Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications
- 781 Downloads
Let X be a subset of n points of the Euclidean space, and let 0 < ɛ < 1. A classical result of Johnson and Lindenstrauss [JL84] states that there is a projection of X onto a subspace of dimension O(ɛ-2 log n), with distortion ≤ 1 + ɛ. Here we show a natural extension of the above result, to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared to [JL84], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show it is possible to embed a subset of size n of the Euclidean space into a O(ɛ-2 klogn)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1+ɛ)s-1. Moreover, distances of points from affine hulls of sets of at most k-1 points in the space do not change by more than a factor of 1 + ɛ. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL84]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbor Search. In particular, it shows a first poly-logarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [Fei00], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.
KeywordsEuclidean Space Dimensionality Reduction Query Point Random Projection Dimensional Euclidean Space
Unable to display preview. Download preview PDF.
- [Ach01]Dimitris Achlioptas. Database-friendly random projections. In roceedings of PODS 01, pages 274–281, 2001.Google Scholar
- [AV99]R.I. Arriaga and S. Vempala. An algorithmic theory of learning: Robust concepts and random projections. In Proceeding of the 31th Symposium on the Theory of Computing, New York, 1999.Google Scholar
- [DG99]S. Dasgupta and A. Gupta. An elementary proof of the johnson-lindenstrauss lemma. In Technical Report TR-99-06, Computer Scinece Institute, Berkeley, CA, 1999.Google Scholar
- [EIO02]L. Engebretsen, P. Indyk, and R. O’Donnell. Derandomized dimensionality reduction with applications. In Proceedings of the 13th Symposium on Discrete Algorithms. IEEE, 2002.Google Scholar
- [HV02]S. Har-Peled and K. R. Varadarajan. Projective clustering in high dimensions using core-sets. In Proc. 18th Annu. ACM Sympos. Comput. Geom., pages 312–318, 2002.Google Scholar
- [IM98]P. Indyk and R. Motwani. Approximate nearest neighbor: towards removing the curse of dimensionality. In Proceedings of the Thirty Second Symposium on Theory of Computing, pages 604–613. ACM, 1998.Google Scholar
- [JL84]W. B. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), pages 189–206. Amer. Math. Soc., Providence, RI, 1984.Google Scholar
- [PC00]Agarwal P. and Procopiuc C. Approximation algortihms for strip cover in the plane. In Proceeding of the 11th ACM-SIAM Symposium on Discrete Algorithms, pages 373–382, 2000.Google Scholar
- [Rao99]S. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 300–306 (electronic), New York, 1999. ACM.Google Scholar
- [Vem98]S. Vempala. Approximating vlsi layout problems. In Proceedings of the thirty ninth Symposium on Foundations of Computer Science. IEEE, 1998.Google Scholar