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.

In this section, we demonstrate the effectiveness of low-rank matrix factorization in recovering the label relationship matrix. We first present four important facts: f1: the rank of the matrix is equivalent to the number of classes. Specifically, this also means that if ˆZi,k = 1, then ˆZj,k = 1. We consider a toy example (without self-loops), ˆZ = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A = 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 (14) In a standard LRMF problem, it is not possible to recover ˆZ from A since no entries are observed for the third and fourth rows. However, we can demonstrate how LRMF effectively performs in this situation. Recovery: We begin by assuming v1 is in class 1, resulting in U1,: = [1, 1, 1] and V1,: = [1,0,0]. By observing A1,4, we know that v4 is also in class 1, resulting in U4,: = [1, 1, 1]and V4,: = [1,0,0](f2). By analyzing A1,2 and A1,3, we determine that v2 and v3 do not belong to class 1.

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Table 9 compares these abstracts to the corresponding examples from Tables I.25-27 in section I of [46]. The abstracts are decoded using greedy algorithm with temperature T = 0.5.