1 Introduction
Given functions , their direct sum is the function given by where addition is in the field . We denote . We study the testability question: given a function test if it is a direct sum, namely if it belongs to the set
We suggest and analyze a fourquery test which we call the “square in a cube” test, and show that it is a strong absolute local test for being a direct sum. By this we mean that neither the number of queries nor the testability constant depend on the parameters and . We also describe a simpler query test, whose easy analysis we defer to section 4.
Our square in a cube test is as follows

Choose uniformly at random

Choose two subsets uniformly at random and let be their symmetric difference.

Accept iff
where is the string whose th coordinate equals if and otherwise.
We prove
Theorem 1 (Main).
There exists an absolute constant s.t. for all and given ,
where are chosen independently and uniformly from the domain of , and are random subsets of , and refers to relative Hamming distance, namely .
Remark 2.
The above theorem is true in a greater generality. Namely, the same proof can be adapted to the case of a function , where the corresponding subspace of direct sums is
Testing if a tensor has rank .
An equivalent way to formulate our question is as a test for whether a dimensional tensor with entries has rank . Indeed moving to multiplicative notation and writing and , we are asking whether there are such that
Denoting
we have
Corollary 3.
There exists an absolute constant s.t. for all and , for every ,
Background.
Direct sum is a natural construction that is often used in complexity for hardness amplification. It is related to the direct product construction: a function is the direct product of as above if for all . The testability of direct products has received attention [GS97, DR06, DG08, IKW12, DS14] as abstraction of certain PCP tests and it was not surprising to find [DDG17] that there is a connection between testing direct products to testing direct sum. However, somewhat unsatisfyingly this connection was confined to testing a certain type of symmetric direct sum. A symmetric direct sum is a function that is a direct product with all components equal; namely such that there is a single such that
In [DDG17] a test was shown for testing if a given is a symmetric direct sum, and the analysis carried out relying on the direct product test. It was left as an open question to devise and analyze a test for the property of being a (not necessarily symmetric) direct sum.
Method.
Our proof, similarly to [DDG17], relies on a combination of the BLR linearity testing theorem [BLR93] and the direct product test of [DS14]. The trick is to find the right combination. We first observe that once we fix , the test is confined to a set of at most points in the domain, and can be viewed as performing a BLR (affinity rather than linearity) test on this piece of the domain. From the BLR theorem we deduce an affine linear function on this piece. The next step is to combine the different affine linear functions, one from each piece, into one global direct sum, and this is done by reducing to direct product.
2 Tensor Product
We refer to as a dimensional binary tensor.
Definition 4.
A dimensional binary tensor is a tensor product, if there exist onedimensional tensors, i.e., vectors, such that .
Definition 5.
A dimensional binary tensor is close to tensor product, if there exists tensor product such that
where is chosen uniformly at random.
In the next two sections we present two different approaches for testing whether a dimensional binary tensor is close to a tensor product.
3 Square in a Cube Test
We start by introducing some notation.
Given two vectors , define

;

the induced subcube is the binary cube ;

the projection map defined for as

the function as .
The following test is the same as the on preceeding the formulation of Theorem 1.
Test 6.
Square in a Cube test. Given query access to a function :

Choose uniformly at random.

Choose uniformly at random.

Query at and .

Accept iff .
Theorem 7.
Suppose a function passes Test 6 with probabilty , then is close to a tensor product.
3.1 The BLR affinity test
The BlumLubyRubinfeld linearity test was introduced in [BLR93], where its remarkable properties were proven. Later a simpler proof via Fourier analysis was presented, e.g. see [BCH95]. Below we give a variation of this test for affine functions, see [O’D14, Chapter 1].
Definition 8.
A function is called affine, if there exists a set and a constant such that for every vector
Definition 9.
A function is said to be close to being affine, if there exists an affine function such that
where is chosen uniformly at random.
Note that (see [O’D14, Exercise 1.26]) a function is affine iff for any two vectors it satisfies
(1) 
The BLR test implies that if a function satisfies (1
) with high probability, then it is close to an affine function.
Test 10.
The BLR affinity test. Given query access to a function :

Choose and independently and uniformly at random.

Query at and .

Accept if .
Theorem 11 ([Blr93]).
Suppose passes the affinity test with probability . In other words, satisfies
Then is close to being affine. ∎
3.2 Direct Product Test
Definition 12.
A function , where , is called a direct product if it is of the form . Given functions , their direct product is a function denoted and defined as .
In [DS14], Dinur and Steurer presented a query test which distinguishes between direct products and functions that are far from direct products with constant probability.
Test 13.
– Twoquery test with intersection . Given query access to a function :

Choose a set of size uniformly at random.

Choose uniformly at random, conditioned .

Query at and .

Accept iff .
Theorem 14.
[DS14, Theorem 1.1] Let be positive integers, let , where , and let . Let be given such that
where are chosen w.r.t. the test distribution . Then there exists a direct product function such that , where stand for the Hamming distance. In particular, when this implies
Remark 15.
Note that Theorem 14 is true for for all , and not just . More precisely, the following statement holds:
If a function passes Test 13 with probability at least for wtih , then passes Test 13 with probability at least for , where is a positive integer.
This reduction shows that Theorem 14 is true as it is stated for for all , as the reduction affects only the constant in the notation.
For a more detailed explanation, see Appendix (Section 6).
3.3 Proof of Theorem 7
For a positive integer , we denote by the distribution on , where each coordinate is equal to with probability and to with probability .
We use the following proposition in the course of the proof.
Proposition 16.
Let be a set and be the corresponding linear function, i.e., . Suppose
then .
Proof.
Consider . Then
Also the following holds
and the statement follows. ∎
Proof.
(of Theorem 7.) Assume Test 6 fails on a function with probability less than , i.e.,
where all distributions are uniform. Recall that is a shorthand for . Then there exists such that
W.l.o.g. we assume that and that . We can assume this, since if needed we can reindex the tensor, and flip it, i.e., add the constant one tensor elementwise. We write for and for . Then for every ,
The BLR theorem (Theorem 11) implies that there exists a subset , such that
Remark.
By the BLR theorem, there should be the “greater or equal to” sign instead of the equality. We assume equality to ease of the proof.
Let be a function defined as follows. For each , the set can be viewed as a subset of , since . Then is defined as the element of corresponding to the set .
We now show that passes Test 13 with high probability and hence is close to a direct product.
Let be chosen uniformly at random, and let be chosen with respect to the following distribution . For each ,
Note that the distribution on pairs , where is chosen uniformly from and w.r.t. , is equaivalent to the following: for each ,
(2) 
In particular, it is symmetric in the sense that choosing uniformly at random first, and then , leads to the same distribution on pairs as the one described above.
For such a pair define distribution on as follows. For a vector ,
Note that the distribution is supported on a binary cube of dimension inside . Denote
We claim that the following holds
(3) 
To see (3) note that since is chosen uniformly, is chosen w.r.t. , and , the resulting distribution for is
which is exactly the uniform distribution on
.We now show that
(4) 
First note that it follows from the definitions that
And by the symmetry of the distribution on pairs ,
Combined together, the previous two equations imply that
and by the Markov inequality, Inequality 4 follows. By the definition of ,
which is equivalent to
Proposition 16 implies that if , then
By Theorem 14, the function is close to a direct product, i.e., there exist functions such that
Therefore,
∎
4 The Shapka Test
In [KL14] Kaufman and Lubotzky showed that coboundary expansion of a dimensional complete simplicial complex implies testability of whether a symmetric matrix is a tensor square of a vector. The following test is inspired by their work and in a way generalizes it.
Given two vectors , for denote by the vector which coincides with in every coordinate except for the th one, where it coincides with , i.e.,
Test 17.
The Shapka Test. Given query access to a function :

Choose uniformly at random.

Define the query set to consist of , for all , and iff is even.

Query at the elements of .

Accept iff .
Remark 18.
Shapka is the Russian word for a winter hat (derived from Old French chape for a cap). The name the Shapka test comes from the fact that the set consists of the two top layers of the induced binary cube (and also the bottom layer if is even).
Theorem 19.
Suppose a function passes Test 17 with probabilty , then is close to a tensor product.
4.1 Proof of Theorem 19
Proof.
Denote by the normalized distance from to the subspace of tensor products, i.e., there exists a tensor product such that
For a vector , for , define a function as follows. For ,
For , the defintion of depends on the parity of and reads as follows. For ,
Given a collection of vectors, , we denote their tensor product by . In other words, for a vector , the following holds
(5) 
In these notations, the following holds for any ,
As is a tensor product, it is at least far from for any vectors , and hence for any ,
(6) 
5 Further Directions

Can the original function be reconstructed by a voting scheme using the Shapka test 17?

It is plausible that the Square in the Cube test 6 be analyzed by the Fourier transofrm approach similarly to the analysis of the BLR test.

Another test in the spirit of the presented above is the following.
Test 20.
Given query access to a function :

Choose uniformly at random.

Choose uniformly at random.

Query at and .

Accept iff .
We conjecture that this test is also good, i.e., if a function passes the test with high probability then it is close to a tensor product.

6 Appendix: Proof of Remark 15
In [DS14], Dinur and Steurer proved Theorem 14 for . The following reduction shows that the theorem is true for all by a reduction from to some . Recall that Test 13 makes two queries according to the distribution , which is the following distribution: (1) Choose a set of size uniformly at random. (2) Choose uniformly at random, conditioned .
Proposition 21.
Let denote the probability that a function passes Test 13 with respect to distribution . If for some , then for , where and .
In addition, is , then also .
Proof.
Fix a function , and suppose for some , i.e.,
We will show that where and . Note that satisfies .
Given a pair of random vectors and a set distributed according to , we construct a sequence of vectors such that for all , the pair is distributed according to .
The complement of has size . Partition it randomly into parts of equal size , . Denote for all .
For each , construct such that it agrees with on the coordinates in and with on the rest of the coordinates . Then for each , agrees with on the set of the size . Therefore,
Hence,
The case of has to be treated separately as it is not covered by Theorem 14. In this case there is a reduction to as follows. Given two vectors distributed w.r.t. construct an intermidiate random vector which agrees on exactly half of the coordinates with both and . ∎
Corollary 22.
Let be positive integers, let , where , and let . Let be given such that
where are chosen w.r.t. the test distribution . Then there exists a direct product function such that , where stand for the Hamming distance. In particular, cwhen this implies
Funding
The first author is supported by ERCCoG grant number 772839. A substantial part of the work was done while the second author held a joint postdoctoral position at The Weizmann Institute and BarIlan University funded by the ERC grant number 33628. Currently, the second author is supported by the SNF grant number 200020_169106.
References
 [BCH95] Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, and Madhu Sudan. Linearity testing in characteristic two. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 2325 October 1995, pages 432–441, 1995.
 [BLR93] Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Selftesting/correcting with applications to numerical problems. Journal of computer and system sciences, 47(3):549–595, 1993.
 [DDG17] Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. SIAM J. Comput., 46(4):1336–1369, 2017.
 [DG08] Irit Dinur and Elazar Goldenberg. Locally testing direct products in the low error range. In Proc. 49th IEEE Symp. on Foundations of Computer Science, 2008.
 [DR06] Irit Dinur and Omer Reingold. Assignment testers: Towards combinatorial proofs of the PCP theorem. SIAM Journal on Computing, 36(4):975–1024, 2006. Special issue on Randomness and Computation.
 [DS14] Irit Dinur and David Steurer. Direct product testing. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 188–196, 2014.
 [GS97] Oded Goldreich and Shmuel Safra. A combinatorial consistency lemma with application to proving the PCP theorem. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science. LNCS, 1997.
 [IKW12] Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New directproduct testers and 2query PCPs. SIAM J. Comput., 41(6):1722–1768, 2012.
 [KL14] Tali Kaufman and Alexander Lubotzky. High dimensional expanders and property testing. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS ’14, pages 501–506, New York, NY, USA, 2014. ACM.
 [O’D14] Ryan O’Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
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