Papers › Multi-level projection with exponential parallel speedup; Application to sparse...
Multi-level projection with exponential parallel speedup; Application to sparse auto-encoders neural networks
Guillaume Perez, Michel Barlaud
The ℓ_(1,∞) norm is an efficient structured projection but the complexity of the best algorithm is unfortunately 𝒪(n m log(n m)) for a matrix in ℝ^(n×m). In this paper, we propose a new bi-level projection method for which we show that the time complexity for the ℓ_(1,∞) norm is only 𝒪(n m ) for a matrix in ℝ^(n×m), and 𝒪(n + m ) with full parallel power. We generalize our method to tensors and we propose a new multi-level projection, having an induced decomposition that yields a linear parallel speedup up to an exponential speedup factor, resulting in a time complexity lower-bounded by the sum of the dimensions, instead of the product of the dimensions. we provide a large base of implementation of our framework for bi-level and tri-level (matrices and tensors) for various norms and provides also the parallel implementation. Experiments show that our projection is $2$ times faster than the actual fastest Euclidean algorithms while providing same accuracy and better sparsity in neural networks applications.
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