This paper introduces the notions of non-interactive and interactive proofs of proximity for properties of distributions, and initiates their study. These notions were studied before in the context of properties of functions, but the two settings are fundamentally different, and this fact is farther demonstrated by the results of this paper. The results are very informative and intriguing, and include the fact that non-interactive proofs do not offer more than a quadratic saving in complexity (which is achievable in some cases), that interactive proofs can be exponentially stronger than that, but only if they use private coins. Indeed, the separation between the power of general interactive proofs (of proximity for distributions) and their public-coin variants is the most striking result in this paper.
Distribution testing is an area of property testing that studies algorithms that receive few samples from a probability distribution D and decide whether D has a certain property or is far (in total variation distance) from all distributions with that property. Most natural properties of distributions, however, require a large number of samples to test, which motivates the question of whether there are natural settings wherein fewer samples suffice.
We initiate a study of proofs of proximity for properties of distributions. In their basic form, these proof systems consist of a tester that not only has sample access to a distribution but also explicit access to a proof string that depends on the distribution. We refer to these as NP distribution testers, or MA distribution testers if the tester is a probabilistic algorithm. We also study the more general notion of IP distribution testers, in which the tester interacts with an all-powerful untrusted prover.
We investigate the power and limitations of proofs of proximity for distributions and chart a landscape that, surprisingly, is significantly different from that of proofs of proximity for functions. Our main results include showing that MA distribution testers can be quadratically stronger than standard distribution testers, but no stronger than that; in contrast, IP distribution testers can be exponentially stronger than standard distribution testers, but when restricted to public coins they can be at best quadratically stronger.
See ECCC TR17-155.