Complexity of network synchronization
Journal of the ACM (JACM)
The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
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
Database-friendly random projections
PODS '01 Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Algorithmic derandomization via complexity theory
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Derandomized dimensionality reduction with applications
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Lectures on Discrete Geometry
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
The intrinsic dimensionality of graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
A tight bound on approximating arbitrary metrics by tree metrics
Journal of Computer and System Sciences - Special issue: STOC 2003
Approximation Algorithms for the 0-Extension Problem
SIAM Journal on Computing
Low-distortion embeddings of general metrics into the line
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the impossibility of dimension reduction in l1
Journal of the ACM (JACM)
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Advances in metric embedding theory
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Embedding ultrametrics into low-dimensional spaces
Proceedings of the twenty-second annual symposium on Computational geometry
Cover trees for nearest neighbor
ICML '06 Proceedings of the 23rd international conference on Machine learning
Optimal-stretch name-independent compact routing in doubling metrics
Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing
Nearest-neighbor-preserving embeddings
ACM Transactions on Algorithms (TALG)
Optimal scale-free compact routing schemes in networks of low doubling dimension
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Embedding metric spaces in their intrinsic dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The geometry of graphs and some of its algorithmic applications
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Metric Embeddings with Relaxed Guarantees
SIAM Journal on Computing
Geometry of Cuts and Metrics
A nonlinear approach to dimension reduction
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
| Hi-index | 0.00 |
We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O(log n)-dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric on n points requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, do doubling metrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion? In this article, we give a positive answer to this question. We show how to embed any doubling metrics into O(log log n) dimensions with O(log n) distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion. This result is one extreme point of our general trade-off between distortion and dimension: given an n-point metric (V,d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we show that the metric embeds into Euclidean space RT with O(log n &sqrt; dimD/T) distortion.