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

Andersen Ang, Jianzhu Ma, Nianjun Liu, Kun Huang, Yijie Wang

Keywords: Capped Simplex, Projection, Newton's Method, Sparse Regression via Boolean Relaxation, Bioinformatics, GWAS

TL;DR: We find that solving the projection onto the capped simplex by Newton's method is FAST.

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.

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Code: https://anonymous.4open.science/r/Projection_Capped_Simplex-BB7C