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We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic factor in the aspect ratio of the polytope. Our algorithms are nearly optimal in two senses: first, their runtimes nearly match those of the most efficient known algorithms for the offline version of the problem. Second, their approximation factors nearly match a lower bound we show against a natural class of geometric streaming algorithms. In contrast to existing works in the streaming setting that compute ellipsoidal roundings only for centrally symmetric convex polytopes, our algorithms apply to general convex polytopes. We also show how to use our algorithms to construct coresets from a stream of points that approximately preserve both the ellipsoidal rounding and the convex hull of the original set of points.
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Streaming Algorithms for Ellipsoidal Approximation of Convex Polytopes
We give efficient deterministic one-pass streaming algorithms for finding an ellipsoidal approximation of a symmetric convex polytope. The algorithms are near-optimal in that their approximation factors differ from that of the optimal offline solution only by a factor sub-logarithmic in the aspect ratio of the polytope
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- PAR ID:
- 10426828
- Editor(s):
- Po-Ling Loh and Maxim Raginsky
- Date Published:
- Journal Name:
- Proceedings of Thirty Fifth Conference on Learning Theory, PMLR
- Volume:
- 178
- Page Range / eLocation ID:
- 3070-3093
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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