Elements of information theory
Elements of information theory
Communication complexity
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On data structures and asymmetric communication complexity
Journal of Computer and System Sciences
The space complexity of approximating the frequency moments
Journal of Computer and System Sciences
On randomized one-round communication complexity
Computational Complexity
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
Finding Frequent Items in Data Streams
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Counting Distinct Elements in a Data Stream
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
An Algorithmic Theory of Learning: Robust Concepts and Random Projection
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
Information Theory Methods in Communication Complexity
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Tight Lower Bounds for the Distinct Elements Problem
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Optimal space lower bounds for all frequency moments
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Tabulation based 4-universal hashing with applications to second moment estimation
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Stable distributions, pseudorandom generators, embeddings, and data stream computation
Journal of the ACM (JACM)
Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The Communication Complexity of Correlation
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Tighter bounds for random projections of manifolds
Proceedings of the twenty-fourth annual symposium on Computational geometry
On variants of the Johnson–Lindenstrauss lemma
Random Structures & Algorithms
Dense Fast Random Projections and Lean Walsh Transforms
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Numerical linear algebra in the streaming model
Proceedings of the forty-first annual ACM symposium on Theory of computing
Communications of the ACM
The Fast Johnson-Lindenstrauss Transform and Approximate Nearest Neighbors
SIAM Journal on Computing
Fast Dimension Reduction Using Rademacher Series on Dual BCH Codes
Discrete & Computational Geometry
A Randomized Algorithm for Principal Component Analysis
SIAM Journal on Matrix Analysis and Applications
How to compress interactive communication
Proceedings of the forty-second ACM symposium on Theory of computing
A sparse Johnson: Lindenstrauss transform
Proceedings of the forty-second ACM symposium on Theory of computing
An optimal algorithm for the distinct elements problem
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Information theory for data management
Proceedings of the 2010 ACM SIGMOD International Conference on Management of data
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Lower bounds for sparse recovery
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Faster least squares approximation
Numerische Mathematik
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Fast moment estimation in data streams in optimal space
Proceedings of the forty-third annual ACM symposium on Theory of computing
Almost optimal explicit Johnson-Lindenstrauss families
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Streaming algorithms for recognizing nearly well-parenthesized expressions
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Sparser Johnson-Lindenstrauss transforms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Efficient and reliable low-power backscatter networks
Proceedings of the ACM SIGCOMM 2012 conference on Applications, technologies, architectures, and protocols for computer communication
Efficient and reliable low-power backscatter networks
ACM SIGCOMM Computer Communication Review - Special october issue SIGCOMM '12
An Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
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The Johnson-Lindenstrauss transform is a dimensionality reduction technique with a wide range of applications to theoretical computer science. It is specified by a distribution over projection matrices from Rn → Rk where k ≪ d and states that k = O(ε−2 log 1/δ) dimensions suffice to approximate the norm of any fixed vector in Rd to within a factor of 1 ± ε with probability at least 1 − δ. In this paper we show that this bound on k is optimal up to a constant factor, improving upon a previous Ω(ε−2 log 1/δ)/log(1/ε) dimension bound of Alon. Our techniques are based on lower bounding the information cost of a novel one-way communication game and yield the first space lower bounds in a data stream model that depend on the error probability δ. For many streaming problems, the most naïve way of achieving error probability δ is to first achieve constant probability, then take the median of O(log 1/δ) independent repetitions. Our techniques show that for a wide range of problems this is in fact optimal! As an example, we show that estimating the lp-distance for any p ∈ [0, 2] requires Ω(ε−2 log n log 1/δ) space, even for vectors in {0, 1}n. This is optimal in all parameters and closes a long line of work on this problem. We also show the number of distinct elements requires Ω(ε−2 log 1/δ + log n) space, which is optimal if ε−2 = Ω(log n). We also improve previous lower bounds for entropy in the strict turnstile and general turnstile models by a multiplicative factor of Ω(log 1/δ). Finally, we give an application to one-way communication complexity under product distributions, showing that unlike in the case of constant δ, the VC-dimension does not characterize the complexity when δ = o(1).