Related papers: Estimating the Spectral Density of Large Implicit …
Many privacy mechanisms reveal high-level information about a data distribution through noisy measurements. It is common to use this information to estimate the answers to new queries. In this work, we provide an approach to solve this…
Spectral methods have myriad applications in high-dimensional statistics and data science, and while previous works have primarily focused on $\ell_2$ or $\ell_{2,\infty}$ eigenvector and singular vector perturbation theory, in many…
Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues…
High-dimensional matrix regression has been studied in various aspects, such as statistical properties, computational efficiency and application to specific instances including multivariate regression, system identification and matrix…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
In high-dimensional statistical inference in which the number of parameters to be estimated is larger than that of the holding data, regularized linear estimation techniques are widely used. These techniques have, however, some drawbacks.…
In this paper we consider regression problems subject to arbitrary noise in the operator or design matrix. This characterization appropriately models many physical phenomena with uncertainty in the regressors. Although the problem has been…
We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial…
Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an…
Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional ($p>n$) regression estimator, (2) the linear regression noise level, and (3) the…
This thesis addresses two persistent and closely related challenges in modern deep learning, reliability and efficiency, through a unified framework grounded in Spectral Geometry and Random Matrix Theory (RMT). As deep networks and large…
This paper proposes a sparse regression strategy for discovery of ordinary differential equations from incomplete and noisy data. Inference is performed over both equation parameters and state variables using a statistically motivated…
Theoretical analysis of biological and artificial neural networks e.g. modelling of synaptic or weight matrices necessitate consideration of the generic real-asymmetric matrix ensembles, those with varying order of matrix elements e.g. a…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
Large language models and deep neural networks achieve strong performance but suffer from reliability issues and high computational cost. This thesis proposes a unified framework based on spectral geometry and random matrix theory to…
We introduce a flexible framework for making inferences about general linear forms of a large matrix based on noisy observations of a subset of its entries. In particular, under mild regularity conditions, we develop a universal procedure…
Linear mixture models are commonly used to represent hyperspectral datacube as a linear combinations of endmember spectra. However, determining of the number of endmembers for images embedded in noise is a crucial task. This paper proposes…
The debiased estimator is a crucial tool in statistical inference for high-dimensional model parameters. However, constructing such an estimator involves estimating the high-dimensional inverse Hessian matrix, incurring significant…
We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special…
The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However,…