Frequency Moments in Noisy Streaming and Distributed Data under Mismatch Ambiguity

2026-03-11

AI Summary

This work addresses the fundamental challenges of efficiently estimating frequency moments (Fₚ) of true data in noisy, distributed streaming environments. The authors propose a novel framework that introduces, for the first time, the concept of “Fₚ-mismatch ambiguity” to formally characterize the intrinsic impact of noise on frequency moment estimation. Leveraging this notion, they establish communication feasibility conditions independent of input size. Within the standard data stream and coordinator model, and by combining information-theoretic lower bounds with sublinear algorithm design, they prove that, in general, F₂ estimation requires polynomial space and cannot achieve polylogarithmic communication. However, when the mismatch ambiguity falls below a certain threshold, both sublinear space complexity and constant communication overhead become achievable.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

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.

Problem

• Frequency Moments
• Noisy Streaming
• Distributed Data
• Mismatch Ambiguity
• Statistical Estimation

Innovation

• frequency moments
• mismatch ambiguity
• noisy streaming
• sublinear space
• distributed estimation