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Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing.…
Solving semidefinite programs (SDP) in a short time is the key to managing various mathematical optimization problems. The matrix-completion primal-dual interior-point method (MC-PDIPM) extracts a sparse structure of input SDP by…
We develop a fast and scalable computational framework to solve large-scale and high-dimensional Bayesian optimal experimental design problems. In particular, we consider the problem of optimal observation sensor placement for Bayesian…
Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical…
A variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity - number of instances…
We study a preconditioner for a Hermitian positive definite linear system, which is obtained as the solution of a matrix nearness problem based on the Bregman log determinant divergence. The preconditioner is of the form of a Hermitian…
Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process…
This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an ill-conditioned linear system $Ax=b$, we show that, if the inverse of the…
The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and…
Approximate dynamic programming is a popular method for solving large Markov decision processes. This paper describes a new class of approximate dynamic programming (ADP) methods- distributionally robust ADP-that address the curse of…
We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice ${\bf Z}^n$. We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the…
Semidefinite programming (SDP) is a fundamental convex optimization problem with wide-ranging applications. However, solving large-scale instances remains computationally challenging due to the high cost of solving linear systems and…
The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem.…
We study infinite-horizon Discounted Markov Decision Processes (DMDPs) under a generative model. Motivated by the Algorithm with Advice framework Mitzenmacher and Vassilvitskii 2022, we propose a novel framework to investigate how a…
Sparse coding--that is, modelling data vectors as sparse linear combinations of basis elements--is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization…
In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of…
Motivated by applications such as sparse PCA, in this paper we present provably-accurate one-pass algorithms for the sparse approximation of the top eigenvectors of extremely massive matrices based on a single compact linear sketch. The…
Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite…
We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding…
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…