We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys \mathcal{O}(d \ln d ) hash computation time. Building on a recent result in (Andoni, Indyk, Laarhoven, Razenshteyn '15), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from \mathcal{O}(d^2) to \mathcal{O}(d \ln d ). Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only \mathcal{O}(\ln^9(d)) random bits.
@InProceedings{kennedy_et_al:LIPIcs.ITCS.2017.53, author = {Kennedy, Christopher and Ward, Rachel}, title = {{Fast Cross-Polytope Locality-Sensitive Hashing}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {53:1--53:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.53}, URN = {urn:nbn:de:0030-drops-81936}, doi = {10.4230/LIPIcs.ITCS.2017.53}, annote = {Keywords: Locality-sensitive hashing, Dimension reduction, Johnson-Lindenstrauss lemma} }
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