LOSoft: ℓ0 Minimization via Soft Thresholding

Mostafa Sadeghi, Fateme Ghayem, Massoud Babaie-Zadeh, Saikat Chatterjee, Mikael Skoglund, Christian Jutten

Abstract: We propose a new algorithm for finding sparse solution of a linear system of equations using I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> minimization. The proposed algorithm relies on approximating the non-smooth I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> (pseudo) norm with a differentiable function. Unlike other approaches, we utilize a particular definition of I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm which states that the I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm of a vector can be computed as the I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of its sign vector. Then, using a smooth approximation of the sign function, the problem is converted to I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> minimization. This problem is solved via iterative proximal algorithms. Our simulations on both synthetic and real data demonstrate the promising performance of the proposed scheme.