Forest Sparsity for Multi-channel Compressive Sensing

Abstract

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 as \emph{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 $\mathcal{O}(Tk+\log (N/k))$ measurements are required for multi-channel data with forest sparsity, where $T$ is the number of channels, $N$ and $k$ are the length and sparsity number of each channel respectively. This result is much better than $\mathcal{O}(Tk+T\log (N/k))$ of tree sparsity, $\mathcal{O}(Tk+k\log (N/k))$ of joint sparsity, and far better than $\mathcal{O}(Tk+Tk\log (N/k))$ 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.

Accepted by IEEE Transactions on Signal Processing, 2014

References in corpus (8)

• Sparsity and Incoherence in Compressive Sampling
• Consistency of the group Lasso and multiple kernel learning
• Learning with Structured Sparsity
• Compressive Imaging using Approximate Message Passing and a Markov-Tree Prior
• C-HiLasso: A Collaborative Hierarchical Sparse Modeling Framework
• Efficient High-Dimensional Inference in the Multiple Measurement Vector Problem
• Collaborative Spectrum Sensing from Sparse Observations in Cognitive Radio Networks
• Graph-Structured Multi-task Regression and an Efficient Optimization Method for General Fused Lasso

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