A fast approximation algorithm for tree-sparse recovery

Sparse signals whose nonzeros obey a tree-like structure occur in a range of applications such as image modeling, genetic data analysis, and compressive sensing. An important problem encountered in recovering signals is that of optimal tree-projection, i.e., finding the closest tree-sparse signal for a given query signal. However, this problem can be computationally very demanding: for optimally projecting a length-n signal onto a tree with sparsity k, the best existing algorithms incur a high runtime of O(nk). This can often be impractical. We suggest an alternative approach to tree-sparse recovery. Our approach is based on a specific approximation algorithm for tree-projection and provably has a near-linear runtime of O(n log(kr)) and a memory cost of O(n), where r is the dynamic range of the signal. We leverage this approach in a fast recovery algorithm for tree-sparse compressive sensing that scales extremely well to high-dimensional datasets. Experimental results on several test cases demonstrate the validity of our approach.