References for the “Algorithmic High Dimensional Geometry” Lectures at the Big Data Boot Camp, Simons Institute, Berkeley
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The Multiplicative Weights Update Method: a Meta Algorithm and Applications
The Multiplicative Weights Update Method: a Meta Algorithm and Applications Sanjeev Arora∗ Elad Hazan Satyen Kale Abstract Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algo- rithm. 1 Introduction Algorithms in varied fields work as follows: a distribution is maintained on a certain set, and at each step the probability assigned to i is multi- plied or divided by (1 + C(i)) where C(i) is some kind of “payoff” for element i. (Rescaling may be needed to ensure that the new values form a distribution.) Some examples include: the Ada Boost algorithm in ma- chine learning [FS97]; algorithms for game playing studied in economics (see below), the Plotkin-Shmoys-Tardos algorithm for packing and covering LPs [PST91], and its improvements in the case of flow problems by Young, Garg-Konneman, and Fleischer [You95, GK98, Fle00]; Impagliazzo’s proof of the Yao XOR lemma [Imp95], etc. The analysis of the running time uses a potential function argument and the final running time is proportional to 1/2. It has been clear to most researchers that these results are very similar, see for instance, Khandekar’s PhD thesis [Kha04]. Here we point out that these are all instances of the same (more general) algorithm. This meta ∗This project supported by David and Lucile Packard Fellowship and NSF grant CCR- 0205594. -
Distance-Sensitive Hashing∗
Distance-Sensitive Hashing∗ Martin Aumüller Tobias Christiani IT University of Copenhagen IT University of Copenhagen [email protected] [email protected] Rasmus Pagh Francesco Silvestri IT University of Copenhagen University of Padova [email protected] [email protected] ABSTRACT tight up to lower order terms. In particular, we extend existing Locality-sensitive hashing (LSH) is an important tool for managing LSH lower bounds, showing that they also hold in the asymmetric high-dimensional noisy or uncertain data, for example in connec- setting. tion with data cleaning (similarity join) and noise-robust search (similarity search). However, for a number of problems the LSH CCS CONCEPTS framework is not known to yield good solutions, and instead ad hoc • Theory of computation → Randomness, geometry and dis- solutions have been designed for particular similarity and distance crete structures; Data structures design and analysis; • Informa- measures. For example, this is true for output-sensitive similarity tion systems → Information retrieval; search/join, and for indexes supporting annulus queries that aim to report a point close to a certain given distance from the query KEYWORDS point. locality-sensitive hashing; similarity search; annulus query In this paper we initiate the study of distance-sensitive hashing (DSH), a generalization of LSH that seeks a family of hash functions ACM Reference Format: Martin Aumüller, Tobias Christiani, Rasmus Pagh, and Francesco Silvestri. such that the probability of two points having the same hash value is 2018. Distance-Sensitive Hashing. In Proceedings of 2018 International Confer- a given function of the distance between them. More precisely, given ence on Management of Data (PODS’18). -
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Arxiv:2102.08942V1 [Cs.DB]
A Survey on Locality Sensitive Hashing Algorithms and their Applications OMID JAFARI, New Mexico State University, USA PREETI MAURYA, New Mexico State University, USA PARTH NAGARKAR, New Mexico State University, USA KHANDKER MUSHFIQUL ISLAM, New Mexico State University, USA CHIDAMBARAM CRUSHEV, New Mexico State University, USA Finding nearest neighbors in high-dimensional spaces is a fundamental operation in many diverse application domains. Locality Sensitive Hashing (LSH) is one of the most popular techniques for finding approximate nearest neighbor searches in high-dimensional spaces. The main benefits of LSH are its sub-linear query performance and theoretical guarantees on the query accuracy. In this survey paper, we provide a review of state-of-the-art LSH and Distributed LSH techniques. Most importantly, unlike any other prior survey, we present how Locality Sensitive Hashing is utilized in different application domains. CCS Concepts: • General and reference → Surveys and overviews. Additional Key Words and Phrases: Locality Sensitive Hashing, Approximate Nearest Neighbor Search, High-Dimensional Similarity Search, Indexing 1 INTRODUCTION Finding nearest neighbors in high-dimensional spaces is an important problem in several diverse applications, such as multimedia retrieval, machine learning, biological and geological sciences, etc. For low-dimensions (< 10), popular tree-based index structures, such as KD-tree [12], SR-tree [56], etc. are effective, but for higher number of dimensions, these index structures suffer from the well-known problem, curse of dimensionality (where the performance of these index structures is often out-performed even by linear scans) [21]. Instead of searching for exact results, one solution to address the curse of dimensionality problem is to look for approximate results. -
January 2011 Prizes and Awards
January 2011 Prizes and Awards 4:25 P.M., Friday, January 7, 2011 PROGRAM SUMMARY OF AWARDS OPENING REMARKS FOR AMS George E. Andrews, President BÔCHER MEMORIAL PRIZE: ASAF NAOR, GUNTHER UHLMANN American Mathematical Society FRANK NELSON COLE PRIZE IN NUMBER THEORY: CHANDRASHEKHAR KHARE AND DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY JEAN-PIERRE WINTENBERGER TEACHING OF MATHEMATICS LEVI L. CONANT PRIZE: DAVID VOGAN Mathematical Association of America JOSEPH L. DOOB PRIZE: PETER KRONHEIMER AND TOMASZ MROWKA EULER BOOK PRIZE LEONARD EISENBUD PRIZE FOR MATHEMATICS AND PHYSICS: HERBERT SPOHN Mathematical Association of America RUTH LYTTLE SATTER PRIZE IN MATHEMATICS: AMIE WILKINSON DAVID P. R OBBINS PRIZE LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT: JOHN WILLARD MILNOR Mathematical Association of America LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION: HENRYK IWANIEC BÔCHER MEMORIAL PRIZE LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH: INGRID DAUBECHIES American Mathematical Society FOR AMS-MAA-SIAM LEVI L. CONANT PRIZE American Mathematical Society FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT: MARIA MONKS LEONARD EISENBUD PRIZE FOR MATHEMATICS AND OR PHYSICS F AWM American Mathematical Society LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION: PATRICIA CAMPBELL RUTH LYTTLE SATTER PRIZE IN MATHEMATICS M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS: American Mathematical Society RHONDA HUGHES ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN: LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION SHERRY GONG Association for Women in Mathematics ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN FOR JPBM Association for Women in Mathematics COMMUNICATIONS AWARD: NICOLAS FALACCI AND CHERYL HEUTON M. -
SIGMOD Flyer
DATES: Research paper SIGMOD 2006 abstracts Nov. 15, 2005 Research papers, 25th ACM SIGMOD International Conference on demonstrations, Management of Data industrial talks, tutorials, panels Nov. 29, 2005 June 26- June 29, 2006 Author Notification Feb. 24, 2006 Chicago, USA The annual ACM SIGMOD conference is a leading international forum for database researchers, developers, and users to explore cutting-edge ideas and results, and to exchange techniques, tools, and experiences. We invite the submission of original research contributions as well as proposals for demonstrations, tutorials, industrial presentations, and panels. We encourage submissions relating to all aspects of data management defined broadly and particularly ORGANIZERS: encourage work that represent deep technical insights or present new abstractions and novel approaches to problems of significance. We especially welcome submissions that help identify and solve data management systems issues by General Chair leveraging knowledge of applications and related areas, such as information retrieval and search, operating systems & Clement Yu, U. of Illinois storage technologies, and web services. Areas of interest include but are not limited to: at Chicago • Benchmarking and performance evaluation Vice Gen. Chair • Data cleaning and integration Peter Scheuermann, Northwestern Univ. • Database monitoring and tuning PC Chair • Data privacy and security Surajit Chaudhuri, • Data warehousing and decision-support systems Microsoft Research • Embedded, sensor, mobile databases and applications Demo. Chair Anastassia Ailamaki, CMU • Managing uncertain and imprecise information Industrial PC Chair • Peer-to-peer data management Alon Halevy, U. of • Personalized information systems Washington, Seattle • Query processing and optimization Panels Chair Christian S. Jensen, • Replication, caching, and publish-subscribe systems Aalborg University • Text search and database querying Tutorials Chair • Semi-structured data David DeWitt, U. -
Lower Bounds on Lattice Sieving and Information Set Decoding
Lower bounds on lattice sieving and information set decoding Elena Kirshanova1 and Thijs Laarhoven2 1Immanuel Kant Baltic Federal University, Kaliningrad, Russia [email protected] 2Eindhoven University of Technology, Eindhoven, The Netherlands [email protected] April 22, 2021 Abstract In two of the main areas of post-quantum cryptography, based on lattices and codes, nearest neighbor techniques have been used to speed up state-of-the-art cryptanalytic algorithms, and to obtain the lowest asymptotic cost estimates to date [May{Ozerov, Eurocrypt'15; Becker{Ducas{Gama{Laarhoven, SODA'16]. These upper bounds are useful for assessing the security of cryptosystems against known attacks, but to guarantee long-term security one would like to have closely matching lower bounds, showing that improvements on the algorithmic side will not drastically reduce the security in the future. As existing lower bounds from the nearest neighbor literature do not apply to the nearest neighbor problems appearing in this context, one might wonder whether further speedups to these cryptanalytic algorithms can still be found by only improving the nearest neighbor subroutines. We derive new lower bounds on the costs of solving the nearest neighbor search problems appearing in these cryptanalytic settings. For the Euclidean metric we show that for random data sets on the sphere, the locality-sensitive filtering approach of [Becker{Ducas{Gama{Laarhoven, SODA 2016] using spherical caps is optimal, and hence within a broad class of lattice sieving algorithms covering almost all approaches to date, their asymptotic time complexity of 20:292d+o(d) is optimal. Similar conditional optimality results apply to lattice sieving variants, such as the 20:265d+o(d) complexity for quantum sieving [Laarhoven, PhD thesis 2016] and previously derived complexity estimates for tuple sieving [Herold{Kirshanova{Laarhoven, PKC 2018]. -
Constraint Clustering and Parity Games
Constrained Clustering Problems and Parity Games Clemens Rösner geboren in Ulm Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn Bonn 2019 1. Gutachter: Prof. Dr. Heiko Röglin 2. Gutachterin: Prof. Dr. Anne Driemel Tag der mündlichen Prüfung: 05. September 2019 Erscheinungsjahr: 2019 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn Abstract Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. We study several clustering objectives. We begin with studying the Euclidean k-center problem. The k-center problem is a classical combinatorial optimization problem which asks to select k centers and assign each input point in a set P to one of the centers, such that the maximum distance of any input point to its assigned center is minimized. The Euclidean k-center problem assumes that the input set P is a subset of a Euclidean space and that each location in the Euclidean space can be chosen as a center. We focus on the special case with k = 1, the smallest enclosing ball problem: given a set of points in m-dimensional Euclidean space, find the smallest sphere enclosing all the points. We combine known results about convex optimization with structural properties of the smallest enclosing ball to create a new algorithm. We show that on instances with rational coefficients our new algorithm computes the exact center of the optimal solutions and has a worst-case run time that is polynomial in the size of the input. -
Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality
THEORY OF COMPUTING, Volume 8 (2012), pp. 321–350 www.theoryofcomputing.org SPECIAL ISSUE IN HONOR OF RAJEEV MOTWANI Approximate Nearest Neighbor: Towards Removing the Curse of Dimensionality Sariel Har-Peled∗ Piotr Indyk † Rajeev Motwani ‡ Received: August 20, 2010; published: July 16, 2012. Abstract: We present two algorithms for the approximate nearest neighbor problem in d high-dimensional spaces. For data sets of size n living in R , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors’ STOC’98 and FOCS’01 papers. It unifies, generalizes and simplifies the results from those papers. ACM Classification: F.2.2 AMS Classification: 68W25 Key words and phrases: approximate nearest neighbor, high dimensions, locality-sensitive hashing 1 Introduction The nearest neighbor (NN) problem is defined as follows: Given a set P of n points in a metric space defined over a set X with distance function D, preprocess P to efficiently answer queries for finding the point in P closest to a query point q 2 X. A particularly interesting case is that of the d-dimensional d Euclidean space where X = R under some `s norm. This problem is of major importance in several ∗Supported by NSF CAREER award CCR-0132901 and AF award CCF-0915984. †Supported by a Stanford Graduate Fellowship, NSF Award CCR-9357849 and NSF CAREER award CCF-0133849. -
On Differentially Private Graph Sparsification and Applications
On Differentially Private Graph Sparsification and Applications Raman Arora Jalaj Upadhyay Johns Hopkins University Rutgers University [email protected] [email protected] Abstract In this paper, we study private sparsification of graphs. In particular, we give an algorithm that given an input graph, returns a sparse graph which approximates the spectrum of the input graph while ensuring differential privacy. This allows one to solve many graph problems privately yet efficiently and accurately. This is exemplified with application of the proposed meta-algorithm to graph algorithms for privately answering cut-queries, as well as practical algorithms for computing MAX-CUT and SPARSEST-CUT with better accuracy than previously known. We also give an efficient private algorithm to learn Laplacian eigenmap on a graph. 1 Introduction Data from social and communication networks have become a rich source to gain useful insights into the social, behavioral, and information sciences. Such data is naturally modeled as observations on a graph, and encodes rich, fine-grained, and structured information. At the same time, due to the seamless nature of data acquisition, often collected through personal devices, the individual information content in network data is often highly sensitive. This raises valid privacy concerns pertaining the analysis and release of such data. We address these concerns in this paper by presenting a novel algorithm that can be used to publish a succinct differentially private representation of network data with minimal degradation in accuracy for various graph related tasks. There are several notions of differential privacy one can consider in the setting described above. Depending on privacy requirements, one can consider edge level privacy that renders two graphs that differ in a single edge as in-distinguishable based on the algorithm’s output; this is the setting studied in many recent works [9, 19, 25, 54]. -
Proof Verification and Hardness of Approximation Problems
Proof Verification and Hardness of Approximation Problems Sanjeev Arora∗ Carsten Lundy Rajeev Motwaniz Madhu Sudanx Mario Szegedy{ Abstract that finding a solution within any constant factor of optimal is also NP-hard. Garey and Johnson The class PCP(f(n); g(n)) consists of all languages [GJ76] studied MAX-CLIQUE: the problem of find- L for which there exists a polynomial-time probabilistic ing the largest clique in a graph. They showed oracle machine that uses O(f(n)) random bits, queries that if a polynomial time algorithm computes MAX- O(g(n)) bits of its oracle and behaves as follows: If CLIQUE within a constant multiplicative factor then x 2 L then there exists an oracle y such that the ma- MAX-CLIQUE has a polynomial-time approximation chine accepts for all random choices but if x 62 L then scheme (PTAS) [GJ78], i.e., for any c > 1 there ex- for every oracle y the machine rejects with high proba- ists a polynomial-time algorithm that approximates bility. Arora and Safra very recently characterized NP MAX-CLIQUE within a factor of c. They used graph as PCP(log n; (log log n)O(1)). We improve on their products to construct \gap increasing reductions" that result by showing that NP = PCP(log n; 1). Our re- mapped an instance of the clique problem into an- sult has the following consequences: other instance in order to enlarge the relative gap of 1. MAXSNP-hard problems (e.g., metric TSP, the clique sizes. Berman and Schnitger [BS92] used MAX-SAT, MAX-CUT) do not have polynomial the ideas of increasing gaps to show that if the clique time approximation schemes unless P=NP. -
Martin Dyer, Mark Jerrum, Marek Karpinski (Editors): Design And
Martin Dyer, Mark Jerrum, Marek Karpinski (editors): Design and Analysis of Randomized and Approximation Algorithms Dagstuhl-Seminar-Report; 309 03.06. - 08.06.2001 (01231) 1 OVERVIEW The Workshop was concerned with the newest development in the design and analysis of randomized and approximation algorithms. The main focus of the workshop was on two specific topics: ap- proximation algorithms for optimization problems, and approximation algorithms for measurement problems, and the various interactions between them. Here, new important paradigms have been discovered recently connecting probabilistic proof verification theory to the theory of approximate computation. Also, some new broadly applicable techniques have emerged recently for designing efficient approximation algorithms for a number of computationally hard optimization and measure- ment problems. This workshop has addressed the above topics and also fundamental insights into the new paradigms and design techniques. The workshop was organized jointly with the RAND- APX meeting on approximation algorithms and intractability and was partially supported by the IST grant 14036 (RAND-APX). The 47 participants of the workshop came from ten countries, thirteen of them came from North America. The 33 lectures delivered at this workshop covered a wide body of research in the above areas. The Program of the meeting and Abstracts of all talks are listed in the subsequent sections of this report. The meeting was hold in a very informal and stimulating atmosphere. Thanks to everybody who contributed to the success of this meeting and made it a very enjoyable event! Martin Dyer Mark Jerrum Marek Karpinski Acknowlegement. We thank Annette Beyer, Christine Marikar, and Angelika Mueller for their continuous support and help in organizing this workshop. -
Quantum Proof Systems for Iterated Exponential Time, and Beyond∗
Quantum Proof Systems for Iterated Exponential Time, and Beyond∗ Joseph Fitzsimons Zhengfeng Ji Horizon Quantum Computing University of Technology Sydney Singapore Australia Thomas Vidick Henry Yuen California Institute of Technology University of Toronto USA Canada ABSTRACT ACM Reference Format: We show that any language solvable in nondeterministic time Joseph Fitzsimons, Zhengfeng Ji, Thomas Vidick, and Henry Yuen. 2019. Quantum Proof Systems for Iterated Exponential Time, and Beyond. In exp¹exp¹· · · exp¹nººº, where the number of iterated exponentials Proceedings of the 51st Annual ACM SIGACT Symposium on the Theory of R¹nº is an arbitrary function , can be decided by a multiprover inter- Computing (STOC ’19), June 23–26, 2019, Phoenix, AZ, USA. ACM, New York, active proof system with a classical polynomial-time verifier and a NY, USA,8 pages. https://doi.org/10.1145/3313276.3316343 constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−C exp¹· · · exp¹nººº, where the number of 1 INTRODUCTION iterated exponentials is R¹nº − 1 and C > 0 is a universal constant. The result was previously known for R = 1 and R = 2; we obtain it The combined study of interactive proof systems and quantum for any time-constructible function R. entanglement has led to multiple discoveries at the intersection The result is based on a compression technique for interactive of theoretical computer science and quantum physics. On the one proof systems with entangled provers that significantly simplifies hand, the study has revealed that quantum entanglement, a fun- and strengthens a protocol compression result of Ji (STOC’17).