In this paper, we investigate a new compressive sensing model for multi-channel sparse data where each channel can be represented as a hierarchical tree and different channels are highly correlated. Therefore, the full data could follow the forest structure and we call this property forest sparsity. It exploits both intra- and inter- channel correlations and enriches the family of existing model-based compressive sensing theories. The proposed theory indicates that only @math measurements are required for multi-channel data with forest sparsity, where @math is the number of channels, @math and @math are the length and sparsity number of each channel, respectively. This result is much better than @math of tree sparsity, @math of joint sparsity, and far better than @math of standard sparsity. In addition, we extend the forest sparsity theory to the multiple measurement vectors problem, where the measurement matrix is a block-diagonal matrix. The result shows that the required measurement bound can be the same as that for dense random measurement matrix, when the data shares equal energy in each channel. A new algorithm is developed and applied on four example applications to validate the benefit of the proposed model. Extensive experiments demonstrate the effectiveness and efficiency of the proposed theory and algorithm.
@cite_19: We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candes and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.
@cite_22: The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity.
@cite_2: Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper, we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters. Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically on the number of subspaces in the signal model. In other words, while unique linear sampling schemes require a small number of samples depending only on the dimension of the subspaces involved, in order to have stable sampling methods, the number of samples depends necessarily logarithmically on the number of subspaces in the model. These results are then applied to two examples, the standard compressed sensing signal model in which the signal has a sparse representation in an orthonormal basis and to a sparse signal model with additional tree structure.
CS result shows that, if @math , a sub-Gaussian random matrix It includes Gaussian and Bernoulli random matrices etc. @cite_22 . @math can satisfy the RIP with high probability @cite_19 @cite_2 .