Frequency Moments in Noisy Streaming and Distributed Data under Mismatch Ambiguity
This addresses challenges in statistical estimation for noisy data in streaming and distributed systems, with incremental improvements over noiseless settings.
The paper tackles the problem of approximating frequency moments (F_p) in noisy streaming and distributed data, showing that it requires polynomial space in the data stream model and generally impossible polylogarithmic communication in the coordinator model, but achieves sublinear space and communication under certain data-dependent conditions.
We propose a novel framework for statistical estimation on noisy datasets. Within this framework, we focus on the frequency moments ($F_p$) problem and demonstrate that it is possible to approximate $F_p$ of the unknown ground-truth dataset using sublinear space in the data stream model and sublinear communication in the coordinator model, provided that the approximation ratio is parameterized by a data-dependent quantity, which we call the $F_p$-mismatch-ambiguity. We also establish a set of lower bounds, which are tight in terms of the input size. Our results yield several interesting insights: (1) In the data stream model, the $F_p$ problem is inherently more difficult in the noisy setting than in the noiseless one. In particular, while $F_2$ can be approximated in logarithmic space in terms of the input size in the noiseless setting, any algorithm for $F_2$ in the noisy setting requires polynomial space. (2) In the coordinator model, in sharp contrast to the noiseless case, achieving polylogarithmic communication in the input size is generally impossible for $F_p$ under noise. However, when the $F_p$ mismatch ambiguity falls below a certain threshold, it becomes possible to achieve communication that is entirely independent of the input size.