l1-norm Penalized Orthogonal Forward Regression

TL;DR

This paper introduces an l1-norm penalized orthogonal forward regression algorithm that efficiently selects regressors by minimizing leave-one-out mean square error, with an analytical approach for regularization and reduced computational cost.

Contribution

The paper proposes a novel l1-POFR algorithm that analytically computes LOOMSE and optimally tunes regularization parameters, improving robustness and efficiency in regression modeling.

Findings

Effective regressor selection demonstrated in examples.

Analytical computation of LOOMSE without data splitting.

Reduced computational cost through adaptive regressor removal.

Abstract

A l1-norm penalized orthogonal forward regression (l1-POFR) algorithm is proposed based on the concept of leaveone- out mean square error (LOOMSE). Firstly, a new l1-norm penalized cost function is defined in the constructed orthogonal space, and each orthogonal basis is associated with an individually tunable regularization parameter. Secondly, due to orthogonal computation, the LOOMSE can be analytically computed without actually splitting the data set, and moreover a closed form of the optimal regularization parameter in terms of minimal LOOMSE is derived. Thirdly, a lower bound for regularization parameters is proposed, which can be used for robust LOOMSE estimation by adaptively detecting and removing regressors to an inactive set so that the computational cost of the algorithm is significantly reduced. Illustrative examples are included to demonstrate the effectiveness of this new l1-POFR approach.