Testing of Index-Invariant Properties in the Huge Object Model

The study of distribution testing has become ubiquitous in the area of property testing, both for its theoretical appeal, as well as for its applications in other fields of Computer Science. The original distribution testing model relies on samples drawn independently from the distribution to be tested. However, when testing distributions over the $n$-dimensional Hamming cube $\left\{0,1\right\}^{n}$ for a large $n$, even reading a few samples is infeasible. To address this, Goldreich and Ron [ITCS 2022] have defined a model called the huge object model, in which the samples may only be queried in a few places. In this work, we initiate a study of a general class of properties in the huge object model, those that are invariant under a permutation of the indices of the vectors in $\left\{0,1\right\}^{n}$, while still not being necessarily fully symmetric as per the definition used in traditional distribution testing. We prove that every index-invariant property satisfying a bounded VC-dimension restriction admits a property tester with a number of queries independent of n. To complement this result, we argue that satisfying only index-invariance or only a VC-dimension bound is insufficient to guarantee a tester whose query complexity is independent of n. Moreover, we prove that the dependency of sample and query complexities of our tester on the VC-dimension is tight. As a second part of this work, we address the question of the number of queries required for non-adaptive testing. We show that it can be at most quadratic in the number of queries required for an adaptive tester of index-invariant properties. This is in contrast with the tight exponential gap for general non-index-invariant properties. Finally, we provide an index-invariant property for which the quadratic gap between adaptive and non-adaptive query complexities for testing is almost tight.