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# Mathematics > Optimization and Control

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

(Submitted on 16 Oct 2021 (v1), last revised 26 Oct 2021 (this version, v4))

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.

## Submission history

From: Man Shun Ang [view email]**[v1]**Sat, 16 Oct 2021 05:03:24 GMT (917kb,D)

**[v2]**Tue, 19 Oct 2021 01:42:06 GMT (917kb,D)

**[v3]**Thu, 21 Oct 2021 19:56:47 GMT (917kb,D)

**[v4]**Tue, 26 Oct 2021 01:57:58 GMT (917kb,D)

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