Related papers: A note on computing the Smallest Conic Singular Va…
Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix…
These lecture notes focus on some numerical linear algebra algorithms in scientific computing. We assume that students are familiar with elementary linear algebra concepts such as vector spaces, systems of equations, matrices, norms,…
Inspired by certain regularization techniques for linear inverse problems, in this work we investigate the convergence properties of the Levenberg-Marquardt method using singular scaling matrices. Under a completeness condition, we show…
In this paper, we present a new approach to linearizing zero-one quadratic minimization problem which has many applications in computer science and communications. Our algorithm is based on the observation that the quadratic term of…
In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters $\omega$ and we are interested in the…
In this paper, we describe a new algorithm to build a few sparse principal components from a given data matrix. Our approach does not explicitly create the covariance matrix of the data and can be viewed as an extension of the Kogbetliantz…
It is proved that the minimal Dirichlet eigenvalue of the Laplacian in an annulus is a monotonically decreasing function of the displacement of the center of the smaller disc. The maximal value of the minimal eigenvalue is attained when the…
We prove explicit lower bounds for the smallest singular value and upper bounds for the condition number of rectangular, multivariate Vandermonde matrices with scattered nodes on the complex unit circle. Analogously to the Shannon-Nyquist…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
In this paper, we propose novel algorithms for inferring the Maximum a Posteriori (MAP) solution of discrete pairwise random field models under multiple constraints. We show how this constrained discrete optimization problem can be…
The minimal (reduced) and extended canonical formulations for (2+1)-dimensional fractional spin particles are considered. We investigate the relationship between them, clearing up the meaning of the coordinates for such particles, and…
I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational…
In this paper, a numerical method is proposed for canonical polyadic (CP) decomposition of small size tensors. The focus is primarily on decomposition of tensors that correspond to small matrix multiplications. Here, rank of the tensors is…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…
Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we…
This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology for dual conic optimization proxies. DLL leverages conic duality and the representation power of ML models to provide high-duality, dual-feasible…
This paper studies the logarithmic moments of the smallest denominator of all rationals in a shrinking interval with random center. Convergence follows from the more general results in [arXiv:2310.11251, Bull. Lond. Math. Soc., to appear],…
We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show…
A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be…
We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355,…