Fixed-time-stable ODE Representation of Lasso

TL;DR

This paper introduces a fixed-time-stable ODE framework for solving Lasso problems, enabling solutions within a user-defined time independent of data, with applications in signal processing, machine learning, and neuroscience.

Contribution

It develops a novel projection-free Newton-based ODE approach for Lasso, ensuring fixed-time convergence regardless of problem data.

Findings

The ODE reaches the optimal solution within the prescribed time.

Numerical experiments confirm fixed-time convergence.

The method is applicable to sparse coding and neurophysiological modeling.

Abstract

Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE) representation of the Lasso problem not only enables its solution on analog computers but also provides a framework for interpreting neurophysiological phenomena. This article proposes a fixed-time-stable ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based ODE representation of the Lasso problem by first transforming it into a smooth nonnegative quadratic program (QP) and then designing a projection-free Newton-based fixed-time-stable ODE system for solving the corresponding Karush-Kuhn-Tucker (KKT) conditions. Moreover, the settling time of the ODE is independent of the problem data and can be arbitrarily prescribed. Numerical experiments verify that the trajectory reaches the optimal solution within the prescribed time.