4,921 research outputs found

    Hashing for Similarity Search: A Survey

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    Similarity search (nearest neighbor search) is a problem of pursuing the data items whose distances to a query item are the smallest from a large database. Various methods have been developed to address this problem, and recently a lot of efforts have been devoted to approximate search. In this paper, we present a survey on one of the main solutions, hashing, which has been widely studied since the pioneering work locality sensitive hashing. We divide the hashing algorithms two main categories: locality sensitive hashing, which designs hash functions without exploring the data distribution and learning to hash, which learns hash functions according the data distribution, and review them from various aspects, including hash function design and distance measure and search scheme in the hash coding space

    Small Width, Low Distortions: Quantized Random Embeddings of Low-complexity Sets

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    Under which conditions and with which distortions can we preserve the pairwise-distances of low-complexity vectors, e.g., for structured sets such as the set of sparse vectors or the one of low-rank matrices, when these are mapped in a finite set of vectors? This work addresses this general question through the specific use of a quantized and dithered random linear mapping which combines, in the following order, a sub-Gaussian random projection in RM\mathbb R^M of vectors in RN\mathbb R^N, a random translation, or "dither", of the projected vectors and a uniform scalar quantizer of resolution δ>0\delta>0 applied componentwise. Thanks to this quantized mapping we are first able to show that, with high probability, an embedding of a bounded set K⊂RN\mathcal K \subset \mathbb R^N in δZM\delta \mathbb Z^M can be achieved when distances in the quantized and in the original domains are measured with the ℓ1\ell_1- and ℓ2\ell_2-norm, respectively, and provided the number of quantized observations MM is large before the square of the "Gaussian mean width" of K\mathcal K. In this case, we show that the embedding is actually "quasi-isometric" and only suffers of both multiplicative and additive distortions whose magnitudes decrease as M−1/5M^{-1/5} for general sets, and as M−1/2M^{-1/2} for structured set, when MM increases. Second, when one is only interested in characterizing the maximal distance separating two elements of K\mathcal K mapped to the same quantized vector, i.e., the "consistency width" of the mapping, we show that for a similar number of measurements and with high probability this width decays as M−1/4M^{-1/4} for general sets and as 1/M1/M for structured ones when MM increases. Finally, as an important aspect of our work, we also establish how the non-Gaussianity of the mapping impacts the class of vectors that can be embedded or whose consistency width provably decays when MM increases.Comment: Keywords: quantization, restricted isometry property, compressed sensing, dimensionality reduction. 31 pages, 1 figur

    k-Nearest Neighbour Classifiers: 2nd Edition (with Python examples)

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    Perhaps the most straightforward classifier in the arsenal or machine learning techniques is the Nearest Neighbour Classifier -- classification is achieved by identifying the nearest neighbours to a query example and using those neighbours to determine the class of the query. This approach to classification is of particular importance because issues of poor run-time performance is not such a problem these days with the computational power that is available. This paper presents an overview of techniques for Nearest Neighbour classification focusing on; mechanisms for assessing similarity (distance), computational issues in identifying nearest neighbours and mechanisms for reducing the dimension of the data. This paper is the second edition of a paper previously published as a technical report. Sections on similarity measures for time-series, retrieval speed-up and intrinsic dimensionality have been added. An Appendix is included providing access to Python code for the key methods.Comment: 22 pages, 15 figures: An updated edition of an older tutorial on kN
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