Vector quantization and signal compression
Vector quantization and signal compression
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Normalized Cuts and Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
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
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
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
Face Recognition Using Laplacianfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
On distance scales, embeddings, and efficient relaxations of the cut cone
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Advances in metric embedding theory
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Unsupervised Learning of Image Manifolds by Semidefinite Programming
International Journal of Computer Vision
Local embeddings of metric spaces
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Embedding metric spaces in their intrinsic dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On low dimensional local embeddings
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Random Projections of Smooth Manifolds
Foundations of Computational Mathematics
Perturbed identity matrices have high rank: Proof and applications
Combinatorics, Probability and Computing
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
A nonlinear approach to dimension reduction
Proceedings of the twenty-second 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
The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Dimension reduction of metric data has become a useful technique with numerous applications. The celebrated Johnson-Lindenstrauss lemma states that any n-point subset of Euclidean space can be embedded in O(ε−2 log n)-dimension with (1 + ε)-distortion. This bound is known to be nearly tight. In many applications the demand that all distances should be nearly preserved is too strong. In this paper we show that indeed under natural relaxations of the goal of the embedding, an improved dimension reduction is possible where the target dimension is independent of n. Our main result can be viewed as a local dimension reduction. There are a variety of empirical situations in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widely-used heuristics such as Isomap and Locally Linear Embedding. Pursuing a line of work begun by Whitney, Nash showed that every C1 manifold of dimension d can be embedded in R2d+2 in such a manner that the local structure at each point is preserved isometrically. Our work is an analog of Nash's for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute near-isometric embedding of neighborhoods of bounded cardinality. We show that any finite subset of Euclidean space can be embedded in O(ε−2 log k)-dimension while preserving with (1 + ε)-distortion the distances within a "core neighborhood" of each point. (The core neighborhood is a metric ball around the point, whose radius is a substantial fraction of the radius of the ball of cardinality k, the k-neighborhood.) When the metric space satisfies a weak growth rate property, the guarantee applies to the entire k-neighborhood (with some dependency of the embedding dimension on the growth rate). We also show how to obtain a global embedding that also keeps distant points well-separated (at the cost of dependency on the doubling dimension of the space). As an application of our methods we obtain an (Assouad-style) dimension reduction for finite subsets of Euclidean space where the metric is raised to some fractional power (the resulting metrics are known as snowflakes). We show that any such metric X can be embedded in dimension Õ(ε−3 dim(X)) with 1 + ε distortion, where dim(X) is the doubling dimension, a measure of the intrinsic dimension of the set. This result improves recent work by Gottlieb and Krauthgamer [20] to a nearly tight bound. The new dimension reduction results are useful for applications such as clustering and distance labeling.