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DIMACS Series in Discrete Mathematics and Theoretical Computer Science DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 65 The Random Projection Method Santosh S. Vempala American Mathematical Society The Random Projection Method https://doi.org/10.1090/dimacs/065 DIMACS Series in Discrete Mathematics and Theoretical Computer Science Volume 65 The Random Projection Method Santosh S. Vempala Center for Discrete Mathematics and Theoretical Computer Science A consortium of Rutgers University, Princeton University, AT&T Labs–Research, Bell Labs (Lucent Technologies), NEC Laboratories America, and Telcordia Technologies (with partners at Avaya Labs, IBM Research, and Microsoft Research) American Mathematical Society 2000 Mathematics Subject Classification. Primary 68Q25, 68W20, 90C27, 68Q32, 68P20. For additional information and updates on this book, visit www.ams.org/bookpages/dimacs-65 Library of Congress Cataloging-in-Publication Data Vempala, Santosh S. (Santosh Srinivas), 1971– The random projection method/Santosh S. Vempala. p.cm. – (DIMACS series in discrete mathematics and theoretical computer science, ISSN 1052- 1798; v. 65) Includes bibliographical references. ISBN 0-8218-2018-4 (alk. paper) 1. Random projection method. 2. Algorithms. I. Title. II. Series. QA501 .V45 2004 518.1–dc22 2004046181 0-8218-3793-1 (softcover) Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2004 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 090807060504 Contents Foreword (by Christos H. Papadimitriou) vii Acknowledgements ix Chapter 1. Random Projection 1 1.1. How to project 1 1.2. Basic properties of random projection 2 1.3. When to project 4 Part 1. Combinatorial Optimization 5 Chapter 2. Rounding via Random Projection 7 2.1. Maximum cut 7 2.2. Maximum k-cut 9 2.3. Coloring 11 Chapter 3. Embedding Metrics in Euclidean Space 15 3.1. Bourgain’s embedding 16 3.2. Finding a minimum distortion embedding 19 3.3. The max-flow min-cut gap for multicommodity flow 20 Chapter 4. Euclidean Embeddings: Beyond Distance Preservation 27 4.1. Metric volume and geometric volume 27 4.2. Preserving point-subset distances. 34 4.3. Ordering and layout problems 40 4.4. The Embed-and-Project rounding algorithm 43 Part 2. Learning Theory 49 Chapter 5. Robust Concepts 51 5.1. A model of cognitive learning 52 5.2. Neuron-friendly random projection 52 5.3. Robust half-spaces 57 Chapter 6. Intersections of Half-Spaces 61 6.1. The complexity of an old problem 61 6.2. A randomized algorithm 63 6.3. Its analysis 67 Part 3. Information Retrieval 75 Chapter 7. Nearest Neighbors 77 v vi CONTENTS 7.1. Euclidean space 77 7.2. Random projection for the hypercube 79 7.3. Back to Euclidean space 83 Chapter 8. Indexing and Clustering 87 8.1. Fast low-rank approximation 87 8.2. Geometric p-median 92 Bibliography 97 Appendix 101 Foreword (by Christos H. Papadimitriou) To project is to throw forward, what a sower does to seeds—an image of delib- erate randomness. The Latin root of the word survives in Caesar’s famous procla- mation on the randomness inherent in life and history, and the decisiveness and skill with which it must be met: “Alea jacta est.” Projection is, of course, a plunge onto a lower-dimensional domain (canvas, retina, screen, projective plane). But it need not be an uncomfortable dimensional compromise; a projection can reveal, enhance, evince, illuminate—as long as it is the right projection. This book is about the radical idea that even a random projection is often useful. Two decades ago, analysts stumbled upon a surprising fact of this sort, the Johnson-Lindenstrauss Lemma, as a crucial tool in their project of extending functions in continuous ways. This result (several variants of which are proved in this book, the first in page 2) says that, if you project n points in some high- dimensional space down to a random O(log n)-dimensional plane, the chances are overwhelming that all distances will be preserved—within a small relative error. So, if distance is all you care about, there is no reason to stay in high dimensions! Over the past ten years, random projection (and its extensions and variants) has been used increasingly often in Computer Science. For example (to mention one application of the technique, due to Luca Trevisan, that is not considered here), the traveling salesman problem in log n dimensions is as hard to approximate as the same problem in any higher dimensional space—that is to say, very hard. Random projection and high-dimensional geometry lie at the heart of several important approximation algorithms for problems of a nongeometric nature. For example, the best known algorithm for approximating the maximum cut of a graph works by first embedding the graph in a high-dimensional space (the “best” such embedding can be found by semidefinite programming). A random hyperplane through the origin will then cut a near-maximum number of edges (cutting by a random hyperplane is, of course, tantamount to projecting on a random line, in a sense the ultimate random projection). There are several other important graph theoretic problems, including multicommodity flow, that are best approximated through a geometric consideration (though not always by random projection), and these constitute the subject of the first part of the book. The second part of the book is about learning—and its geometry. How is it, asks Vempala, that we can learn complex concepts by focusing on the right aspects of the world, ignoring thousands of irrelevant attributes? He speculates that the key to the mystery lies in geometry. What is shown here is that, if concepts are geometrically well separated, or if the natural ways to separate them use criteria (hyperplanes) that are highly interdependent algebraically, then certain natural algorithms converge quickly to the right answer. (It would be very intriguing to verify that the genres of concepts in the learning of which we excel have indeed such vii viii FOREWORD (BY CHRISTOS H. PAPADIMITRIOU) positive properties.) These arguments involve, among other interesting geometric and algebraic techniques, a simpler variant of random projection that is expressly plausible in the context of learning and networks of neurons. The last part deals with what the author calls “information retrieval”: The nearest neighbor problem, for which random projection works so well, and his own geometric take on clustering and principal components analysis (using yet another variant of random projection that is best adapted to the current application). This is an elegant monograph, dense in ideas and techniques, diverse in its applications, and above all vibrant with the author’s enthusiasm for the area. I believe that randomized projection and related geometric techniques will continue to have more and more applications in Computer Science—and that this book, by making the area accessible to graduate students and researchers (sowing the seeds, so to speak), will contribute importantly in this regard. Acknowledgements The idea for this book originated from the DIMACS special year on computa- tional intractability (1999-2000). I would like to thank the special year committee and especially the chair, Mike Saks, for his support. Mike provided me with three referees: Avrim Blum, Jon Kleinberg and Satish Rao. I thank them for their comments and suggestions. Special thanks to Christos Papadimitriou for a thoughtful foreword and his infectious enthusiasm. Many students and colleagues read earlier drafts (or parts of them) and gave me useful feedback. I am grateful to Augustine Bompadre, John Dunagan, Uri Feige, Adam Kalai, Piyush Kumar, Mark Mercer, Alantha Newman, Yuval Rabani, Luis Rademacher, Sebastian Roch, Mike Saks, Adrian Vetta, Grant Wang and Kevin Zatloukal. I thank the staff at AMS, in particular, Shirley Hill and Tom Costa for help with preparing the manuscript. I also acknowledge the NSF and the Sloan foundation. I am grateful to my parents for their encouragement (and periodically inquiring about the status of the book). Most of all, I thank Rosa, my life, in whose projection Iflourish. ix Bibliography [1] D. Achlioptas, Database friendly random projections, Proc. Principles of Database Systems (PODS), 274–281, 2001. [2] D. Achlioptas and F. McSherry, Fast Computation of Low Rank Approximations,Proc.of STOC, 611–618, 2001. [3] S. Agmon, The relaxation method for linear inequalities, Canadian Journal of Mathematics 6(3), 382–392, 1954. [4] F. Alizadeh, Interior point methods in semidefinite programming with applications to combi- natorial optimization, Siam J. Optimization 5(1), 13–51, 1995. [5] N. Alon, N. Kahale, Approximating the independence number via the ϑ–function, Math. Pro- gramming 80, 253–264, 1998. [6] N. Alon and J. Spencer, The Probabilistic Method, Wiley, 2000. [7]S.Arora,S.RaoandU.Vazirani,Expander flows, geometric embeddings and graph partition- ing, Proc. STOC, 2004. [8]R.I.ArriagaandS.Vempala,An algorithmic theory of learning: Robust concepts and random projection, Proc.
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