Related papers: Large deviations for the largest eigenvalue of Gau…

We consider preferential attachment random graphs which may be obtained as follows: It starts with a single node. If a new node appears, it is linked by an edge to one or more existing node(s) with a probability proportional to function of…

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…

Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and…

Let $M_n$ be the maximum of $n$ zero-mean gaussian variables $X_1,..,X_n$ with covariance matrix of minimum eigenvalue $\lambda$ and maximum eigenvalue $\Lambda$. Then, for $n \ge 70$, $$\Pr\{M_n \ge \lambda \left (2 \log n - 2.5 - \log(2…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that…

We study the near-critical behavior of the sparse Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$ on $n\gg1$ vertices, where the connection probability $p$ satisfies $np = 1+\theta(b_n^2/n)^{1/3}$, with $n^{3/10}\ll {b_n}\ll n^{1/2}$, and…

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated…

We investigate theoretical guarantees for the false-negative rate (FNR) -- the fraction of true causal edges whose orientation is not recovered, under single-variable random interventions and an $\epsilon$-interventional faithfulness…

This paper develops a general approach to characterize the long-time trajectory behavior of nonconvex gradient descent in generalized single-index models in the large aspect ratio regime. In this regime, we show that for each iteration the…

We consider the branching random walk on the real line where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. It is well known that the normalized empirical measure…

This article studies the infinite-width limit of deep feedforward neural networks whose weights are dependent, and modelled via a mixture of Gaussian distributions. Each hidden node of the network is assigned a nonnegative random variable…

There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as…

In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $\mathbb Z$. While sleeping frogs do not move, the active ones move as…

Recent findings by Cohen et al., 2021, demonstrate that when training neural networks using full-batch gradient descent with a step size of $\eta$, the largest eigenvalue $\lambda_{\max}$ of the full-batch Hessian consistently stabilizes…

Symmetric tridiagonal matrices appear ubiquitously in mathematical physics, serving as the matrix representation of discrete random Schr\"odinger operators. In this work we investigate the top eigenvalue of these matrices in the large…

Whilst deep neural networks have shown great empirical success, there is still much work to be done to understand their theoretical properties. In this paper, we study the relationship between random, wide, fully connected, feedforward…

We consider the Erd\"{o}s--R\'{e}nyi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et…

We study the probability distribution of the index ${\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\mathcal{N}_+$ with the fraction…

The problem of community detection receives great attention in recent years. Many methods have been proposed to discover communities in networks. In this paper, we propose a Gaussian stochastic blockmodel that uses Gaussian distributions to…