arXiv subjects

Himasish Talukdar

Publications and source records attributed to Himasish Talukdar.

6 recordsLinked to original sources

arXiv2026

Semicircular law with a few independent entries in a random matrix

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

math.PR↗

arXiv2026

Approximation Theory for Neural Networks: Old and New

Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. This survey presents several glimpses into this theory. We review classical density results for single-hidden-layer networks, as well as quantitative bounds that relate approximation error to network size and smoothness assumptions on target functions. Particular emphasis is placed on depth--width trade-offs and on results demonstrating that deeper architectures can achieve superior parameter efficiency for structured function classes. In addition to standard feedforward neural networks, we also review recent developments on Kolmogorov--Arnold Networks (KANs), which offer an alternative architectural paradigm and whose approximation-theoretic properties have begun to attract significant theoretical attention.

cs.LG↗

arXiv2026

Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$

Elephant random walks were studied recently in \cite{mukherjee2025elephant} on the groups $\mathbb{Z}^{*d_1} * \mathbb{Z}_2^{*d_2}$ whose Cayley graphs are infinite $d$-regular trees with $d = 2d_1 + d_2$. It was found that for $d \ge 3$, the elephant walk is ballistic with the same asymptotic speed $\frac{d - 2}{d}$ as the simple random walk and the memory parameter appears only in the rate of convergence to the limiting speed. In the $d = 2$ case, there are two such groups, both having the bi-infinite path as their Cayley graph. For $(d_1, d_2) = (1, 0)$, the walk is the usual elephant random walk on $\mathbb{Z}$, which exhibits anomalous diffusion. In this article, we study the other case, namely $(d_1, d_2) = (0, 2)$, which corresponds to the infinite dihedral group $D_\infty \cong \mathbb{Z}_2 * \mathbb{Z}_2$. Unlike the classical ERW on $\mathbb{Z}$, which is a time-inhomogeneous Markov chain, the ERW on $D_{\infty}$ is non-Markovian. We show that the first and second order behaviours of the \emph{signed location} of the walker agree with those of the simple symmetric random walk on $\mathbb{Z}$, with the memory parameter essentially manifesting itself via a lower order correction term that can be written as an explicit functional of the elephant walk on $\mathbb{Z}$. Our result demonstrates that unlike the simple random walk, the elephant walk is sensitive to local algebraic relations. Indeed, although $D_{\infty}$ is virtually abelian, containing $\mathbb{Z}$ as a finite-index subgroup, the involutive nature of its generators effectively neutralises memory, thereby ruling out any potential superdiffusive behaviour, in contrast to the superdiffusion observed on its abelian cousin $\mathbb{Z}$.

math.PR↗

arXiv2025

Spectra of contractions of the Gaussian Orthogonal Tensor Ensemble

In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of order $r$ and dimension $n$ drawn from the density \[ f(\mathcal{G}) \propto \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg). \] For $\mathbf{w} \in \mathbb{S}^{n - 1}$, the unit-sphere in $\mathbb{R}^n$, we consider the matrix-valued contraction $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain semi-circle bulk-limits in all regimes, generalising the works of Goulart et al. (2022); Au and Garza-Vargas (2023); Bonnin (2024) in the fixed-$r$ setting. We also study the edge-spectrum. We obtain a Baik-Ben Arous-Péché phase-transition for the largest and the smallest eigenvalues at $r = 4$, generalising a result of Mukherjee et al. (2024) in the context of adjacency matrices of random hypergraphs. We also show that the extreme eigenvectors of $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ contain non-trivial information about the contraction direction $\mathbf{w}$. Finally, we report some results, in the case $r = 4$, on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$, $\mathbf{u}, \mathbf{v} \in \mathbb{S}^{n - 1}$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these two matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Furthermore, it turns out that there are no outlier eigenvalues in the spectrum of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ when $\langle \mathbf{u}, \mathbf{v} \rangle = o(1)$.

math.PR↗

arXiv2024

Spectra of adjacency and Laplacian matrices of Erdős-Rényi hypergraphs

We study adjacency and Laplacian matrices of Erdős-Rényi $r$-uniform hypergraphs on $n$ vertices with hyperedge inclusion probability $p$, in the setting where $r$ can vary with $n$ such that $r / n \to c \in [0, 1)$. Adjacency matrices of hypergraphs are contractions of adjacency tensors and their entries exhibit long range correlations. We show that under the Erdős-Rényi model, the expected empirical spectral distribution of an appropriately normalised hypergraph adjacency matrix converges weakly to the semi-circle law with variance $(1 - c)^2$ as long as $\frac{d_{\avg}}{r^7} \to \infty$, where $d_{\avg} = \binom{n-1}{r-1} p$. In contrast with the Erdős-Rényi random graph ($r = 2$), two eigenvalues stick out of the bulk of the spectrum. When $r$ is fixed and $d_{\avg} \gg n^{r - 2} \log^4 n$, we uncover an interesting Baik-Ben Arous-Péché (BBP) phase transition at the value $r = 3$. For $r \in \{2, 3\}$, an appropriately scaled largest (resp. smallest) eigenvalue converges in probability to $2$ (resp. $-2$), the right (resp. left) end point of the support of the standard semi-circle law, and when $r \ge 4$, it converges to $\sqrt{r - 2} + \frac{1}{\sqrt{r - 2}}$ (resp. $-\sqrt{r - 2} - \frac{1}{\sqrt{r - 2}}$). Further, in a Gaussian version of the model we show that an appropriately scaled largest (resp. smallest) eigenvalue converges in distribution to $\frac{c}{2} ζ+ \big[\frac{c^2}{4}ζ^2 + c(1 - c)\big]^{1/2}$ (resp. $\frac{c}{2} ζ- \big[\frac{c^2}{4}ζ^2 + c(1 - c)\big]^{1/2}$), where $ζ$ is a standard Gaussian. We also establish analogous results for the bulk and edge eigenvalues of the associated Laplacian matrices.

math.PR↗

arXiv2024

Bulk Spectra of Truncated Sample Covariance Matrices

Determinantal Point Processes (DPPs), which originate from quantum and statistical physics, are known for modelling diversity. Recent research [Ghosh and Rigollet (2020)] has demonstrated that certain matrix-valued $U$-statistics (that are truncated versions of the usual sample covariance matrix) can effectively estimate parameters in the context of Gaussian DPPs and enhance dimension reduction techniques, outperforming standard methods like PCA in clustering applications. This paper explores the spectral properties of these matrix-valued $U$-statistics in the null setting of an isotropic design. These matrices may be represented as $X L X^\top$, where $X$ is a data matrix and $L$ is the Laplacian matrix of a random geometric graph associated to $X$. The main mathematically interesting twist here is that the matrix $L$ is dependent on $X$. We give complete descriptions of the bulk spectra of these matrix-valued $U$-statistics in terms of the Stieltjes transforms of their empirical spectral measures. The results and the techniques are in fact able to address a broader class of kernelised random matrices, connecting their limiting spectra to generalised Marčenko-Pastur laws and free probability.

math.ST↗