The Johnson-Lindenstrauss Lemma and the sphericity of some graphs
Journal of Combinatorial Theory Series A
Approximating the bandwidth via volume respecting embeddings (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
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
Random Projection: A New Approach to VLSI Layout
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Embeddings of surfaces, curves, and moving points in euclidean space
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Dimensionality Reductions in ℓ2 that Preserve Volumes and Distance to Affine Spaces
Discrete & Computational Geometry
Nearest-neighbor-preserving embeddings
ACM Transactions on Algorithms (TALG)
Fast dimension reduction using Rademacher series on dual BCH codes
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Tighter bounds for random projections of manifolds
Proceedings of the twenty-fourth annual symposium on Computational geometry
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
On multiplicative λ-approximations and some geometric applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Discovering diverse and salient threads in document collections
EMNLP-CoNLL '12 Proceedings of the 2012 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning
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
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Let Pbe a set of npoints in Euclidean space and let 0 茂戮驴Ponto a subspace of dimension $\mathcal{O}(\epsilon^{-2} \log n)$ such that distances change by at most a factor of 1 + 茂戮驴. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs but all subsets of at most kpoints maintain their volume approximately. More precisely, we require that sets of size s≤ kpreserve their volumes within a factor of (1 + 茂戮驴)s茂戮驴 1. We show that this can be achieved using $\mathcal{O}(\max\{\frac{k}{\epsilon},\epsilon^{-2}\log n\})$ dimensions. This in particular means that for $k = \mathcal{O}(\log n/\epsilon)$ we require no more dimensions (asymptotically) than the special case k= 2, handled by Johnson and Lindenstrauss. Our work improves on a result of Magen (that required as many as $\mathcal{O}(k\epsilon^{-2}\log n)$ dimensions) and is tight up to a factor of $\mathcal{O}(1/\epsilon)$. Another outcome of our work is an alternative and greatly simplified proof of the result of Magen showing that all distances between points and affine subspaces spanned by a small number of points are approximately preserved when projecting onto $\mathcal{O}(k\epsilon^{-2}\log n)$ dimensions.