Computational geometry

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

This paper describes ANN-Benchmarks, a tool for evaluating the performance of in-memory approximate nearest neighbor algorithms It provides a standard interface for measuring the performance and quality achieved by nearest neighbor algorithms on different standard data sets It supports several different ways of integrating k-NN algorithms, and its configuration system automatically tests a range of parameter settings for each algorithm Algorithms are compared with respect to many different (approximate) quality measures, and adding more is easy and fast; the included plotting front-ends can visualise these as images, Open image in new window plots, and websites with interactive plots ANN-Benchmarks aims to provide a constantly updated overview of the current state of the art of k-NN algorithms In the short term, this overview allows users to choose the correct k-NN algorithm and parameters for their similarity search task; in the longer term, algorithm designers will be able to use this overview to test and refine automatic parameter tuning The paper gives an overview of the system, evaluates the results of the benchmark, and points out directions for future work Interestingly, very different approaches to k-NN search yield comparable quality-performance trade-offs The system is available at http://sssprojectsitudk/ann-benchmarks/

ANN-Benchmarks: A benchmarking tool for approximate nearest neighbor algorithms

Term frequency is a common method for identifying the importance of a term in a query or document. But it is a weak signal, especially when the frequency distribution is flat, such as in long queries or short documents where the text is of sentence/passage-length. This paper proposes a Deep Contextualized Term Weighting framework that learns to map BERT's contextualized text representations to context-aware term weights for sentences and passages. When applied to passages, DeepCT-Index produces term weights that can be stored in an ordinary inverted index for passage retrieval. When applied to query text, DeepCT-Query generates a weighted bag-of-words query. Both types of term weight can be used directly by typical first-stage retrieval algorithms. This is novel because most deep neural network based ranking models have higher computational costs, and thus are restricted to later-stage rankers. Experiments on four datasets demonstrate that DeepCT's deep contextualized text understanding greatly improves the accuracy of first-stage retrieval algorithms.

Context-Aware Sentence/Passage Term Importance Estimation For First Stage Retrieval

Entity Resolution (ER), a core task of Data Integration, detects different entity profiles that correspond to the same real-world object. Due to its inherently quadratic complexity, a series of techniques accelerate it so that it scales to voluminous data. In this survey, we review a large number of relevant works under two different but related frameworks: Blocking and Filtering. The former restricts comparisons to entity pairs that are more likely to match, while the latter identifies quickly entity pairs that are likely to satisfy predetermined similarity thresholds. We also elaborate on hybrid approaches that combine different characteristics. For each framework we provide a comprehensive list of the relevant works, discussing them in the greater context. We conclude with the most promising directions for future work in the field.

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Blocking and Filtering Techniques for Entity Resolution: A Survey

We consider the problem of approximate set similarity search under Braun-Blanquet similarity B(x, y) = |x ∩ y| / max(|x|, |y|). The (b1, b2)-approximate Braun-Blanquet similarity search problem is to preprocess a collection of sets P such that, given a query set q, if there exists x E P with B(q, x) ≥ b1, then we can efficiently return x′ E P with B(q, x′) > b2. We present a simple data structure that solves this problem with space usage O(n1+ρlogn + ∑x e P|x|) and query time O(|q|nρ logn) where n = |P| and ρ = log(1/b1)/log(1/b2). Making use of existing lower bounds for locality-sensitive hashing by O'Donnell et al. (TOCT 2014) we show that this value of ρ is tight across the parameter space, i.e., for every choice of constants 0 b2 b1 In the case where all sets have the same size our solution strictly improves upon the value of ρ that can be obtained through the use of state-of-the-art data-independent techniques in the Indyk-Motwani locality-sensitive hashing framework (STOC 1998) such as Broder's MinHash (CCS 1997) for Jaccard similarity and Andoni et al.'s cross-polytope LSH (NIPS 2015) for cosine similarity. Surprisingly, even though our solution is data-independent, for a large part of the parameter space we outperform the currently best data-dependent method by Andoni and Razenshteyn (STOC 2015).

Set similarity search beyond MinHash

Set similarity join is a fundamental and well-studied database operator. It is usually studied in the exact setting where the goal is to compute all pairs of sets that exceed a given similarity threshold (measured e.g. as Jaccard similarity). But set similarity join is often used in settings where 100% recall may not be important — indeed, where the exact set similarity join is itself only an approximation of the desired result set. We present a new randomized algorithm for set similarity join that can achieve any desired recall up to 100%, and show theoretically and empirically that it significantly improves on existing methods. The present state-of-the-art exact methods are based on prefix-filtering, the performance of which depends on the data set having many rare tokens. Our method is robust against the absence of such structure in the data. At 90% recall our algorithm is often more than an order of magnitude faster than state-of-the-art exact methods, depending on how well a data set lends itself to prefix filtering. Our experiments on benchmark data sets also show that the method is several times faster than comparable approximate methods. Our algorithm makes use of recent theoretical advances in high-dimensional sketching and indexing that we believe to be of wider relevance to the data engineering community.

Scalable and Robust Set Similarity Join

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 that satisfies certain locality-sensitivity properties, we can construct a dynamic data structure that solves the approximate near neighbor problem in d-dimensional space with query time dnρq+o(1), update time dnρu+o(1), and space usage dn+n1+ρu+o(1) where n denotes the number of points in the data structure. The space-time tradeoff is tied to the tradeoff between query time and update time (insertions/deletions), controlled by the exponents ρq, ρu that are determined by the filter family.Locality-sensitive filtering was introduced by Becker et al. (SODA 2016) together with a framework yielding a single, balanced, tradeoff between query time and space, further relying on the assumption of an efficient oracle for the filter evaluation algorithm. We extend the LSF framework to support space-time tradeoffs and through a combination of existing techniques we remove the oracle assumption.Laarhoven (arXiv 2015), building on Becker et al., introduced a family of filters with space-time tradeoffs for the high-dimensional unit sphere under inner product similarity and analyzed it for the important special case of random data. We show that a small modification to the family of filters gives a simpler analysis that we use, together with our framework, to provide guarantees for worst-case data. Through an application of Bochner's Theorem from harmonic analysis by Rahimi & Recht (NIPS 2007), we are able to extend our solution on the unit sphere to ℝd under the class of similarity measures corresponding to real-valued characteristic functions. For the characteristic functions of s-stable distributions we obtain a solution to the (r, cr)-near neighbor problem in lds-spaces with query and update exponents [EQUATION] and [EQUATION] where λ ∈ [−1, 1] is a tradeoff parameter. This result improves upon the space-time tradeoff of Kapralov (PODS 2015) and is shown to be optimal in the case of a balanced tradeoff, matching the LSH lower bound by O'Donnell et al. (ITCS 2011) and a similar LSF lower bound proposed in this paper. Finally, we show a lower bound for the space-time tradeoff on the unit sphere that matches Laarhoven's and our own upper bound in the case of random data.

A framework for similarity search with space-time tradeoffs using locality-sensitive filtering

The generation of pseudorandom elements over finite fields is fundamental to the time, space and randomness complexity of randomized algorithms and data structures. We consider the problem of generating k-independent random values over a finite field F in a word RAM model equipped with constant time addition and multiplication in F, and present the first nontrivial construction of a generator that outputs each value in constant time, not dependent on k. Our generator has period length |F| poly log k and uses k poly (log k) log |F| bits of space, which is optimal up to a poly log k factor. We are able to bypass Siegel's lower bound on the time-space tradeoff for k-independent functions by a restriction to sequential evaluation.

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Generating k-Independent Variables in Constant Time

We consider the following fundamental problems: (1) Constructing $k$-independent hash functions with a space-time tradeoff close to Siegel's lower bound. (2) Constructing representations of unbalanced expander graphs having small size and allowing fast computation of the neighbor function. It is not hard to show that these problems are intimately connected in the sense that a good solution to one of them leads to a good solution to the other one. In this paper we exploit this connection to present efficient, recursive constructions of $k$-independent hash functions (and hence expanders with a small representation). While the previously most efficient construction (Thorup, FOCS 2013) needed time quasipolynomial in Siegel's lower bound, our time bound is just a logarithmic factor from the lower bound.

Tobias Christiani

Papers

Set similarity search beyond MinHash

Scalable and Robust Set Similarity Join

A framework for similarity search with space-time tradeoffs using locality-sensitive filtering

Generating k-Independent Variables in Constant Time

From Independence to Expansion and Back Again