# Generative Models of Huge Objects

## TL;DR

This paper systematically studies explicit distributions that are indistinguishable from a single large combinatorial object, extending previous work on uniform indistinguishability and exploring applications in learning, graph theory, and fairness.

## Contribution

It introduces a framework for generative models of huge objects indistinguishable from a single object, with algorithms for dense and sparse structures under global constraints.

## Key findings

- Developed a learning algorithm for indistinguishable objects in various settings.
- Extended the weak regularity lemma to sparse graphs with global properties.
- Generalized pseudorandom objects and notions in algorithmic fairness.

## Abstract

This work initiates the systematic study of explicit distributions that are indistinguishable from a single exponential-size combinatorial object. In this we extend the work of Goldreich, Goldwasser and Nussboim (SICOMP 2010) that focused on the implementation of huge objects that are indistinguishable from the uniform distribution, satisfying some global properties (which they coined truthfulness). Indistinguishability from a single object is motivated by the study of generative models in learning theory and regularity lemmas in graph theory. Problems that are well understood in the setting of pseudorandomness present significant challenges and at times are impossible when considering generative models of huge objects. We demonstrate the versatility of this study by providing a learning algorithm for huge indistinguishable objects in several natural settings including: dense functions and graphs with a truthfulness requirement on the number of ones in the function or edges in the graphs, and a version of the weak regularity lemma for sparse graphs that satisfy some global properties. These and other results generalize basic pseudorandom objects as well as notions introduced in algorithmic fairness. The results rely on notions and techniques from a variety of areas including learning theory, complexity theory, cryptography, and game theory.