Fast Projection onto the Capped Simplex with Applications to Sparse Regression in Bioinformatics

TL;DR

This paper introduces a Newton's method-based algorithm for fast projection onto the k-capped simplex, significantly reducing computational time for large-scale sparse regression problems in bioinformatics.

Contribution

The paper presents a novel, efficient Newton's method algorithm for projecting onto the k-capped simplex, outperforming existing sorting-based methods in speed and accuracy.

Findings

Achieves high-precision projection with O(n) complexity.

Outperforms state-of-the-art methods by 6-8 times in runtime.

Enables large-scale sparse regression in bioinformatics with 3-6 times faster results.

Abstract

We consider the problem of projecting a vector onto the so-called k-capped simplex, which is a hyper-cube cut by a hyperplane. For an n-dimensional input vector with bounded elements, we found that a simple algorithm based on Newton's method is able to solve the projection problem to high precision with a complexity roughly about O(n), which has a much lower computational cost compared with the existing sorting-based methods proposed in the literature. We provide a theory for partial explanation and justification of the method. We demonstrate that the proposed algorithm can produce a solution of the projection problem with high precision on large scale datasets, and the algorithm is able to significantly outperform the state-of-the-art methods in terms of runtime (about 6-8 times faster than a commercial software with respect to CPU time for input vector with 1 million variables or more). We further illustrate the effectiveness of the proposed algorithm on solving sparse regression in a bioinformatics problem. Empirical results on the GWAS dataset (with 1,500,000 single-nucleotide polymorphisms) show that, when using the proposed method to accelerate the Projected Quasi-Newton (PQN) method, the accelerated PQN algorithm is able to handle huge-scale regression problem and it is more efficient (about 3-6 times faster) than the current state-of-the-art methods.