Invariance in Property Testing

Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, one could ask how it can be tested by so few samples? We suggest that for “natural” properties, this should happen because the property is invariant under “nice” set of “relabellings” of the data. We refer to this set of relabellings as the “invariance class” of the property and advocate explicit identification of the invariance class of locally testable properties. Our hope is the explicit knowledge of the invariance class may lead to more general, broader, results. After pointing out the invariance classes associated with some the basic classes of testable properties, we focus on “algebraic properties” which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain. We survey recent results (obtained with Tali Kaufman, Elena Grigorescu and Eli Ben-Sasson) that give broad conditions that are sufficient for local testability among this class of properties, and some structural theorems that attempt to describe which properties exhibit the sufficient conditions.