Subspace-Sparse Representation
Given an overcomplete dictionary $A$ and a signal $b$ that is a linear combination of a few linearly independent columns of $A$, classical sparse recovery theory deals with the problem of recovering the unique sparse representation $x$ such that $b = A x$. It is known that under certain conditions on $A$, $x$ can be recovered by the Basis Pursuit (BP) and the Orthogonal Matching Pursuit (OMP) algorithms. In this work, we consider the more general case where $b$ lies in a low-dimensional subspace spanned by some columns of $A$, which are possibly linearly dependent. In this case, the sparsest solution $x$ is generally not unique, and we study the problem that the representation $x$ identifies the subspace, i.e. the nonzero entries of $x$ correspond to dictionary atoms that are in the subspace. Such a representation $x$ is called subspace-sparse. We present sufficient conditions for guaranteeing subspace-sparse recovery, which have clear geometric interpretations and explain properties of subspace-sparse recovery. We also show that the sufficient conditions can be satisfied under a randomized model. Our results are applicable to the traditional sparse recovery problem and we get conditions for sparse recovery that are less restrictive than the canonical mutual coherent condition. We also use the results to analyze the sparse representation based classification (SRC) method, for which we get conditions to show its correctness.
Code (0)
등록된 구현이 없습니다.
Tasks
Sparse Representation-based ClassificationSimilar Papers 제목 키워드 기반
Learning with $\ell^{0}$-Graph: $\ell^{0}$-Induced Sparse Subspace Clustering
Sparse subspace clustering methods, such as Sparse Subspace Clustering (SSC) \cite{ElhamifarV13} and $\ell^{1}$-graph \cite{YanW09,ChengYYFH10}, are effective in partitioning the data that lie in a union of subspaces. Mo…
ClusteringStochastic Sparse Subspace Clustering
State-of-the-art subspace clustering methods are based on self-expressive model, which represents each data point as a linear combination of other data points. By enforcing such representation to be sparse, sparse subspa…
ClusteringSegmentation of Subspaces in Sequential Data
We propose Ordered Subspace Clustering (OSC) to segment data drawn from a sequentially ordered union of subspaces. Similar to Sparse Subspace Clustering (SSC) we formulate the problem as one of finding a sparse represent…
ClusteringSegmentationSparse Subspace Clustering: Algorithm, Theory, and Applications
In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensio…
ClusteringFace ClusteringImage ClusteringMotion SegmentationBasis Pursuit and Orthogonal Matching Pursuit for Subspace-preserving Recovery: Theoretical Analysis
Given an overcomplete dictionary $A$ and a signal $b = Ac^*$ for some sparse vector $c^*$ whose nonzero entries correspond to linearly independent columns of $A$, classical sparse signal recovery theory considers the pro…