Direct sum testing. [Extended abstract]

The k-fold direct sum encoding of a string a {0; 1}ⁿ is a function f_a that takes as input sets S [n] of size k and outputs f_a(S) = Σ_iεS 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 (ε) functions f that are "-far from being a direct sum encoding. 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 k'th 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 ≤ √n) to a tensor power. Thus our theorem implies a three query test for deciding if a given tensor f {-1; 1}^nk is a tensor power of a single dimensional vector a {-1; 1}ⁿ, i.e. whether there is some a such that f = a^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 (STOC '90). 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 (CCC '2014). The k = n/2 case is proven via a new proof for linearity testing on the hypercube, which we extend to the restricted domain of the n/2-th layer of the hypercube.