Related papers: A note on computing the Smallest Conic Singular Va…

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…

This paper is essentially an exercise in studying the minima of a certain least squares optimization using the second partial derivative test. The motivation is to gain insight into an optimization-based solution to the problem of tracking…

We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of…

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances…

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution…

We are interested in matrices of minors of order p of a invertible matrix. Special cases are studied when this matrix is in SL(n) or SO(n)

These notes recall central elements of the cone calculus. The focus lies on conically degenerate differential operators and the Laplace-Beltrami operator with respect to a conically degenerate metric as a prototypical example. The topics…

The aim of this paper is to revisit some duality results in conic linear programming and to answer an open problem related to the duality gap function for Gale's example.

This mini-paper presents a fast and simple algorithm to compute the projection onto the canonical simplex $\triangle^n$. Utilizing the Moreau's identity, we show that the problem is essentially a univariate minimization and the objective…

The singular value decomposition (SVD) allows to write a matrix as a product of a left singular vectors matrix, a nonnegative singular values diagonal matrix and a right singular vectors matrix. Among the applications of the SVD are the…

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value…

The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a…

Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…

An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical…

We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address the generic minimal rank problem.

The first step to study lower bounds for a stochastic process is to prove a special property - Sudakov minoration. The property means that if a certain number of points from the index set are well separated then we can provide an optimal…

This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a problem with quadratic objective and 0-1 integer variables.…

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…