16,098 research outputs found

    Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search

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    The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distribution H\mathcal{H} over locality-sensitive hash functions that partition space. For a collection of nn points, after preprocessing, the query time is dominated by O(nρlogn)O(n^{\rho} \log n) evaluations of hash functions from H\mathcal{H} and O(nρ)O(n^{\rho}) hash table lookups and distance computations where ρ(0,1)\rho \in (0,1) is determined by the locality-sensitivity properties of H\mathcal{H}. It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to O(log2n)O(\log^2 n), leaving the query time to be dominated by O(nρ)O(n^{\rho}) distance computations and O(nρlogn)O(n^{\rho} \log n) additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework, making it a viable replacement in practice. Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to O(nρ)O(n^\rho).Comment: 15 pages, 3 figure

    Faster tuple lattice sieving using spherical locality-sensitive filters

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    To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension dd in time 20.3717d+o(d)2^{0.3717d + o(d)}, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time 20.3588d+o(d)2^{0.3588d + o(d)}. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology ePrint Archive 2017/228, available at https://ia.cr/2017/122

    Scalable Image Retrieval by Sparse Product Quantization

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    Fast Approximate Nearest Neighbor (ANN) search technique for high-dimensional feature indexing and retrieval is the crux of large-scale image retrieval. A recent promising technique is Product Quantization, which attempts to index high-dimensional image features by decomposing the feature space into a Cartesian product of low dimensional subspaces and quantizing each of them separately. Despite the promising results reported, their quantization approach follows the typical hard assignment of traditional quantization methods, which may result in large quantization errors and thus inferior search performance. Unlike the existing approaches, in this paper, we propose a novel approach called Sparse Product Quantization (SPQ) to encoding the high-dimensional feature vectors into sparse representation. We optimize the sparse representations of the feature vectors by minimizing their quantization errors, making the resulting representation is essentially close to the original data in practice. Experiments show that the proposed SPQ technique is not only able to compress data, but also an effective encoding technique. We obtain state-of-the-art results for ANN search on four public image datasets and the promising results of content-based image retrieval further validate the efficacy of our proposed method.Comment: 12 page

    Lower Bounds for Oblivious Near-Neighbor Search

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    We prove an Ω(dlgn/(lglgn)2)\Omega(d \lg n/ (\lg\lg n)^2) lower bound on the dynamic cell-probe complexity of statistically oblivious\mathit{oblivious} approximate-near-neighbor search (ANN\mathsf{ANN}) over the dd-dimensional Hamming cube. For the natural setting of d=Θ(logn)d = \Theta(\log n), our result implies an Ω~(lg2n)\tilde{\Omega}(\lg^2 n) lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for ANN\mathsf{ANN}. This is the first super-logarithmic unconditional\mathit{unconditional} lower bound for ANN\mathsf{ANN} against general (non black-box) data structures. We also show that any oblivious static\mathit{static} data structure for decomposable search problems (like ANN\mathsf{ANN}) can be obliviously dynamized with O(logn)O(\log n) overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page
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