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We design a new sparse projection method for a set of vectors that guaranteesa desired average sparsity level measured leveraging the popular Hoyer measure(an affine function of the ratio of the $\ell_1$ and $\ell_2$ norms). Existingapproaches either project each vector individually or require the use of aregularization parameter which implicitly maps to the average $\ell_0$-measureof sparsity. Instead, in our approach we set the sparsity level for the wholeset explicitly and simultaneously project a group of vectors with the sparsitylevel of each vector tuned automatically. We show that the computationalcomplexity of our projection operator is linear in the size of the problem.Additionally, we propose a generalization of this projection by replacing the$\ell_1$ norm by its weighted version. We showcase the efficacy of our approachin both supervised and unsupervised learning tasks on image datasets includingCIFAR10 and ImageNet. In deep neural network pruning, the sparse modelsproduced by our method on ResNet50 have significantly higher accuracies atcorresponding sparsity values compared to existing competitors. In nonnegativematrix factorization, our approach yields competitive reconstruction errorsagainst state-of-the-art algorithms.

Explicit Group Sparse Projection with Applications to Deep Learning and NMF