arXiv subjects

Simona Diaconu

Publications and source records attributed to Simona Diaconu.

8 recordsLinked to original sources

arXiv2026

On Empirical Spectral Distributions for Random Tensor Product Models

In statistics, assuming samples are independent is reasonable. However, this property can fail to hold for the features, a distinction that has led to several lines of work aiming to remove the latter assumption of independence present in the early literature, while preserving the original conclusions. Empirical spectral distributions of covariance matrices are key for understanding the data, and their almost sure convergence is oftentimes desirable. The random tensor product model, $X=(x_{i_1}x_{i_2}...x_{i_d})_{1 \leq i_1<...<i_d \leq n}$ for $x_1,x_2,\hspace{0.05cm}...\hspace{0.05cm},x_n$ i.i.d., introduced by the machine learning community, has a dependence structure for its features far from trivial and has been studied in recent years. When $x_1 \in \mathbb{R}, \mathbb{E}[x_1^4]<\infty, \frac{d}{n^{1/3}}=o(1),$ the empirical spectral distributions of the covariance matrices were proved to converge almost surely to Marchenko-Pastur laws in the random matrix theory regime. This work extends this result to the range $\frac{d}{n^{1/2}}=~o(1)$ when $x_1$ is symmetric with a subgaussian norm slowly growing in $n$ (the aforesaid range arises naturally, and the result failing when $\frac{d}{n^{1/2}} \to \infty$ appears to be a plausible claim) and shows that similarly to the case with independent features, the almost sure convergence holds under more general conditions on the covariance structure than the isotropic case. The latter result provides a means of deriving convergence for empirical spectral distributions of random matrices, applicable to other models as well so long as their entries exhibit a certain degree of concentration.

math.PR↗

arXiv2025

Admissible Sequences for Talagrand's $\gamma_2$-functional

Suprema of random processes appear naturally in a plethora of disciplines, and Talagrand's majorizing theorem yields a geometric interpretation for them: for a centered Gaussian random process $(X_t)_{t \in T},$ $\mathbb{E}[\sup_{t \in T}{X_t}]$ is comparable to the $\gamma_2$-functional of $T,$ a quantity that depends solely on the space $(T,d),$ where $d$ denotes the pseudometric $d(u,v)=\sqrt{\mathbb{E}[(X_u-X_v)^2]}.$ Despite the explicit definition of this functional, an infimum over admissible sequences, this tool tends to be used exclusively as a means to bound the expectation of the supremum of a random process by that of another. This work considers the $\gamma_2$-functional as a proxy for the quantity of interest by constructing admissible sequences that are close to being optimal, and aims to provide a promising avenue towards understanding expectations of suprema of Gaussian random processes.

math.PR↗

arXiv2025

Moment Inequalities for Suprema of Gaussian Random Processes

Suppose $(X_t)_{t \in T}$ is a Gaussian process indexed by some arbitrary set $T:$ the random variable $\sup_{t \in T}{X_t}$ can be very intricate and bounding its expectation is a natural step towards understanding it. Sudakov-Fernique inequality allows to order expectations of suprema of such random processes: if $(X_t)_{t \in T},(Y_t)_{t \in T}$ are centered Gaussian random processes satisfying $\mathbb{E}[(X_t-X_s)^2] \leq \mathbb{E}[(Y_t-Y_s)^2]$ for all $t,s \in T,$ then $\mathbb{E}[\sup_{t \in T}{X_t}] \leq \mathbb{E}[\sup_{t \in T}{Y_t}].$ This work obtains similar results for higher moments under a slightly stronger condition than the one aforementioned.

math.PR↗

arXiv2023

Universality for Random Matrices

Traces of large powers of real-valued Wigner matrices are known to have Gaussian fluctuations: for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n}\in \mathbb{R}^{n \times n}, A=A^T$ with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d., symmetric, subgaussian, $\mathbb{E}[a^{2}_{11}]=1,$ and $p=o(n^{2/3}),$ as $n,p \to \infty,$ $\frac{\sqrt{\pi}}{2^{p}}(tr(A^p)-\mathbb{E}[tr(A^p)]) \Rightarrow N(0,1).$ This work shows the entries of $A^{2p},$ properly scaled, also have asymptotically normal laws when $n \to \infty, p=n^{o(1)}:$ the normalizations of the diagonal entries depend on $\mathbb{E}[a_{11}^4],$ contributions that become negligible as $p \to \infty,$ whereas their counterparts in $A^{2p+1}$ depend on all the moments of $a_{11}$ when $p$ is bounded or the moments grow fast relatively to $p.$ This result demonstrates large powers of Wigner matrices are roughly Wigner matrices with normal entries when $a_{11} \overset{d}{=} -a_{11},\mathbb{E}[a^{2}_{11}]=1, \mathbb{E}[|a_{11}|^{8+\epsilon_0}] \leq C(\epsilon_0),$ providing another perspective on eigenvector universality, which until now has been justified primarily via local laws. The last part of this paper finds the first-order terms of traces of Wishart matrices in the random matrix theory regime, rendering yet another connection between Wigner and Wishart ensembles, as well as an avenue to extend the results herein for the former to the latter. The primary tools employed %behind the entry CLTs are the method of moments and a simple identity the Catalan numbers satisfy.

math.PR↗

arXiv2022

More Limiting Distributions for Eigenvalues of Wigner Matrices

The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue $λ_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=1,$ and the tail of $a_{11}$ decays sufficiently fast: $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=0,$ whereas when the law of $a_{11}$ is regularly varying with index $α\in (0,4),$ $c_a(n)n^{1/2-2/α}λ_1(A)$ converges to a Fréchet distribution for $c_a:(0,\infty) \to (0,\infty)$ slowly varying and depending solely on the law of $a_{11}.$ This paper considers a family of edge cases, $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=c \in (0,\infty),$ and unveils a new type of limiting behavior for $λ_1(A):$ a continuous function of a Fréchet distribution in which $2,$ the almost sure limit of $λ_1(A)$ in the light-tailed case, plays a pivotal role: $f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$

math.PR↗

arXiv2022

Two CLTs for Sparse Random Matrices

Let $G=G(n,p_n)$ be a homogeneous Erd\"os-R\'enyi graph, and $A$ its adjacency matrix with eigenvalues $\lambda_1(A) \geq \lambda_2(A) \geq ... \geq \lambda_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit fundamentally different behaviors: Tracy-Widom ($p_n \gg n^{-2/3}$), normal ($n^{-7/9} \ll p_n \ll~n^{-2/3}$), and a mix of both ($p_n=cn^{-2/3}$). Additionally, this technique renders the largest eigenvalue $\lambda_1(A),$ separated from the rest of the spectrum for $p_n \gg n^{-1},$ has Gaussian fluctuations when $p_n \geq n^{-1}(\log{n})^{6+c}$ for some $c>0.$ This paper shows this remains true in the range $Bn^{-1}(\log{n})^4 \leq p_n \leq 1-Bn^{-1}(\log{n})^4$ with $B>0$ universal, the tool behind it being a central limit theorem for the eigenvalue statistics of $A$ that is justified via the method of moments.

math.PR↗

arXiv2022

Finite Rank Perturbations of Heavy-Tailed Wigner Matrices

One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+θvv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with centered standard normal distributions, and $θ>0, v \in \mathbb{S}^{n-1}.$ It is well known $λ_1(P),$ the largest eigenvalue of $P,$ has a phase transition at $θ_0=1:$ when $θ\leq 1,$ $λ_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $θ> 1,$ $λ_1(P) \xrightarrow[]{a.s.} θ+θ^{-1}.$ Under more general conditions, the limiting behavior of $λ_1(P),$ appropriately normalized, has also been established: it is normal if $||v||_{\infty}=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $α\in (0,4),$ the fluctuations are shown to be universal and dependent on $θ$ but not on $v,$ whereas a subfamily of the edge case $α=4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at $θ_0=1, θ_0 \in [1,\frac{128}{89}],$ respectively. These results build on our previous which analyzes the asymptotic behavior of $λ_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.

math.PR↗

arXiv2021

On the Eigenstructure of Covariance Matrices with Divergent Spikes

For a generalization of Johnstone's spiked model, a covariance matrix with eigenvalues all one but $M$ of them, the number of features $N$ comparable to the number of samples $n: N=N(n), M=M(n), γ^{-1} \leq \frac{N}{n} \leq γ$ where $γ\in (0,\infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: \lim_{n \to \infty}{\frac{\sqrt{\log{n}}}{\log{\frac{n}{M(n)}}}}=0.$ Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1/6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.

math.ST↗