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Homomorphic encryption (HE) is a promising technique used for privacy-preserving computation. Since HE schemes only support primitive polynomial operations, homomorphic evaluation of polynomial approximations for non-polynomial functions…

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently…

In this work, we study the generalization capability of algorithms from an information-theoretic perspective. It has been shown that the expected generalization error of an algorithm is bounded from above by a function of the relative…

In this paper, we present a generalized Cuppen's divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We extend the Cuppen's work to the rank two modifications of the form $A =T +\beta_1\bw_1\bw_1^T +…

We derive computable formulas for the structured backward errors of a complex number $\lambda$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric,…

Optimization of unitary transformations in Variational Quantum Algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the…

We unravel the Rodeo Algorithm to determine the eigenstates and eigenvalues spectrum for a general Hamiltonian considering arbitrary initial states. By presenting a novel methodology, we detail the original method and show how to define all…

The iterative algorithm recently proposed by Waxman for solving eigenvalue problems, which relies on the method of moments, has been modified to improve its convergence considerably without sacrificing its benefits or elegance. The…

This thesis introduces stochastic generalized routing problem model and proposes exact and heuristic algorithms to solve it efficiently, in a wide range of problem sizes. At first, the classic routing problem with its common variations in…

Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value…

Quadratic eigenvalue problems (QEP) and more generally polynomial eigenvalue problems (PEP) are among the most common types of nonlinear eigenvalue problems. Both problems, especially the QEP, have extensive applications. A typical approach…

Affine rank minimization algorithms typically rely on calculating the gradient of a data error followed by a singular value decomposition at every iteration. Because these two steps are expensive, heuristic approximations are often used to…

The Maximum Cut (MaxCut) problem is NP-Complete, and obtaining its optimal solution is NP-hard in the worst case. As a result, heuristic-based algorithms are commonly used, though their design often requires significant domain expertise.…

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural…

This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved…

Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum…

In this paper, we develop two new randomized block-coordinate optimistic gradient algorithms to approximate a solution of nonlinear equations in large-scale settings, which are called root-finding problems. Our first algorithm is…

The solution, $x$, of the linear system of equations $A x\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel $H(s,t)$ is considered. The Tikhonov regularized solution $ x(\lambda)$ is…

We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for…

In this work, we observe that many existing self-supervised learning algorithms can be both unified and generalized when seen through the lens of equivariant representations. Specifically, we introduce a general framework we call…