BALSON: Bayesian Least Squares Optimization with Nonnegative L1-Norm Constraint

TL;DR

BALSON introduces a Bayesian framework for polynomial fitting that explicitly incorporates nonnegative L1-norm constraints on parameters, using Dirichlet distributions and sampling methods to improve parameter estimation.

Contribution

It proposes a novel Bayesian approach with Dirichlet-based posterior approximation for constrained polynomial fitting, outperforming traditional methods.

Findings

Effective parameter reconstruction in polynomial fitting

Better performance than conventional methods

Utilization of multiple sampling techniques

Abstract

A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parameters is equivalent to finding the mode of the posterior distribution. In order to explicitly characterize the nonnegative L1-norm constraint of the parameters, we further approximate the true posterior distribution by a Dirichlet distribution. We estimate the statistics of the approximating Dirichlet posterior distribution by sampling methods. Four sampling methods have been introduced. With the estimated posterior distributions, the original parameters can be effectively reconstructed in polynomial fitting problems, and the BALSON framework is found to perform better than conventional methods.