Block Coordinate Descent for Sparse NMF
This addresses the computational bottleneck in sparse NMF for data analysis, making it more suitable for large-scale applications, though it is incremental as it improves speed rather than introducing a new paradigm.
The paper tackled the problem of slow optimization for sparse nonnegative matrix factorization (NMF) with mixed norm L1/L2 sparsity constraints, and the result was a new algorithm that performs an order of magnitude faster than state-of-the-art solvers on real-world datasets.
Nonnegative matrix factorization (NMF) has become a ubiquitous tool for data analysis. An important variant is the sparse NMF problem which arises when we explicitly require the learnt features to be sparse. A natural measure of sparsity is the L$_0$ norm, however its optimization is NP-hard. Mixed norms, such as L$_1$/L$_2$ measure, have been shown to model sparsity robustly, based on intuitive attributes that such measures need to satisfy. This is in contrast to computationally cheaper alternatives such as the plain L$_1$ norm. However, present algorithms designed for optimizing the mixed norm L$_1$/L$_2$ are slow and other formulations for sparse NMF have been proposed such as those based on L$_1$ and L$_0$ norms. Our proposed algorithm allows us to solve the mixed norm sparsity constraints while not sacrificing computation time. We present experimental evidence on real-world datasets that shows our new algorithm performs an order of magnitude faster compared to the current state-of-the-art solvers optimizing the mixed norm and is suitable for large-scale datasets.