We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.
@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2020.32, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Simonov, Kirill}, title = {{Low-Rank Binary Matrix Approximation in Column-Sum Norm}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.32}, URN = {urn:nbn:de:0030-drops-126355}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.32}, annote = {Keywords: Binary Matrix Factorization, PTAS, Column-sum norm} }
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