When applying nonnegative matrix factorization (NMF), generally the rank parameter is unknown. Such rank in NMF, called the nonnegative rank, is usually estimated heuristically since computing the exact value of it is NP-hard. In this work, we propose an approximation method to estimate such rank while solving NMF on-the-fly. We use sum-of-norm (SON), a group-lasso structure that encourages pairwise similarity, to reduce the rank of a factor matrix where the rank is overestimated at the beginning. On various datasets, SON-NMF is able to reveal the correct nonnegative rank of the data without any prior knowledge nor tuning. SON-NMF is a nonconvx nonsmmoth non-separable non-proximable problem, solving it is nontrivial. First, as rank estimation in NMF is NP-hard, the proposed approach does not enjoy a lower computational complexity. Using a graph-theoretic argument, we prove that the complexity of the SON-NMF is almost irreducible. Second, the per-iteration cost of any algorithm solving SON-NMF is possibly high, which motivated us to propose a first-order BCD algorithm to approximately solve SON-NMF with a low per-iteration cost, in which we do so by the proximal average operator. Lastly, we propose a simple greedy method for post-processing. SON-NMF exhibits favourable features for applications. Beside the ability to automatically estimate the rank from data, SON-NMF can deal with rank-deficient data matrix, can detect weak component with small energy. Furthermore, on the application of hyperspectral imaging, SON-NMF handle the issue of spectral variability naturally.
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