A Lagrange Programming Neural Network Approach with an ℓ0-Norm Sparsity Measurement for Sparse Recovery and Its Circuit Realization

Abstract

Many analog neural network approaches for sparse recovery were based on using ℓ1-norm as the surrogate of ℓ0-norm. This paper proposes an analog neural network model, namely the Lagrange programming neural network with ℓp objective and quadratic constraint (LPNN-LPQC), with an ℓ0-norm sparsity measurement for solving the constrained basis pursuit denoise (CBPDN) problem. As the ℓ0-norm is non-differentiable, we first use a differentiable ℓp-norm-like function to approximate the ℓ0-norm. However, this ℓp-norm-like function does not have an explicit expression and, thus, we use the locally competitive algorithm (LCA) concept to handle the nonexistence of the explicit expression. With the LCA approach, the dynamics are defined by the internal state vector. In the proposed model, the thresholding elements are not conventional analog elements in analog optimization. This paper also proposes a circuit realization for the thresholding elements. In the theoretical side, we prove that the equilibrium points of our proposed method satisfy Karush Kuhn Tucker (KKT) conditions of the approximated CBPDN problem, and that the equilibrium points of our proposed method are asymptotically stable. We perform a large scale simulation on various algorithms and analog models. Simulation results show that the proposed algorithm is better than or comparable to several state-of-art numerical algorithms, and that it is better than state-of-art analog neural models.

One-Rank Linear Transformations and Fejer-Type Methods: An Overview

“…Responding to new challenges in nondifferential optimization requires novel solution techniques. Combining artificial intelligence-based approaches with classical nondifferential techniques is an interesting area for future research [62,63].…”

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

“…In [415], Boolean matrix factorization is solved through a collaborative neurodynamic approach, which uses a population of Boltzmann machines for a scattered search of factorization solutions and particle swarm optimization for re-initializing the Boltzmann machines upon local convergence. Some other examples of neurodynamics-based methods for sparse signal reconstruction include a smoothing neurodynamic neural network modeled [416] for L p -norm 2 ≥ p ≥ 1, a projected neurodynamic neural network for L 0 -norm [417], and a Lagrange programming neural network with L p -norm [418]. Similarly, a discrete-time projection neural network for sparse NMF is presented in [419].…”