Direct Sum Testing

The k-fold direct sum encoding of a string a is an element of {0, 1}(n) is a function f(a) that takes as input sets S subset of [n] of size k and outputs f(a)(S) = Sigma(i is an element of s)a(i) (mod 2). In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form f(a) for some a and rejects with probability Omega(epsilon) functions f that are s-far from being a direct sum encoding, where the constant behind the Omega notation is independent of k. This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of [n] with vectors in {0, 1}(n) in the natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the kth layer of the hypercube (i.e., the set of n-bit strings with Hamming weight k). Tensor power testing: By moving to -1,1 notation, the direct sum encoding is equivalent (up to a difference that is negligible when k << root n) to a tensorpower. Thus our theorem implies a three query test for deciding if a given tensor f is an element of {-1,1}(nk) is a tensor power of a single dimensional vector alpha is an element of {-1, 1}(n), i.e., whether there is some a such that f = alpha circle times(k). We also provide a four query test for checking if a given +/- 1 matrix has rank 1. Our test naturally extends the linearity test of Blum, Luby, and Rubinfeld [T. Comput. Syst. Sci., 47 (1993), pp. 549-595]. Our analysis proceeds by first handling the k = n/2 case and then reducing this case to the general k < n/2 case, using a recent direct product testing theorem of Dinur and Steurer [Proceedings of CCC '2014]. The k = n/2 case is proved via a new proof for linearity testing on the hypercube, which we extend to the restricted domain of the n/2th layer of the hypercube.