and extended some of the ideas introduced here in order to provide the best known upper bound for the well studied problem of estimating the singularity probability of random symmetric {±1}-valued matrices, and in upcoming work [11], the second named author uses some of the results in this paper to study the non-asymptotic behavior of the least singular value of different models of discrete random matrices. In another upcoming work of the second named author [10], it is shown how to extend the techniques introduced here and in [11] to study not-necessarily-discrete models of random matrices. We also anticipate that the techniques presented here (along with some additional combinatorial ideas) should suffice to provide an 'exponential-type' upper bound on the probability of singularity of the adjacency matrix of a dense random regular digraph, thereby making substantial progress towards a conjecture of Cook [4,Conjecture 1.7].…”

Section: Further Directions and Related Workmentioning

confidence: 99%

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On the counting problem in inverse Littlewood--Offord theory

Ferber¹,

Jain²,

Luh³

et al. 2019

Preprint

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Let ǫ 1 , . . . , ǫ n be i.i.d. Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a = (a 1 , . . . , a n ), its concentration probability is the quantityThe Littlewood-Offord problem asks for bounds on ρ(a) under various hypotheses on a, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors a for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors a belonging to a specified set have large ρ(a)? The motivation for our study is that in typical applications, the inverse Littlewood-Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first 'exponential-type' (i.e., exp(−n c ) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best known bound is O(n −1/4 ) due to Cook; and (ii) dense rowregular {0, 1}-matrices, for which the previous best known bound is O C (n −C ) for any constant C > 0 due to Nguyen.

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Section: Further Directions and Related Workmentioning

confidence: 99%

Section: Further Directions and Related Workmentioning

confidence: 99%

On the counting problem in inverse Littlewood--Offord theory

Ferber¹,

Jain²,

Luh³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

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“…The study of such matrices was initiated by Nguyen [19], who proved that if d = n/2 then Q is asymptotically almost surely nonsingular (where n → ∞ along the even integers). Strengthenings of Nguyen's theorem have been proved by several authors; see for example [2,10,12,13,23]. Recently, Aigner-Horev and Person [2] conjectured an analogue of Theorem 1.1 for sparse random combinatorial matrices, which we prove in this note.…”

Section: Introductionmentioning

confidence: 54%

Singularity of sparse random matrices: simple proofs

Ferber¹,

Kwan²,

Sauermann³

2020

Preprint

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Consider a random n × n zero-one matrix with "density" p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the "Bernoulli" model), or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the "combinatorial" model). We give simple proofs of the (essentially bestpossible) fact that in both models, if min(p, 1 − p) ≥ (1 + ε) log n/n for any constant ε > 0, then our random matrix is nonsingular with probability 1 − o(1). In the Bernoulli model this fact was already well-known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

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“…We often deal with a model in which the variables to which we wish to apply hypercontractivity are not independent, but constrained to have a fixed sum. Rather than use hypercontractivity on the slice, we use a trick of Jain [16] which allows one to transfer bounds from the independent model to a fixed sum model via a simple conditioning argument. See the proof of [16, Lemma 5.4] for an example of this trick.…”

Section: Preliminariesmentioning

confidence: 99%

Local limit theorems for subgraph counts

Sah

Sawhney

2022

Journal of London Math Soc

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We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between 'fixed-size' and 'independent' models. We also adapt a notion of 'graph factors' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erdős-Renyi random graph 𝐺(𝑛, 𝑝), building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answer a question of Fox, Kwan, and Sauermann. We also derive a local central limit theorem for induced subgraph counts, as long as 𝑝 is bounded away from a set of 'problematic' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant 𝑝, and finding a graph 𝐻 for which anticoncentration for induced subgraph counts fails in 𝐺(𝑛, 1∕2). These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of 𝑘-term arithmetic progressions in subsets of ℤ∕𝑛ℤ and deduce a local limit theorem wherein the behavior is Gaussian at a global

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Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

Cited by 7 publications

References 32 publications

On the counting problem in inverse Littlewood--Offord theory

On the counting problem in inverse Littlewood--Offord theory

Singularity of sparse random matrices: simple proofs

Local limit theorems for subgraph counts

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