Testing tensor products.

A function $f:[n]^d\to\mathbb{F}_2$ is a direct sum if it is of the form $ f\left((a_1,\dots,a_d)\right) = f_1(a_1)+\dots + f_d (a_d),$ for some $d$ functions $f_1,\dots,f_d:[n]\to\mathbb{F}_2$. We present a $4$-query test which distinguishes between direct sums and functions that are far from them. The test relies on the BLR linearity test and on the direct product test constructed by Dinur and Steurer. We also present a different test, which queries the function $(d+1)$ times, but is easier to analyze. In multiplicative $\pm 1$ notation, the above reads as follows. A $d$-dimensional tensor with $\pm 1$ entries is called a tensor product if it is a tensor product of $d$ vectors with $\pm 1$ entries. In other words, it is a tensor product if it is of rank $1$. The presented tests check whether a given tensor is close to a tensor product.