A Fast Coordinate Descent Method for High-Dimensional Non-Negative Least Squares using a Unified Sparse Regression Framework

TL;DR

This paper introduces a unified theoretical framework connecting various constrained regression methods and proposes a fast coordinate descent solver for high-dimensional non-negative least squares, demonstrating significant speed improvements.

Contribution

The paper establishes a theoretical link across multiple regression methods and develops a novel, efficient coordinate descent algorithm for NNLS in high dimensions.

Findings

Achieves at least 5x speed-up over existing solvers.

Provides a theoretical basis for sparsity in constrained least squares.

Demonstrates effectiveness on simulated and real data.

Abstract

We develop theoretical results that establish a connection across various regression methods such as the non-negative least squares, bounded variable least squares, simplex constrained least squares, and lasso. In particular, we show in general that a polyhedron constrained least squares problem admits a locally unique sparse solution in high dimensions. We demonstrate the power of our result by concretely quantifying the sparsity level for the aforementioned methods. Furthermore, we propose a novel coordinate descent based solver for NNLS in high dimensions using our theoretical result as motivation. We show through simulated data and a real data example that our solver achieves at least a 5x speed-up from the state-of-the-art solvers.