Matrix polynomials appear in many areas of computational algebra, control systems theory, di‚erential equations, and mechanics, typically with real or complex coecients. Because of numerical error and instability, a matrix polynomial may appear of considerably higher rank (generically full rank), while being very close to a rank-de€cient matrix. “Close” is de€ned naturally under the Frobenius ...

This paper addresses the challenging problem of automatically evaluating output from machine translation (MT) systems in order to support the developers of these systems. Conventional approaches to the problem include methods that automatically assign a rank such as A, B, C, or D to MT output according to a single edit distance between this output and a correct translation example. The single e...

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M , the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) such that the Frobenius distance ‖M −M ′‖ is small. We propose a Newton-like iteration for solving ...

We present efficient streaming algorithms to compute two commonly used anomaly measures: the rank-k leverage scores (aka Mahalanobis distance) and the rank-k projection distance, in the rowstreaming model. We show that commonly used matrix sketching techniques such as the Frequent Directions sketch and random projections can be used to approximate these measures. Our main technical contribution...

In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino ...

The Euclidean distance matrix for n distinct points in ℝ r is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.

This paper introduces a new distance and mean on the set of positive semi-definite matrices of fixed-rank. The proposed distance is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is ...

A lower bound for the interleaving distance on persistence vector spaces is given in terms of rank invariants. This offers an alternative proof of the stability of rank invariants.

Although distance metric learning has been successfully applied to many real-world applications, learning a distance metric from large-scale and high-dimensional data remains a challenging problem. Due to the PSD constraint, the computational complexity of previous algorithms per iteration is at least O(d) where d is the dimensionality of the data. In this paper, we develop an efficient stochas...

A T -Direct code is defined as the set of T F -ary linear codes Γ1,Γ2, . . . ,ΓT such that Γi ∩ Γi = {0}, where Γi = Γ1 ⊕ Γ2⊕ · · · ⊕ Γi−1⊕ Γi+1 ⊕ · · · ⊕ ΓT is the dual of Γi with respect to the direct sum Λ = Γ1 ⊕ Γ2⊕ · · · ⊕ ΓT for each i = 1, 2, . . . , T . In this paper, a construction to a class of T -Direct codes with the constituent codes from the class of 2-Cyclic Maximum Rank Distance...