Sparse Signal Reconstruction with Hierarchical Decomposition

In this project, we develop a Hierarchical Decomposition algorithm to solve the `1-Regularized Least Square problem: min x∈Rn {||x||`1 ∣∣Ax = b}. With the new approach, we show a systemic approach on how to select a family of regularization parameters λ’s in order to improve accuracy while retaining the sparsity of our approximation. 1 Background: A Constrained `0 Minimization In the recent decades, there have been a considerate amount of interests put into solving a minimization problem originated in Compressed Sensing. The problem is asking for the possibility of which is to ask for effective and efficient approaches to encode a large and sparse signal (reconstruct) with a relatively small number of linear measurements (acquire). Mathematically speaking, we are looking for a solution of the following minimization problem: min x∈Rn {||x||`0 ∣∣Ax = b} (1) Where the || · ||`0 measures the number of non-zero element in a vector x, A ∈ Rm×n is a m × n matrix over real and a measurement vector b ∈ R, with m n. Two possible scenarios exit for choosing the matrix A: it is either prescribed by a specific transformation or chosen by the user to recover x using least amount of information possible. The system Ax = b is underdetermined; when A has full rank, an infinite number of solutions exists. One can then find a x with the minimal `0-norm. However, it is shown in [10] that (1) is NP hard and requires techniques ∗[email protected] †[email protected]