Phase transitions with structured sparsity

While phase transition phenomena in compressed sensing have been rigorously established for simple sparse signals by Donoho and others, the behavior of structured sparse signals—such as block or tree patterns common in real-world applications—remains theoretically underexplored. This paper addresses this critical gap by extending phase transition analysis to structured sparsity models through the geometric lens of high-dimensional convex polytope projections. Our investigation reveals that weak thresholds, representing the proportion of faces lost after random projection, remain invariant across both simple and structured sparsity frameworks. In contrast, strong thresholds, which determine exact recovery guarantees, vary significantly according to structure type. We derive explicit mathematical expressions for these thresholds in both block-structured and tree-structured signals, demonstrating how additional structural constraints modify recovery boundaries. For block-sparse signals, we prove that the strong threshold rises as the number of blocks increases. Similarly, tree-sparse signals exhibit distinct threshold behaviors depending on whether sparsity falls below or exceeds the thresholds. These findings provide theoretical justification for the empirical success of structured sparsity models in applications ranging from medical imaging to radar systems, where they consistently outperform traditional compressed sensing approaches.