From L1 Minimization to Entropy Minimization: A Novel Approach for Sparse Signal Recovery in Compressive Sensing

The groundbreaking theory of compressive sensing (CS) enables reconstructing many common classes or real-world signals from a number of samples that is well below that prescribed by the Shannon sampling theorem, which exclusively relates to the bandwidth of the signal. Differently, CS takes profit of the sparsity or compressibility of the signals in an appropriate basis to reconstruct them from few measurements. A large number of algorithms exist for solving the sparse recovery problem, which can be roughly classified in greedy pursuits and l1 minimization algorithms. Chambolle and Pock's (C&P) primal-dual l1minimization algorithm has shown to deliver state-of-the-art results with optimal convergence rate. In this work we present an algorithm for l1 minimization that operates in the null space of the measurement matrix and follows a Nesterov-accelerated gradient descent structure. Restriction to the null space allows the algorithm to operate in a minimal-dimension subspace. A further novelty lies on the fact that the cost function is no longer the l1 norm of the temporal solution, but a weighted sum of its entropy and its l1 norm. The inclusion of the entropy pushes the $l_{1}$ minimization towards a de facto quasi-10 minimization, while the l1 norm term avoids divergence. Our algorithm globally outperforms C&P and other recent approaches for $l_{1}$ minimization in terms of l2reconstruction error, including a different entropy-based method.