Algebraic property testing: the role of invariance

TLDR: This work considers (F-)linear properties that are invariant under linear transformations of the domain and proves that an O(1)-local "characterization" is a necessary and sufficient condition for O( 1)-local testability, and shows that local formal characterizations essentially imply local testability.

Abstract: We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space Kn over a field K to a subfield F. We consider (F-)linear properties that are invariant under linear transformations of the domain and prove that an O(1)-local "characterization" is a necessary and sufficient condition for O(1)-local testability. when |K| = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)-local constraint implies O(1)-local testability. These results generalize and extend the class of algebraic properties, most notably linearity and low-degree-ness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)-locally testable. Our results are proved by introducing a new notion that we term "formal characterizations". Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations essentially imply local testability. We then investigate properties that are linear-invariant and attempt to understand their local formal characterizability. Our results here give coarse upper and lower bounds on the locality of constraints and characterizations for linear-invariant properties in terms of some structural parameters of the property we introduce. The lower bounds rule out any characterization, while the upper bounds give formal characterizations. Combining the two gives a test for all linear-invariant properties with local characterizations. We believe that invariance of properties is a very interesting notion to study in the context of property testing in general and merits a systematic study. In particular, the class of linear-invariant and affine-invariant properties exhibits a rich variety among algebraic properties and offer better intuition about algebraic properties than the more limited class of low-degree functions.