相关论文: Higher-Rank Numerical Ranges and Compression Probl…
The class of complex random vectors whose covariance matrix is linearly parameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, and the maximum compression ratios that preserve all second-order information are derived…
A presentation of numerical range for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Then we are extended to the treatment of rank-k numerical range.
With lowrank approximation the storage requirements for dense data are reduced down to linear complexity and with the addition of hierarchy this also works for data without global lowrank properties. However, the lowrank factors itself are…
Sequence representations supporting queries $access$, $select$ and $rank$ are at the core of many data structures. There is a considerable gap between the various upper bounds and the few lower bounds known for such representations, and how…
Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this…
Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken…
In this paper we consider the low-rank matrix completion problem with specific application to forecasting in time series analysis. Briefly, the low-rank matrix completion problem is the problem of imputing missing values of a matrix under a…
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right…
We consider the problem of noisy matrix completion, in which the goal is to reconstruct a structured matrix whose entries are partially observed in noise. Standard approaches to this underdetermined inverse problem are based on assuming…
We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
The notion of the higher rank numerical range $\Lambda_{k}(L(\lambda))$ for matrix polynomials $L(\lambda)=A_{m}\lambda^{m}+...+A_{1}\lambda+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further,…
An exact solution to the problem of parametric level statistics in non-Gaussian ensembles of N by N Hermitian random matrices with either soft or strong level confinement is formulated within the framework of the orthogonal polynomial…
Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and…
We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…
Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and…
We introduce a notion of nuclear numerical range defined as the set of expectation values of a given operator $A$ among normalized pure states, which belong to the nucleus of an auxiliary operator $Z$. This notion proves to be applicable to…