Direct sum testing

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_i∈Sa_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 Ω(ϵ) functions f that are ϵ-far from being a direct sum encoding, where the constant behind the Ω 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 ≪ √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 [J. 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.