The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
The space complexity of approximating the frequency moments
Journal of Computer and System Sciences
Dimensionality reduction techniques for proximity problems
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Database-friendly random projections
PODS '01 Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Learning mixtures of arbitrary gaussians
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
A lower bound on the distortion of embedding planar metrics into Euclidean space
Proceedings of the eighteenth annual symposium on Computational geometry
Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces
SIAM Journal on Computing
Dimension Reduction in the \ell _1 Norm
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Learning Mixtures of Gaussians
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
An Algorithmic Theory of Learning: Robust Concepts and Random Projection
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Polynomial time approximation schemes for geometric k-clustering
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Stable distributions, pseudorandom generators, embeddings and data stream computation
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Vertex cuts, random walks, and dimension reduction in series-parallel graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Very sparse stable random projections for dimension reduction in lα (0
Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining
Estimators and tail bounds for dimension reduction in lα (0
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Ordinal Embedding: Approximation Algorithms and Dimensionality Reduction
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
Maximum Gradient Embeddings and Monotone Clustering
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Learning and inferencing in user ontology for personalized Semantic Web search
Information Sciences: an International Journal
OntoSearch: a full-text search engine for the semantic web
AAAI'06 proceedings of the 21st national conference on Artificial intelligence - Volume 2
Ultra-low-dimensional embeddings for doubling metrics
Journal of the ACM (JACM)
Nonlinear estimators and tail bounds for dimension reduction in l1 using Cauchy random projections
COLT'07 Proceedings of the 20th annual conference on Learning theory
Subspace embeddings for the L1-norm with applications
Proceedings of the forty-third annual ACM symposium on Theory of computing
Low distortion metric embedding into constant dimension
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Limitations on quantum dimensionality reduction
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Indexing the earth mover's distance using normal distributions
Proceedings of the VLDB Endowment
Dimension reduction for finite trees in l1
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A nonlinear approach to dimension reduction
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Efficient point-to-subspace query in ℓ1 with application to robust face recognition
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part IV
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The Johnson--Lindenstrauss lemma shows that any n points in Euclidean space (i.e., ℝn with distances measured under the ℓ2 norm) may be mapped down to O((log n)/ε2) dimensions such that no pairwise distance is distorted by more than a (1 + ε) factor. Determining whether such dimension reduction is possible in ℓ1 has been an intriguing open question. We show strong lower bounds for general dimension reduction in ℓ1. We give an explicit family of n points in ℓ1 such that any embedding with constant distortion D requires nΩ(1/D2) dimensions. This proves that there is no analog of the Johnson--Lindenstrauss lemma for ℓ1; in fact, embedding with any constant distortion requires nΩ(1) dimensions. Further, embedding the points into ℓ1 with (1+ε) distortion requires n½−O(ε log(1/ε)) dimensions. Our proof establishes this lower bound for shortest path metrics of series-parallel graphs. We make extensive use of linear programming and duality in devising our bounds. We expect that the tools and techniques we develop will be useful for future investigations of embeddings into ℓ1.