Direct Sum Testing

Abstract

The $k$-fold direct sum encoding of a string $a \in \{0,1\}^n$ is a function $f_a$ that takes as input sets $S \subseteq [n]$ of size $k$ and outputs $f_a(S) = \sum_{i \in S} a_i \pmod 2$. 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 $\Omega(\varepsilon)$ functions $f$ that are $\varepsilon$-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 $k$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 thatis negligible when $k\ll \sqrt n$) to a tensor power. Thus our theorem implies a three query test for deciding if a given tensor $f\in \{-1,1\}^{n^k}$ is a tensor power of a single dimensional vector $a\in \{-1,1\}^n$, i.e., whether there is some $a$ such that $f = a^{\otimes k}$. We also provide a four query test for checking if a given $\pm 1$ 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 $n/2$th layer of the hypercube.