ﻻ يوجد ملخص باللغة العربية
Matrices satisfying the Restricted Isometry Property (RIP) play an important role in the areas of compressed sensing and statistical learning. RIP matrices with optimal parameters are mainly obtained via probabilistic arguments, as explicit constructions seem hard. It is therefore interesting to ask whether a fixed matrix can be incorporated into a construction of restricted isometries. In this paper, we construct a new broad ensemble of random matrices with dependent entries that satisfy the restricted isometry property. Our construction starts with a fixed (deterministic) matrix $X$ satisfying some simple stable rank condition, and we show that the matrix $XR$, where $R$ is a random matrix drawn from various popular probabilistic models (including, subgaussian, sparse, low-randomness, satisfying convex concentration property), satisfies the RIP with high probability. These theorems have various applications in signal recovery, random matrix theory, dimensionality reduction, etc. Additionally, motivated by an application for understanding the effectiveness of word vector embeddings popular in natural language processing and machine learning applications, we investigate the RIP of the matrix $XR^{(l)}$ where $R^{(l)}$ is formed by taking all possible (disregarding order) $l$-way entrywise products of the columns of a random matrix $R$.
This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary $mathcal{D}$ in a Hilbert space $mathcal{H}$. Given an element $fin mathcal{H}$, OMP generates a sequence of approximations $f_n$, $n
The multilabel learning problem with large number of labels, features, and data-points has generated a tremendous interest recently. A recurring theme of these problems is that only a few labels are active in any given datapoint as compared to the to
We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in stro
When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i.e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaran
Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $delta$. If $delta$ is too large, however, then counterexamples containing spurious local minima