A property of features is named location-invariant (or symmetric) if it may be characterised when it comes to the frequencies wherein every worth happens within the perform, whatever the places wherein every worth happens. It’s recognized that the (question) complexity of testing location-invariant properties of features is intently associated to the (pattern) complexity of testing the (corresponding properties of the) corresponding distributions. The principle message of the present work is that this shut relationship isn’t maintained within the context of verification. This holds each when contemplating verification by common interactive proofs of proximity (i.e., IPPs) and when proscribing consideration to doubly-sublinear IPPs (ds-IPPs). Alternatively, one could view this work as a subsequent step within the examine of doubly-sublinear IPPs (of properties of features), the place we are saying that an IPP is doubly-sublinear if (1) the question complexity of the verifier is sublinear within the question complexity of testing the property, and (2) the question complexity of the trustworthy prover is sublinear within the question complexity of studying a perform within the property. Particularly, we current doubly-sublinear IPPs for a number of pure location-invariant properties. Our outcomes embody: (1) We current doubly-sublinear IPPs for the set of features from [m] to [n] wherein every worth happens m/n occasions: For each α ∈ (0, 0.5), the question complexity of the verifier is O(n^{0.5−α}), and the question complexity of the trustworthy prover is e^{O(n^{0.5+α}/ε^2)}. (2) We current doubly-sublinear IPPs for the set of features from [m] to [n] wherein every worth happens both m/okay occasions or by no means: For each α ∈ (0, 1/3), the question complexity of the verifier is poly(1/ε) · okay^{(2/3)−2α}, and the question complexity of the prover is poly(1/ε) · e^{O(okay^{(2/3)+α})}. In distinction, in each circumstances, it’s recognized that the corresponding properties of distributions don’t have any doubly-efficient IPP (see Herman and Rothblum, 2025). Truly, the primary property of distributions (i.e., uniformity over [n]) doesn’t even have an IPP wherein the verifier makes use of o(n^{1/2}) samples, no matter different complexity measures.
- † Weizmann Institute of Science
