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We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…

A new method is proposed to improve the numeri- cal simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space…

We consider the low rank matrix completion problem over finite fields. This problem has been extensively studied in the domain of real/complex numbers, however, to the best of authors' knowledge, there exists merely one efficient algorithm…

The aim of this paper is to implement some new techniques, based on conjugate duality in convex optimization, for proving the existence of global error bounds for convex inequality systems. We deal first of all with systems described via…

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…

This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper…

Spark plays a great role in studying uniqueness of sparse solutions of the underdetermined linear equations. In this article, we derive a new lower bound of spark. As an application, we obtain a new criterion for the uniqueness of sparse…

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to…

When machine learning systems meet real world applications, accuracy is only one of several requirements. In this paper, we assay a complementary perspective originating from the increasing availability of pre-trained and regularly…

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization…

We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…

Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the…

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation…

We derive improved bounds on the error and erasure rate for spherical codes and for binary linear codes under Forney's erasure/list decoding scheme and prove some related results.

We show new upper and lower bounds for the complexity of implementation of a sequence of Boolean matrices proposed by Kaski et al. (arXiv:1208.0554) with additive circuits.

We prove lower bounds for the worst case error of quadrature formulas that use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets $\X_n$, but also prove lower bounds that hold with high…

In the field of sampling algorithms, MCMC (Markov Chain Monte Carlo) methods are widely used when direct sampling is not possible. However, multimodality of target distributions often leads to slow convergence and mixing. One common…

This note further addresses the global optimization problem for max-plus linear systems considered in [Automatica 119 (2020) 109104]. Firstly, the operations between infinity elemens and real numbers involved in the formulas of solving…