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We consider the problem of projecting a vector onto the so-called k-cappedsimplex, which is a hyper-cube cut by a hyperplane. For an n-dimensional inputvector with bounded elements, we found that a simple algorithm based onNewton's method is able to solve the projection problem to high precision witha complexity roughly about O(n), which has a much lower computational costcompared with the existing sorting-based methods proposed in the literature. Weprovide a theory for partial explanation and justification of the method.  We demonstrate that the proposed algorithm can produce a solution of theprojection problem with high precision on large scale datasets, and thealgorithm is able to significantly outperform the state-of-the-art methods interms of runtime (about 6-8 times faster than a commercial software withrespect to CPU time for input vector with 1 million variables or more).  We further illustrate the effectiveness of the proposed algorithm on solvingsparse regression in a bioinformatics problem. Empirical results on the GWASdataset (with 1,500,000 single-nucleotide polymorphisms) show that, when usingthe proposed method to accelerate the Projected Quasi-Newton (PQN) method, theaccelerated PQN algorithm is able to handle huge-scale regression problem andit is more efficient (about 3-6 times faster) than the current state-of-the-artmethods.

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