The Huge Object model of property testing [Goldreich and Ron, TheoretiCS 23] concerns properties of distributions supported on $\{0,1\}^n$, where $n$ is so large that even reading a single sampled string is unrealistic. Instead, query access is provided to the samples, and the efficiency of the algorithm is measured by the total number of queries that were made to them. Index-invariant properties under this model were defined in [Chakraborty et al., COLT 23], as a compromise between enduring the full intricacies of string testing when considering unconstrained properties, and giving up completely on the string structure when considering label-invariant properties. Index-invariant properties are those that are invariant through a consistent reordering of the bits of the involved strings. Here we provide an adaptation of Szemer\'edi's regularity method for this setting, and in particular show that if an index-invariant property admits an $\epsilon$-test with a number of queries depending only on the proximity parameter $\epsilon$, then it also admits a distance estimation algorithm whose number of queries depends only on the approximation parameter.

翻译：Goldreich与Ron在TheoretiCS 23中提出的性质测试巨型对象模型关注定义在$\{0,1\}^n$上的分布性质，其中$n$极大，以至于读取单个采样字符串都不现实。该模型提供对样本的查询访问，算法效率通过对其进行的查询总数来衡量。Chakraborty等人在COLT 23中定义了该模型下的索引不变性质，作为两种极端情况的折中：既避免考虑无约束性质时承受字符串测试的全部复杂性，又避免考虑标签不变性质时完全放弃字符串结构。索引不变性质指那些在涉及字符串的比特位进行一致重排时保持不变的性质。本文为此场景提供了Szemerédi正则性方法的适配版本，特别证明了若某个索引不变性质可通过查询次数仅依赖于邻近参数$\epsilon$的$\epsilon$-测试进行检验，则该性质同样可通过查询次数仅依赖于近似参数的距离估计算法进行处理。