Testing Distributions of Huge Objects

TL;DR

This paper introduces a hybrid property testing model for distributions over large objects like long strings, analyzing query complexity and relating it to standard distribution testing, with focus on natural and classical properties.

Contribution

It develops a new model for testing properties of distributions over huge objects, connecting it to existing models and analyzing complexity for key properties.

Findings

Query complexity bounds established for the new model.

Relation between new model's complexity and standard distribution testing.

Complexity results for uniform and identical distribution testing.

Abstract

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings. Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings). Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings) as well as for the distance between objects (resp., strings), and the distance between distributions is defined as the earth mover's distance with respect to the relative Hamming distance between strings. We study the query complexity of testing in this new model, focusing on three directions. First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution testing model. Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings). Third, we consider the complexity of testing properties that were extensively studied in the standard distribution testing model: Two such cases are uniform distributions and pairs of identical distributions.