Partitioned Least Squares

TL;DR

This paper introduces a novel partitioned least squares model that allows grouping features for similar contribution, providing methods for exact and approximate solutions, and analyzing their efficiency and correctness.

Contribution

It presents a new non-convex formulation of least squares with feature partitioning and offers both an exact reformulation and an approximate alternating least squares method.

Findings

The exact method guarantees optimal solutions.

The approximate method is faster and nearly as accurate.

The problem is NP-complete, indicating computational difficulty.

Abstract

In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. The output allows practitioners to assess the importance of each group and of each variable in the group. We formally show that the new formulation is not convex and provide two alternative methods to deal with the problem: one non-exact method based on an alternating least squares approach; and one exact method based on a reformulation of the problem using an exponential number of sub-problems whose minimum is guaranteed to be the optimal solution. We formally show the correctness of the exact method and also compare the two solutions showing that the exact solution provides better results in a fraction of the time required by the alternating least squares solution (assuming that the number of partitions is small). For the sake of completeness, we also provide an alternative branch and bound algorithm that can be used in place of the exact method when the number of partitions is too large, and a proof of NP-completeness of the optimization problem introduced in this paper.