We consider a new construction of locality-sensitive hash functions for Hamming space that is \emph{covering} in the sense that is it guaranteed to produce a collision for every pair of vectors within a given radius $r$. The construction is \emph{efficient} in the sense that the expected number of hash collisions between vectors at distance~$cr$, for a given $c>1$, comes close to that of the best possible data independent LSH without the covering guarantee, namely, the seminal LSH construction of Indyk and Motwani (STOC '98). The efficiency of the new construction essentially \emph{matches} their bound when the search radius is not too large --- e.g., when $cr = o(\log(n)/\log\log n)$, where $n$ is the number of points in the data set, and when $cr = \log(n)/k$ where $k$ is an integer constant. In general, it differs by at most a factor $\ln(4)$ in the exponent of the time bounds. As a consequence, LSH-based similarity search in Hamming space can avoid the problem of false negatives at little or no cost in efficiency.
Locality-sensitive hashing (LSH) has emerged as the dominant algorithmic technique for similarity search with strong performance guarantees in high-dimensional spaces. A drawback of traditional LSH schemes is that they may have \emph{false negatives}, i.e., the recall is less than 100\%. This limits the applicability of LSH in settings requiring precise performance guarantees. Building on the recent theoretical "CoveringLSH" construction that eliminates false negatives, we propose a fast and practical covering LSH scheme for Hamming space called \emph{Fast CoveringLSH (fcLSH)}. Inheriting the …
This paper introduces "Multi-Level Spherical LSH": parameter-free, a multi-level, data-dependant Locality Sensitive Hashing data structure for solving the Approximate Near Neighbors Problem (ANN). This data structure uses a modified version of a multi-probe adaptive querying algorithm, with the potential of achieving a $O(n^p + t)$ query run time, for all inputs n where $t <= n$.
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 $\mathcal{H}$ over locality-sensitive hash functions that partition space. For a collection of $n$ points, after preprocessing, the query time is dominated by $O(n^ρ \log n)$ evaluations of hash functions from $\mathcal{H}$ and $O(n^ρ)$ hash table lookups and distance computations where $ρ\in (0,1)$ is determined by the locality-sensitivity properties of $\mathcal{H}$. It follows from a recent resul…
In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius $R$ in $d$ dimensional space equipped with $l_p$ norm when $p \in [1,\infty]$. Furthermore, we show how to use these hash functions to solve the $c$-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance $R$, we will certainly report it and points at distance greater than $cR$ will not be reported for $c=Ω(\sqrt{d},d^{1-\frac{1}{p}})$. The constructed algorithms work:…
Locality sensitive hashing (LSH) was introduced by Indyk and Motwani (STOC `98) to give the first sublinear time algorithm for the c-approximate nearest neighbor (ANN) problem using only polynomial space. At a high level, an LSH family hashes "nearby" points to the same bucket and "far away" points to different buckets. The quality of measure of an LSH family is its LSH exponent, which helps determine both query time and space usage. In a seminal work, Andoni and Indyk (FOCS `06) constructed an LSH family based on random ball partitioning of space that achieves an LSH exponent of 1/c^2 for the…
Edit distance similarity search, also called approximate pattern matching, is a fundamental problem with widespread database applications. The goal of the problem is to preprocess $n$ strings of length $d$, to quickly answer queries $q$ of the form: if there is a database string within edit distance $r$ of $q$, return a database string within edit distance $cr$ of $q$. Previous approaches to this problem either rely on very large (superconstant) approximation ratios $c$, or very small search radii $r$. Outside of a narrow parameter range, these solutions are not competitive with trivially sear…
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 (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), 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, …
In this paper, we examine the hash functions expressed as scalar products, i.e., $f(x)=<v,x>$, for some bounded random vector $v$. Such hash functions have numerous applications, but often there is a need to optimize the choice of the distribution of $v$. In the present work, we focus on so-called anti-concentration bounds, i.e. the upper bounds of $\mathbb{P}\left[|<v,x>| < α\right]$. In many applications, $v$ is a vector of independent random variables with standard normal distribution. In such case, the distribution of $<v,x>$ is also normal and it is easy to approximate $\mathbb{P}\left[|<…
Locality-Sensitive Hashing (LSH) is one of the most popular methods for $c$-Approximate Nearest Neighbor Search ($c$-ANNS) in high-dimensional spaces. In this paper, we propose a novel LSH scheme based on the Longest Circular Co-Substring (LCCS) search framework (LCCS-LSH) with a theoretical guarantee. We introduce a novel concept of LCCS and a new data structure named Circular Shift Array (CSA) for $k$-LCCS search. The insight of LCCS search framework is that close data objects will have a longer LCCS than the far-apart ones with high probability. LCCS-LSH is \emph{LSH-family-independent}, an…
We present a new data structure for the c-approximate near neighbor problem (ANN) in the Euclidean space. For n points in R^d, our algorithm achieves O(n^ρ + d log n) query time and O(n^{1 + ρ} + d log n) space, where ρ<= 7/(8c^2) + O(1 / c^3) + o(1). This is the first improvement over the result by Andoni and Indyk (FOCS 2006) and the first data structure that bypasses a locality-sensitive hashing lower bound proved by O'Donnell, Wu and Zhou (ICS 2011). By a standard reduction we obtain a data structure for the Hamming space and \ell_1 norm with ρ<= 7/(8c) + O(1/c^{3/2}) + o(1), which is the …
Similarity joins are important operations with a broad range of applications. In this paper, we study the problem of vector similarity join size estimation (VSJ). It is a generalization of the previously studied set similarity join size estimation (SSJ) problem and can handle more interesting cases such as TF-IDF vectors. One of the key challenges in similarity join size estimation is that the join size can change dramatically depending on the input similarity threshold. We propose a sampling based algorithm that uses the Locality-Sensitive-Hashing (LSH) scheme. The proposed algorithm LSH-SS u…
We prove a tight lower bound for the exponent $ρ$ for data-dependent Locality-Sensitive Hashing schemes, recently used to design efficient solutions for the $c$-approximate nearest neighbor search. In particular, our lower bound matches the bound of $ρ\le \frac{1}{2c-1}+o(1)$ for the $\ell_1$ space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC'15]. In recent years it emerged that data-dependent hashing is strictly superior to the classical Locality-Sensitive Hashing, when the hash function is data-independent. In the latter setting, the best exponent has been already known:…
A Locality-Sensitive Hash (LSH) function is called $(r,cr,p_1,p_2)$-sensitive, if two data-points with a distance less than $r$ collide with probability at least $p_1$ while data points with a distance greater than $cr$ collide with probability at most $p_2$. These functions form the basis of the successful Indyk-Motwani algorithm (STOC 1998) for nearest neighbour problems. In particular one may build a $c$-approximate nearest neighbour data structure with query time $\tilde O(n^ρ/p_1)$ where $ρ=\frac{\log1/p_1}{\log1/p_2}\in(0,1)$. That is, sub-linear time, as long as $p_1$ is not too small. …
We study the problem of approximate near neighbor (ANN) search and show the following results: - An improved framework for solving the ANN problem using locality-sensitive hashing, reducing the number of evaluations of locality-sensitive hash functions and the word-RAM complexity compared to the standard framework. - A framework for solving the ANN problem with space-time tradeoffs as well as tight upper and lower bounds for the space-time tradeoff of framework solutions to the ANN problem under cosine similarity. - A novel approach to solving the ANN problem on sets along with a matching lowe…
We present a data structure for *spherical range reporting* on a point set $S$, i.e., reporting all points in $S$ that lie within radius $r$ of a given query point $q$. Our solution builds upon the Locality-Sensitive Hashing (LSH) framework of Indyk and Motwani, which represents the asymptotically best solutions to near neighbor problems in high dimensions. While traditional LSH data structures have several parameters whose optimal values depend on the distance distribution from $q$ to the points of $S$, our data structure is parameter-free, except for the space usage, which is configurable by…
Random binning features, introduced in the seminal paper of Rahimi and Recht (2007), are an efficient method for approximating a kernel matrix using locality sensitive hashing. Random binning features provide a very simple and efficient way of approximating the Laplace kernel but unfortunately do not apply to many important classes of kernels, notably ones that generate smooth Gaussian processes, such as the Gaussian kernel and Matern kernel. In this paper, we introduce a simple weighted version of random binning features and show that the corresponding kernel function generates Gaussian proce…
Recently it was shown that the problem of Maximum Inner Product Search (MIPS) is efficient and it admits provably sub-linear hashing algorithms. Asymmetric transformations before hashing were the key in solving MIPS which was otherwise hard. In the prior work, the authors use asymmetric transformations which convert the problem of approximate MIPS into the problem of approximate near neighbor search which can be efficiently solved using hashing. In this work, we provide a different transformation which converts the problem of approximate MIPS into the problem of approximate cosine similarity s…
We consider variations of set reconciliation problems where two parties, Alice and Bob, each hold a set of points in a metric space, and the goal is for Bob to conclude with a set of points that is close to Alice's set of points in a well-defined way. This setting has been referred to as robust set reconciliation. More specifically, in one variation we examine the goal is for Bob to end with a set of points that is close to Alice's in earth mover's distance, and in another the goal is for Bob to have a point that is close to each of Alice's. The first problem has been studied before; our resul…
We investigate the problem of finding reverse nearest neighbors efficiently. Although provably good solutions exist for this problem in low or fixed dimensions, to this date the methods proposed in high dimensions are mostly heuristic. We introduce a method that is both provably correct and efficient in all dimensions, based on a reduction of the problem to one instance of $\e$-nearest neighbor search plus a controlled number of instances of {\em exhaustive $r$-\pleb}, a variant of {\em Point Location among Equal Balls} where all the $r$-balls centered at the data points that contain the query…
We propose a new class of data-independent locality-sensitive hashing (LSH) algorithms based on the fruit fly olfactory circuit. The fundamental difference of this approach is that, instead of assigning hashes as dense points in a low dimensional space, hashes are assigned in a high dimensional space, which enhances their separability. We show theoretically and empirically that this new family of hash functions is locality-sensitive and preserves rank similarity for inputs in any `p space. We then analyze different variations on this strategy and show empirically that they outperform existing …
We present Falconn++, a novel locality-sensitive filtering approach for approximate nearest neighbor search on angular distance. Falconn++ can filter out potential far away points in any hash bucket \textit{before} querying, which results in higher quality candidates compared to other hashing-based solutions. Theoretically, Falconn++ asymptotically achieves lower query time complexity than Falconn, an optimal locality-sensitive hashing scheme on angular distance. Empirically, Falconn++ achieves higher recall-speed tradeoffs than Falconn on many real-world data sets. Falconn++ is also competiti…
Department of Computer Science, University of Southern California（南加州大学计算机科学系）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
We present a GPU-based Locality Sensitive Hashing (LSH) algorithm to speed up beam search for sequence models. We utilize the winner-take-all (WTA) hash, which is based on relative ranking order of hidden dimensions and thus resilient to perturbations in numerical values. Our algorithm is designed by fully considering the underling architecture of CUDA-enabled GPUs (Algorithm/Architecture Co-design): 1) A parallel Cuckoo hash table is applied for LSH code lookup (guaranteed O(1) lookup time); 2) Candidate lists are shared across beams to maximize the parallelism; 3) Top frequent words are merg…
We show that approximate similarity (near neighbour) search can be solved in high dimensions with performance matching state of the art (data independent) Locality Sensitive Hashing, but with a guarantee of no false negatives. Specifically, we give two data structures for common problems. For $c$-approximate near neighbour in Hamming space we get query time $dn^{1/c+o(1)}$ and space $dn^{1+1/c+o(1)}$ matching that of \cite{indyk1998approximate} and answering a long standing open question from~\cite{indyk2000dimensionality} and~\cite{pagh2016locality} in the affirmative. By means of a new deter…
We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an $n$-point data set in a $d$-dimensional space our data structure achieves query time $O(d n^{ρ+o(1)})$ and space $O(n^{1+ρ+o(1)} + dn)$, where $ρ=\tfrac{1}{2c^2-1}$ for the Euclidean space and approximation $c>1$. For the Hamming space, we obtain an exponent of $ρ=\tfrac{1}{2c-1}$. Our result completes the direction set forth in [AINR14] who gave a proof-of-concept that data-dependent hashing can outperform classical Locality Sensitive Hashing (LSH). In contrast to [AINR14], the new bound is not …
Rice University（莱斯大学）；Institute for Advanced Study（高等研究院）；Princeton University（普林斯顿大学）
We present the first provable Least-Squares Value Iteration (LSVI) algorithms that have runtime complexity sublinear in the number of actions. We formulate the value function estimation procedure in value iteration as an approximate maximum inner product search problem and propose a locality sensitive hashing (LSH) [Indyk and Motwani STOC'98, Andoni and Razenshteyn STOC'15, Andoni, Laarhoven, Razenshteyn and Waingarten SODA'17] type data structure to solve this problem with sublinear time complexity. Moreover, we build the connections between the theory of approximate maximum inner product sea…
Locality-sensitive hashing (LSH) is an important tool for managing high-dimensional noisy or uncertain data, for example in connection with data cleaning (similarity join) and noise-robust search (similarity search). However, for a number of problems the LSH framework is not known to yield good solutions, and instead ad hoc solutions have been designed for particular similarity and distance measures. For example, this is true for output-sensitive similarity search/join, and for indexes supporting annulus queries that aim to report a point close to a certain given distance from the query point.…
In this paper we present a lock-free version of Hopscotch Hashing. Hopscotch Hashing is an open addressing algorithm originally proposed by Herlihy, Shavit, and Tzafrir, which is known for fast performance and excellent cache locality. The algorithm allows users of the table to skip or jump over irrelevant entries, allowing quick search, insertion, and removal of entries. Unlike traditional linear probing, Hopscotch Hashing is capable of operating under a high load factor, as probe counts remain small. Our lock-free version improves on both speed, cache locality, and progress guarantees of the…
Microsoft Research（微软研究院）；The University of Queensland（昆士兰大学）；Tsinghua University（清华大学）
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…
In recent, deep learning has become the most popular direction in machine learning and artificial intelligence. However, preparation of training data is often a bottleneck in the lifecycle of deploying a deep learning model for production or research. Reusing models for inferencing a dataset can greatly save the human costs required for training data creation. Although there exist a number of model sharing platform such as TensorFlow Hub, PyTorch Hub, DLHub, most of these systems require model uploaders to manually specify the details of each model and model downloaders to screen keyword searc…
Carpooling, or sharing a ride with other passengers, holds immense potential for urban transportation. Ridesharing platforms enable such sharing of rides using real-time data. Finding ride matches in real-time at urban scale is a difficult combinatorial optimization task and mostly heuristic approaches are applied. In this work, we mathematically model the problem as that of finding near-neighbors and devise a novel efficient spatio-temporal search algorithm based on the theory of locality sensitive hashing for Maximum Inner Product Search (MIPS). The proposed algorithm can find $k$ near-optim…
We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}^d under the Hamming distance. Recall that here H is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y in {0,1}^d with dist(x,y) at most r have probability at least p of collision under a randomly chosen h in H, whereas all pairs x, y in {0,1}^d with dist(x,y) at least cr have probability at most q of collision. Typically, one considers d tending to infinity, with c fixed and q bounded away from 0. For its applications to approximate nearest neighbor search in high dimensio…
Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let $ s(x,y) $ be the similarity between two points $ x $ and $ y $. We define a similarity between a set $ Q$ and a point $ x $ by aggregating the similarities $ s(p,x) $ for all $ p\in Q $. For example, we can take $ s(p,x) $ to be the angular similarity between $ p $ and $ x $ (i.e., $1-{\angle (x,p)}/…
Computing approximate nearest neighbors in high dimensional spaces is a central problem in large-scale data mining with a wide range of applications in machine learning and data science. A popular and effective technique in computing nearest neighbors approximately is the locality-sensitive hashing (LSH) scheme. In this paper, we aim to develop LSH schemes for distance functions that measure the distance between two probability distributions, particularly for f-divergences as well as a generalization to capture mutual information loss. First, we provide a general framework to design LHS scheme…
We study data structures for storing a set of polygonal curves in ${\rm R}^d$ such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fréchet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. We give solutions for different types of alignments including constrained and unconstrained versions. For uncons…
Can we sense our location in an unfamiliar environment by taking a sublinear-size sample of our surroundings? Can we efficiently encrypt a message that only someone physically close to us can decrypt? To solve this kind of problems, we introduce and study a new type of hash functions for finding shifts in sublinear time. A function $h:\{0,1\}^n\to \mathbb{Z}_n$ is a $(d,δ)$ {\em locality-preserving hash function for shifts} (LPHS) if: (1) $h$ can be computed by (adaptively) querying $d$ bits of its input, and (2) $\Pr [ h(x) \neq h(x \ll 1) + 1 ] \leq δ$, where $x$ is random and $\ll 1$ denote…
Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. represent polygons by their "turning function" - a function which follows the angle between the polygon's tangent and the $ x $-axis while traversing the perimeter of the polygon. T…
Sister Nibedita Govt. General Degree College for Girls（尼贝迪塔修女女子政府普通学位学院）；Indian Statistical Institute（印度统计研究所）
Single Molecule Real-Time (SMRT) sequencing is a recent advancement of Next Gen technology developed by Pacific Bio (PacBio). It comes with an explosion of long and noisy reads demanding cutting edge research to get most out of it. To deal with the high error probability of SMRT data, a novel contextual Locality Sensitive Hashing (conLSH) based algorithm is proposed in this article, which can effectively align the noisy SMRT reads to the reference genome. Here, sequences are hashed together based not only on their closeness, but also on similarity of context. The algorithm has $\mathcal{O}(n^{…
Given a collection of objects and an associated similarity measure, the all-pairs similarity search problem asks us to find all pairs of objects with similarity greater than a certain user-specified threshold. Locality-sensitive hashing (LSH) based methods have become a very popular approach for this problem. However, most such methods only use LSH for the first phase of similarity search - i.e. efficient indexing for candidate generation. In this paper, we present BayesLSH, a principled Bayesian algorithm for the subsequent phase of similarity search - performing candidate pruning and similar…
We present PSEUDo, an adaptive feature learning technique for exploring visual patterns in multi-track sequential data. Our approach is designed with the primary focus to overcome the uneconomic retraining requirements and inflexible representation learning in current deep learning-based systems. Multi-track time series data are generated on an unprecedented scale due to increased sensors and data storage. These datasets hold valuable patterns, like in neuromarketing, where researchers try to link patterns in multivariate sequential data from physiological sensors to the purchase behavior of p…
Hash tables are a ubiquitous class of dictionary data structures. However, standard hash table implementations do not translate well into the external memory model, because they do not incorporate locality for insertions. Iacono and Patracsu established an update/query tradeoff curve for external hash tables: a hash table that performs insertions in $O(λ/B)$ amortized IOs requires $Ω(\log_λN)$ expected IOs for queries, where $N$ is the number of items that can be stored in the data structure, $B$ is the size of a memory transfer, $M$ is the size of memory, and $λ$ is a tuning parameter. They p…
Many bioinformatics applications involve bucketing a set of sequences where each sequence is allowed to be assigned into multiple buckets. To achieve both high sensitivity and precision, bucketing methods are desired to assign similar sequences into the same bucket while assigning dissimilar sequences into distinct buckets. Existing $k$-mer-based bucketing methods have been efficient in processing sequencing data with low error rate, but encounter much reduced sensitivity on data with high error rate. Locality-sensitive hashing (LSH) schemes are able to mitigate this issue through tolerating t…
We present a framework for similarity search based on Locality-Sensitive Filtering (LSF), generalizing the Indyk-Motwani (STOC 1998) Locality-Sensitive Hashing (LSH) framework to support space-time tradeoffs. Given a family of filters, defined as a distribution over pairs of subsets of space with certain locality-sensitivity properties, we can solve the approximate near neighbor problem in $d$-dimensional space for an $n$-point data set with query time $dn^{ρ_q+o(1)}$, update time $dn^{ρ_u+o(1)}$, and space usage $dn + n^{1 + ρ_u + o(1)}$. The space-time tradeoff is tied to the tradeoff betwee…
We present the first provably sublinear time algorithm for approximate \emph{Maximum Inner Product Search} (MIPS). Our proposal is also the first hashing algorithm for searching with (un-normalized) inner product as the underlying similarity measure. Finding hashing schemes for MIPS was considered hard. We formally show that the existing Locality Sensitive Hashing (LSH) framework is insufficient for solving MIPS, and then we extend the existing LSH framework to allow asymmetric hashing schemes. Our proposal is based on an interesting mathematical phenomenon in which inner products, after indep…
To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehlé [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 $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we gener…
We consider the problem of approximate set similarity search under Braun-Blanquet similarity $B(\mathbf{x}, \mathbf{y}) = |\mathbf{x} \cap \mathbf{y}| / \max(|\mathbf{x}|, |\mathbf{y}|)$. The $(b_2, b_2)$-approximate Braun-Blanquet similarity search problem is to preprocess a collection of sets $P$ such that, given a query set $\mathbf{q}$, if there exists $\mathbf{x} \in P$ with $B(\mathbf{q}, \mathbf{x}) \geq b_1$, then we can efficiently return $\mathbf{x}' \in P$ with $B(\mathbf{q}, \mathbf{x}') > b_2$. We present a simple data structure that solves this problem with space usage $O(n^{1+ρ}…
All-pairs set similarity is a widely used data mining task, even for large and high-dimensional datasets. Traditionally, similarity search has focused on discovering very similar pairs, for which a variety of efficient algorithms are known. However, recent work highlights the importance of finding pairs of sets with relatively small intersection sizes. For example, in a recommender system, two users may be alike even though their interests only overlap on a small percentage of items. In such systems, some dimensions are often highly skewed because they are very popular. Together these two prop…
The probability Jaccard similarity was recently proposed as a natural generalization of the Jaccard similarity to measure the proximity of sets whose elements are associated with relative frequencies or probabilities. In combination with a hash algorithm that maps those weighted sets to compact signatures which allow fast estimation of pairwise similarities, it constitutes a valuable method for big data applications such as near-duplicate detection, nearest neighbor search, or clustering. This paper introduces a class of one-pass locality-sensitive hash algorithms that are orders of magnitude …
We present a new locality sensitive hashing (LSH) algorithm for $c$-approximate nearest neighbor search in $\ell_p$ with $1<p<2$. For a database of $n$ points in $\ell_p$, we achieve $O(dn^ρ)$ query time and $O(dn+n^{1+ρ})$ space, where $ρ\le O((\ln c)^2/c^p)$. This improves upon the previous best upper bound $ρ\le 1/c$ by Datar et al. (SOCG 2004), and is close to the lower bound $ρ\ge 1/c^p$ by O'Donnell, Wu and Zhou (ITCS 2011). The proof is a simple generalization of the LSH scheme for $\ell_2$ by Andoni and Indyk (FOCS 2006).
[See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the $c$-Approximate Near Neighbor Search problem. For the $d$-dimensional Euclidean space and $n$-point datasets, we develop a data structure with space $n^{1 + ρ_u + o(1)} + O(dn)$ and query time $n^{ρ_q + o(1)} + d n^{o(1)}$ for every $ρ_u, ρ_q \geq 0$ such that: \begin{equation} c^2 \sqrt{ρ_q} + (c^2 - 1) \sqrt{ρ_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor $c > 1$, improving…
