Constrained Least Absolute Deviation Neural Networks

Abstract

It is well known that least absolute deviation (LAD) criterion or L 1 -norm used for estimation of parameters is characterized by robustness, i.e., the estimated parameters are totally resistant (insensitive) to large changes in the sampled data. This is an extremely useful feature, especially, when the sampled data are known to be contaminated by occasionally occurring outliers or by spiky noise. In our previous works, we have proposed the least absolute deviation neural network (LADNN) to solve unconstrained LAD problems. The theoretical proofs and numerical simulations have shown that the LADNN is Lyapunov-stable and it can globally converge to the exact solution to a given unconstrained LAD problem. We have also demonstrated its excellent application value in timedelay estimation. More generally, a practical LAD application problem may contain some linear constraints, such as a set of equalities and/or inequalities, which is called constrained LAD problem, whereas the unconstrained LAD can be considered as a special form of the constrained LAD. In this paper, we present a new neural network called constrained least absolute deviation neural network (CLADNN) to solve general constrained LAD problems. Theoretical proofs and numerical simulations demonstrate that the proposed CLADNN is Lyapunov stable and globally converges to the exact solution to a given constrained LAD problem, independent of initial values. The numerical simulations have also illustrated that the proposed CLADNN can be used to robustly estimate parameters for nonlinear curve fitting, which is extensively used in signal and image processing.

Improved Image Quality for Static BLADE Magnetic Resonance Imaging Using the Total-Variation Regularized Least Absolute Deviation Solver

“…However, recent studies have shown that the disadvantages of TV-related artifacts can be mitigated by using least absolute deviation (LAD) data fidelity [16][17][18][19][20] as opposed to common ordinary least squares (OLS) data fidelity. A well characterized property of LAD is its robustness to outliers and sampling noise, and LAD has been applied to image interpolation [21], linear regression [22][23][24], neural networks [25], and the problem of regularized image restoration [22], but its application to MRI reconstruction requires further evaluation. It is worth noting that the integration of LAD and TV may generate an efficient algorithm for clinical purpose.…”