# Binary embeddings with structured hashed projections

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Anna Choromanska,
Krzysztof Choromanski,
Mariusz Bojarski,
Tony Jebara,
Sanjiv Kumar,
Yann LeCun
;

Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:344-353, 2016.

#### Abstract

We consider the hashing mechanism for constructing binary embeddings, that involves pseudo-random projections followed by nonlinear (sign function) mappings. The pseudo-random projection is described by a matrix, where not all entries are independent random variables but instead a fixed “budget of randomness” is distributed across the matrix. Such matrices can be edfficiently stored in sub-quadratic or even linear space, provide reduction in randomness usage (i.e. number of required random values), and very often lead to computational speed ups. We prove several theoretical results showing that projections via various structured matrices followed by nonlinear mappings accurately preserve the angular distance between input high-dimensional vectors. To the best of our knowledge, these results are the first that give theoretical ground for the use of general structured matrices in the nonlinear setting. We empirically verify our theoretical findings and show the dependence of learning via structured hashed projections on the performance of neural network as well as nearest neighbor classifier.

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