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We outline refined versions of two major quantum algorithms for performing principal component analysis and solving linear equations. Our methods are exponentially faster than their classical counterparts and even previous quantum…

We show that the core reasons that complex and hypercomplex valued neural networks offer improvements over their real-valued counterparts is the weight sharing mechanism and treating multidimensional data as a single entity. Their algebra…

In this paper, we study the strong convergence of an algorithm to solve the variational inequality problem which extends(Thong et al, Numerical Algorithms. 78, 1045-1060 (2018)). We have reduced and refined some of their algorithm's…

Establishing a fast rate of convergence for optimization methods is crucial to their applicability in practice. With the increasing popularity of deep learning over the past decade, stochastic gradient descent and its adaptive variants…

Learning rate schedulers have been widely adopted in training deep neural networks. Despite their practical importance, there is a discrepancy between its practice and its theoretical analysis. For instance, it is not known what schedules…

This paper studies the inference problem in quantile regression (QR) for a large sample size $n$ but under a limited memory constraint, where the memory can only store a small batch of data of size $m$. A natural method is the na\"ive…

Embedded computer vision applications increasingly require the speed and power benefits of single-precision (32 bit) floating point. However, applications which make use of Levenberg-like optimization can lose significant accuracy when…

We present an overlapping Schwarz decomposition algorithm for constrained quadratic programs (QPs). Schwarz algorithms have been traditionally used to solve linear algebra systems arising from partial differential equations, but we have…

Modern datasets arising from social media, genomics, and biomedical informatics are often heterogeneous and (ultra) high-dimensional, creating substantial challenges for conventional modeling techniques. Quantile regression (QR) not only…

Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA at sublinear cost -- by using much fewer memory cells and…

The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…

Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or…

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is…

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the "Sender-Receiver" model, we propose quantum algorithms for matrix operations such as matrix-vector product,…

Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies…

Sequential Quadratic Programming (SQP) is a powerful class of algorithms for solving nonlinear optimization problems. Local convergence of SQP algorithms is guaranteed when the Hessian approximation used in each Quadratic Programming…

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we…

This paper presents and analyzes the first matrix optimization model which allows general coordinate and spectral constraints. The breadth of problems our model covers is exemplified by a lengthy list of examples from the literature,…

Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually…