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Fast Binary Embeddings with Gaussian Circulant Matrices: Improved Bounds

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Abstract

We consider the problem of encoding a finite set of vectors into a small number of bits while approximately retaining information on the angular distances between the vectors. By deriving improved variance bounds related to binary Gaussian circulant embeddings, we largely fix a gap in the proof of the best known fast binary embedding method. Our bounds also show that well-spreadness assumptions on the data vectors, which were needed in earlier work on variance bounds, are unnecessary. In addition, we propose a new binary embedding with a faster running time on sparse data.

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Acknowledgements

The authors would like to thank the reviewers for valuable comments, in particular for a suggestion that substantially shortened the proof of Lemma 2.3. A. Stollenwerk acknowledges funding by the European Research Council through ERC Starting Grant StG 258926. S. Dirksen and A. Stollenwerk acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) through the project Quantized Compressive Spectrum Sensing (QuaCoSS), which is part of the priority program SPP 1798 Compressed Sensing in Information Processing (COSIP).

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  1. Lehrstuhl C für Mathematik (Analysis), RWTH Aachen University, Pontdriesch 10, 52062, Aachen, Germany

    Sjoerd Dirksen & Alexander Stollenwerk

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  1. Sjoerd Dirksen
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  2. Alexander Stollenwerk
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Correspondence to Sjoerd Dirksen.

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Dirksen, S., Stollenwerk, A. Fast Binary Embeddings with Gaussian Circulant Matrices: Improved Bounds. Discrete Comput Geom 60, 599–626 (2018). https://doi.org/10.1007/s00454-017-9964-x

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  • DOI: https://doi.org/10.1007/s00454-017-9964-x

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