Persistent Homological Structures in Compressed Sensing and Sparse Likelihood

In this paper, we explore hidden persistent homological structures in sparse regressions. Sparse regressions are usually parameterized by tuning parameters that determine the sparsity of solutions. By treating the tuning parameters as additional dimension, we can have multi-scale representations. We can show there exist hidden persistent homological structures in these dimensions. By exploiting the hidden topological structures further, it is possible to completely bypass the computational bottlenecks that usually occur in solving the sparse regressions.