High-Dimensional Geometric Streaming in Polynomial Space
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Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once d≥log n, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and Löwner-John ellipsoids of n points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to poly(d,log n) bits of space by trading off with a poly(d,log n) factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for ℓ_∞ subspace embeddings with poly(d,log n) space and poly(d,log n) distortion. Our algorithm also gives similar guarantees in the online coreset model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a log n dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For ℓ_p subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using O(d^2log n) space and O((dlog n)^1/2-1/p) distortion for p>2, whereas previous deterministic algorithms incurred a poly(n) factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
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