arXiv subjects

Asad Lodhia

Publications and source records attributed to Asad Lodhia.

8 recordsLinked to original sources

arXiv2023

Positivity in Linear Gaussian Structural Equation Models

We study a notion of positivity of Gaussian directed acyclic graphical models corresponding to a non-negativity constraint on the coefficients of the associated structural equation model. We prove that this constraint is equivalent to the distribution being conditionally increasing in sequence (CIS), a well-known subclass of positively associated random variables. These distributions require knowledge of a permutation, a CIS ordering, of the nodes for which the constraint of non-negativity holds. We provide an algorithm and prove in the noise-less setting that a CIS ordering can be recovered when it exists. We extend this result to the noisy setting and provide assumptions for recovering the CIS orderings. In addition, we provide a characterization of Markov equivalence for CIS DAG models. Further, we show that when a CIS ordering is known, the corresponding class of Gaussians lies in a family of distributions in which maximum likelihood estimation is a convex problem.

math.ST↗

arXiv2022

Covariance Kernel of Linear Spectral Statistics for Half-Heavy tailed Wigner Matrices

In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $α$ moments for $2<α< 4$. We obtain a closed form $α$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covariance as an integral kernel acting on bounded continuous test functions. The resulting formulation allows us to offer a heuristic interpretation of the impact the typical large eigenvalues of this matrix ensemble have on the covariance structure.

math.PR↗

arXiv2021

Matrix Means and a Novel High-Dimensional Shrinkage Phenomenon

Many statistical settings call for estimating a population parameter, most typically the population mean, based on a sample of matrices. The most natural estimate of the population mean is the arithmetic mean, but there are many other matrix means that may behave differently, especially in high dimensions. Here we consider the matrix harmonic mean as an alternative to the arithmetic matrix mean. We show that in certain high-dimensional regimes, the harmonic mean yields an improvement over the arithmetic mean in estimation error as measured by the operator norm. Counter-intuitively, studying the asymptotic behavior of these two matrix means in a spiked covariance estimation problem, we find that this improvement in operator norm error does not imply better recovery of the leading eigenvector. We also show that a Rao-Blackwellized version of the harmonic mean is equivalent to a linear shrinkage estimator studied previously in the high-dimensional covariance estimation literature, while applying a similar Rao-Blackwellization to regularized sample covariance matrices yields a novel nonlinear shrinkage estimator. Simulations complement the theoretical results, illustrating the conditions under which the harmonic matrix mean yields an empirically better estimate.

math.ST↗

arXiv2021

Recovering shared structure from multiple networks with unknown edge distributions

In increasingly many settings, data sets consist of multiple samples from a population of networks, with vertices aligned across these networks. For example, brain connectivity networks in neuroscience consist of measures of interaction between brain regions that have been aligned to a common template. We consider the setting where the observed networks have a shared expectation, but may differ in the noise structure on their edges. Our approach exploits the shared mean structure to denoise edge-level measurements of the observed networks and estimate the underlying population-level parameters. We also explore the extent to which edge-level errors influence estimation and downstream inference. We establish a finite-sample concentration inequality for the low-rank eigenvalue truncation of a random weighted adjacency matrix that may be of independent interest. The proposed approach is illustrated on synthetic networks and on data from an fMRI study of schizophrenia.

math.ST↗

arXiv2019

Harmonic Means of Wishart Random Matrices

We use free probability to compute the limiting spectral properties of the harmonic mean of $n$ i.i.d. Wishart random matrices $\mathbf{W}_i$ whose limiting aspect ratio is $γ\in (0,1)$ when $\mathbb{E}[\mathbf{W}_i] = \mathbf{I}$. We demonstrate an interesting phenomenon where the harmonic mean $\mathbf{H}$ of the $n$ Wishart matrices is closer in operator norm to $\mathbb{E}[\mathbf{W}_i]$ than the arithmetic mean $\mathbf{A}$ for small $n$, after which the arithmetic mean is closer. We also prove some results for the general case where the expectation of the Wishart matrices are not the identity matrix.

math.PR↗

arXiv2017

Marčenko-Pastur Law for Kendall's Tau

We prove that Kendall's Rank correlation matrix converges to the Marčenko-Pastur law, under the assumption that the observations are i.i.d random vectors $X_1$, $\dots$, $X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.

math.ST↗

arXiv2016

Mesoscopic central limit theorem for general $β$-ensembles

We prove that the linear statistics of eigenvalues of $β$-log gasses satisfying the one-cut and off-critical assumption with a potential $V \in C^6(\mathbb{R})$ satisfy a central limit theorem at all mesoscopic scales $α\in (0; 1)$. We prove this for compactly supported test functions $f \in C^5(\mathbb{R})$ using loop equations at all orders along with rigidity estimates.

math.PR↗

arXiv2016

Fractional Gaussian fields: a survey

We discuss a family of random fields indexed by a parameter $s\in \mathbb{R}$ which we call the fractional Gaussian fields, given by \[ \mathrm{FGF}_s(\mathbb{R}^d)=(-Δ)^{-s/2} W, \] where $W$ is a white noise on $\mathbb{R}^d$ and $(-Δ)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized by their Hurst parameter $H = s-d/2$. In one dimension, examples of $\mathrm{FGF}_s$ processes include Brownian motion ($s = 1$) and fractional Brownian motion ($1/2 < s < 3/2$). Examples in arbitrary dimension include white noise ($s = 0$), the Gaussian free field ($s = 1$), the bi-Laplacian Gaussian field ($s = 2$), the log-correlated Gaussian field ($s = d/2$), Lévy's Brownian motion ($s = d/2 + 1/2$), and multidimensional fractional Brownian motion ($d/2 < s < d/2 + 1$). These fields have applications to statistical physics, early-universe cosmology, finance, quantum field theory, image processing, and other disciplines. We present an overview of fractional Gaussian fields including covariance formulas, Gibbs properties, spherical coordinate decompositions, restrictions to linear subspaces, local set theorems, and other basic results. We also define a discrete fractional Gaussian field and explain how the $\mathrm{FGF}_s$ with $s \in (0,1)$ can be understood as a long range Gaussian free field in which the potential theory of Brownian motion is replaced by that of an isotropic $2s$-stable Lévy process.

math.PR↗