Metric Embeddings with Relaxed Guarantees Export

@article{citeulike:3081677, abstract = {We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [29], we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into \ell 1 which exhibit gracefully degrading distortion: these is a single embedding into \ell 1 that achieves distortion at most O(\log \frac{1}{\varepsilon }) on all but at most an \varepsilon fraction of distances, simultaneously for all \varepsilon \> 0. We extend this with distortion O(\log \frac{1}{\varepsilon }) to maps into general \ell p, p \ge 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for \varepsilon-slack embeddings from lower bounds for low-distortion embeddings.}, address = {Los Alamitos, CA, USA}, author = {Abraham, Ittai and Bartal, Yair and Chan, Hubert T. H. and Dhamdhere, Kedar D. and Gupta, Anupam and Kleinberg, Jon and Neiman, Ofer and Slivkins, Aleksandrs}, citeulike-article-id = {3081677}, citeulike-linkout-0 = {http://doi.ieeecomputersociety.org/10.1109/SFCS.2005.51}, citeulike-linkout-1 = {http://dx.doi.org/10.1109/SFCS.2005.51}, doi = {10.1109/SFCS.2005.51}, journal = {focs}, keywords = {embeddings, metrics, slack}, pages = {83--100}, posted-at = {2008-08-04 16:12:26}, priority = {2}, publisher = {IEEE Computer Society}, title = {Metric Embeddings with Relaxed Guarantees}, url = {http://dx.doi.org/10.1109/SFCS.2005.51}, volume = {00}, year = {2005} }

TY - JOUR ID - citeulike:3081677 L3 - citeulike-article-id:3081677 N2 - We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [29], we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into \ell 1 which exhibit gracefully degrading distortion: these is a single embedding into \ell 1 that achieves distortion at most O(\log \frac{1}{\varepsilon }) on all but at most an \varepsilon fraction of distances, simultaneously for all \varepsilon > 0. We extend this with distortion O(\log \frac{1}{\varepsilon }) to maps into general \ell p, p \ge 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for \varepsilon-slack embeddings from lower bounds for low-distortion embeddings. JF - focs AD - Los Alamitos, CA, USA EP - 100 TI - Metric Embeddings with Relaxed Guarantees PB - IEEE Computer Society VL - 00 SP - 83 KW - embeddings KW - metrics KW - slack AU - Abraham, Ittai AU - Bartal, Yair AU - Chan, Hubert AU - Dhamdhere, Kedar AU - Gupta, Anupam AU - Kleinberg, Jon AU - Neiman, Ofer AU - Slivkins, Aleksandrs PY - 2005/// UR - http://dx.doi.org/10.1109/SFCS.2005.51 ER -