{'fraction_non_alphanumeric': 0.02483343428225318, 'fraction_numerical': 0.0, 'mean_word_length': 5.233962264150944, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Low-Rank Matrix Completion in the Presence of High Coherence

Prevalent matrix completion methods capture only the low-rank property which gives merely a constraint that the data points lie on some low-dimensional subspace, but generally ignore the extra structures (beyond low-rank) that specify in more detail how the data points lie on the subspace. Whenever the data points are not uniformly distributed on the low-dimensional subspace, the row-coherence of the target matrix to recover could be considerably high and, accordingly, prevalent methods might fail even if the target matrix is fairly low-rank. To relieve this challenge, we suggest to consider a model termed low-rank factor decomposition (LRFD), which imposes an additional restriction that the data points must be represented as linear, compressive combinations of the bases in a given dictionary. We show that LRFD can effectively mitigate the challenges of high row-coherence, provided that its dictionary is configured properly. Namely, it is mathematically proven that if the dictionary is well-conditioned and low-rank, then LRFD can weaken the dependence on the row-coherence. In particular, if the dictionary itself is low-rank, then the dependence on the row-coherence can be entirely removed. Subsequently, we devise two practical algorithms to obtain proper dictionaries in unsupervised environments: one uses the existing matrix completion methods to construct the dictionary in LRFD, and the other tries to learn a proper dictionary from the data given. Experiments on randomly generated matrices and motion datasets show superior performance of our proposed algorithms.