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.

翻译：本文开创性地系统研究了与单个指数级规模组合对象不可区分的显式分布。本研究扩展了Goldreich、Goldwasser与Nussboim（SICOMP 2010）的工作，该工作聚焦于实现与均匀分布不可区分且满足某些全局性质（他们称之为"真实性"）的巨物。与单个对象不可区分的概念源于学习理论中生成模型和图论中正则引理的研究。在伪随机性背景下已得到充分理解的问题，在考虑巨物生成模型时会面临重大挑战，甚至在某些情况下无法实现。我们通过为多种自然场景中的巨量不可区分对象提供学习算法，展现了该研究的广泛适用性，具体包括：具有一维函数中"1"的个数或图中边数真实性要求的稠密函数与图，以及满足某些全局性质的稀疏图弱正则引理的变体版本。这些成果及其他结论不仅推广了基本伪随机对象，还推广了算法公平性中引入的概念。本研究成果依赖于来自学习理论、复杂性理论、密码学和博弈论等多个领域的概念与技术。