# Direct Sum Testing: The General Case

## TL;DR

This paper introduces a 4-query test to efficiently distinguish direct sum functions from those far from such functions, extending linearity testing to higher dimensions and tensor products.

## Contribution

It presents a novel 4-query test for direct sums, generalizing the BLR linearity test and agreement tests to higher-dimensional tensor product functions.

## Key findings

- The test distinguishes direct sums from far functions with high probability.
- The approach extends linearity testing to tensor product structures.
- An alternative, simpler test with up to (d+2) queries is also proposed.

## Abstract

A function $f:[n_1]\times\dots\times[n_d]\to\mathbb{F}_2$ is a direct sum if it is of the form $f\left(a_1,\dots,a_d\right) = f_1(a_1)\oplus\dots \oplus f_d (a_d),$ for some $d$ functions $f_i:[n_i]\to\mathbb{F}_2$ for all $i=1,\dots, d$, and where $n_1,\dots,n_d\in\mathbb{N}$. 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 (Blum, Luby, Rubinfeld, 1993) and on an agreement test which slightly generalizes the direct product test (Dinur, Steurer, 2014). In multiplicative $\pm 1$ notation, our result 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, or equivalently, if it is of rank $1$. The presented tests can be read as tests for distinguishing between tensor products and tensors that are far from being tensor products. We also present a different test, which queries the function at most $(d+2)$ times, but is easier to analyze.