Incoherent dictionaries and the statistical restricted isometry property

Restricted $q$-Isometry Properties Adapted to Frames for Nonconvex $l_q$-Analysis

This paper discusses reconstruction of signals from few measurements in the situation that signals are sparse or approximately sparse in terms of a general frame via the $l_q$-analysis optimization with $0<q\leq 1$. We first introduce a notion of restricted $q$-isometry property ($q$-RIP) adapted to a dictionary, which is a natural extension of the standard $q$-RIP, and establish a generalized $q$-RIP condition for approximate reconstruction of signals via the $l_q$-analysis optimization. We then determine how many random, Gaussian measurements are needed for the condition to hold with high pr…

On some deterministic dictionaries supporting sparsity

We describe a new construction of an incoherent dictionary, referred to as the oscillator dictionary, which is based on considerations in the representation theory of finite groups. The oscillator dictionary consists of order of p^5 unit vectors in a Hilbert space of dimension p, where p is an odd prime, whose pairwise inner products have magnitude of at most 4/sqrt(p). An explicit algorithm to construct a large portion of the oscillator dictionary is presented.

具有相干和冗余字典的压缩感知

Compressed Sensing with Coherent and Redundant Dictionaries

针对信号在冗余字典而非正交基下稀疏的情况，提出测量矩阵的广义限制等距性质，保证通过L1分析优化问题实现精确恢复，填补了压缩感知在相干冗余字典领域的空白。

Greedy Signal Space Recovery Algorithm with Overcomplete Dictionaries in Compressive Sensing

Compressive Sensing (CS) is a new paradigm for the efficient acquisition of signals that have sparse representation in a certain domain. Traditionally, CS has provided numerous methods for signal recovery over an orthonormal basis. However, modern applications have sparked the emergence of related methods for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary, particularly due to their potential to generate different kinds of sparse representation of signals. To this end, we apply a signal space greedy method, which relies on the ability t…

压缩感知与冗余字典及结构化测量

Compressive Sensing with Redundant Dictionaries and Structured Measurements

本文研究了在冗余字典和结构化测量下信号恢复问题，提出通过子采样固定正交矩阵保证D-RIP条件，从而为结构化测量和冗余字典的压缩感知提供理论保障。

Signal Space CoSaMP for Sparse Recovery with Redundant Dictionaries

Compressive sensing (CS) has recently emerged as a powerful framework for acquiring sparse signals. The bulk of the CS literature has focused on the case where the acquired signal has a sparse or compressible representation in an orthonormal basis. In practice, however, there are many signals that cannot be sparsely represented or approximated using an orthonormal basis, but that do have sparse representations in a redundant dictionary. Standard results in CS can sometimes be extended to handle this case provided that the dictionary is sufficiently incoherent or well-conditioned, but these app…

通过受限等距性质改进的Johnson-Lindenstrauss嵌入

New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property

本文证明，通过随机化具有受限等距性质的矩阵的列符号，能以高概率将任意固定点集嵌入到低维空间，且范数失真受控，从而在N的对数因子内实现最优的Johnson-Lindenstrauss嵌入。

用于不相干性和更广泛领域的贪心信号空间方法

Greedy Signal Space Methods for incoherence and beyond

本文提出了一种新的信号空间贪心方法分析，适用于不相干或结构相干字典，改进了传统压缩采样文献中的结果。

Note on RIP-based Co-sparse Analysis

Over the past years, there are increasing interests in recovering the signals from undersampling data where such signals are sparse under some orthogonal dictionary or tight framework, which is referred to be sparse synthetic model. More recently, its counterpart, i.e., the sparse analysis model, has also attracted researcher's attentions where many practical signals which are sparse in the truly redundant dictionary are concerned. This short paper presents important complement to the results in existing literatures for treating sparse analysis model. Firstly, we give the natural generalizatio…