Stable Distributions, Pseudorandom Generators, Embeddings and Data Stream Computation

In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular: —We show that, for any p ∈ (0, 2], one can maintain (using only O(log n/ 2 ) words of storage) a sketch C(q) of a point q ∈ ln p under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate q −s p up to a factor of (1+ ) with large probability. This solves the main open problem of Feigenbaum et al. [1999]. —We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ n and y ∈ m , estimates Ax − y p in O(n + m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p = 2. —We obtain another sketch function C which probabilistically embeds ln 1 into a normed space lm 1 . The embedding guarantees that, if we ...

• Type: Conference
• Year: 2000
• Where: FOCS
• Authors: Piotr Indyk
• Keywords: Aforementioned Sketching Approach, Algorithms, Dimensionality Reduction, FOCS 2000, Theoretical Computer Science