## Direct Sum Testing

by Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar

#### Oded's comments

My view of this work is as addressing the question
of testing linearity over a partial domain, which is not
a linear subspace: The standard question
of testing linearity refers to testing a homomorphism between
two groups, but the archetypical case is when the first group
is a vector space over the two-element field (and the second
group is that field). The current work studies testing linearity
over a non-linear subset of the vector space, specifically, the
set of all vectors of fixed weight. The work contains several
highly appealing results about this type of question.

#### The original abstract

For a string $a \in \{0,1\}^n$ its $k$-fold direct sum encoding is a
function $f_a$ that takes as input sets $S \subseteq [n]$ of
size $k$ and outputs $f_a(S) = \sum_{i \in S} a_i$.
In this paper we are interested in the Direct Sum Testing Problem,
where we are given a function $f$, and our goal is
to test whether $f$ is close to a direct sum encoding,
i.e., whether there exists some $a \in \{0,1\}^n$ such that
$f(S) = \sum_{i \in S} a_i$ for most inputs $S$.
By identifying the subsets of $[n]$ with vectors
in $\{0,1\}^n$ in the natural way, this problem can be thought of as
linearity testing of functions whose domain is restricted to the
$k$'th layer of the hypercube.

We first consider the case $k=n/2$, and analyze for it a
variant of the natural 3-query linearity test introduced
by Blum, Luby, and Rubinfeld (STOC '90). Our analysis
proceeds via a new proof for linearity testing on
the hypercube, which extends also to our setting.

We then reduce the Direct Sum Testing Problem for general $k < n/2$ to
the case $k = n/2$, and use a recent result on Direct Product Testing
of Dinur and Steurer in order to analyze the test.

See ECCC TR14-002

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