Let be an affine invariant property of functions Fnp[R] for fixed p and R. We show that if is locally testable with a constant number of queries, then one can estimate the distance of a function f from with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over F2.

Our test is simple: take a restriction of f to a constant dimensional affine subspace, and measure its distance from . We show that by choosing the dimension large enough, this approximates with high probability the global distance of f from \cP. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].