Forest Sparsity for Multi-channel Compressive Sensing

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.

💡 Research Summary

This paper introduces a novel structured sparsity model called forest sparsity for multi‑channel signals, where each channel exhibits a hierarchical tree‑structured sparsity and the channels are highly correlated. By treating the collection of channels as a “forest” of trees that share common roots or adjacent nodes, the authors capture both intra‑channel (tree) and inter‑channel (joint) dependencies in a single mathematical framework.

The main theoretical contribution is a measurement bound that dramatically improves upon existing model‑based compressive sensing results. For a signal consisting of (T) channels, each of length (N) and tree‑(k)‑sparse, the number of linear measurements required for exact recovery with a random Gaussian matrix is
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