title: Fast Projection onto the Capped Simplex with Applications to Sparse Regression in Bioinformatics
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.
authors: Andersen Ang Jianzhu Ma Nianjun Liu Kun Huang Yijie Wang
hIndex: Jianzhu Ma 20