Characterizing Direct Product Testing via Coboundary Expansion

A d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution µ over pairs of k-faces (A,A′), and given query access to F: X(k)→{0,1}k it samples (A,A′)∼ µ and checks that F[A]|A∩ A′ = F[A′]|A∩ A′. The tester should have (1) the ”completeness property”, meaning that any assignment F which is a direct product assignment passes the test with probability 1, and (2) the ”soundness property”, meaning that if F passes the test with probability s, then F must be correlated with a direct product function. Dinur and Kaufman showed that a sufficiently good spectral expanding complex X admits a direct product tester in the ”high soundness” regime where s is close to 1. They asked whether there are high dimensional expanders that support direct product tests in the ”low soundness”, when s is close to 0. We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as ”Unique-Games coboundary expanders”. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution.


INTRODUCTION
The problem of testing direct product functions lies at the intersection of many areas within theoretical computer science, such as error correcting codes, probabilistically checkable proofs (PCPs), hardness ampli cation and property testing.In its purest form, one wishes to encode a function : [ ] → {0, 1} using local views in a way that admits local testability/local correction.More precisely, given a parameter 1 ⩽ < , the encoding of using subsets of size can be viewed as : [ ] → {0, 1} that to each subset ⊆ [ ] of size assigns a vector of length describing the restriction of to . 1 We refer to this encoding as the direct product encoding of according to the Johnson graph (for reasons that will become apparent shortly).The obvious downside of this encoding scheme is, of course, that its length is much larger than the description of (roughly vs Θ( )).However, as this encoding contains many redundancies, one hopes that it more robustly stores the information in the function , thereby being more resilient against corruptions.
1.1 Direct Product Testing with 2 Queries Indeed, one of the primary bene ts of the above direct product encoding is that it admits local testers using a few queries.These testing algorithms also go by the name "agreements testers" or "direct product testers", and are often very natural to design.A direct product tester for the above encoding, which we parameterize by a natural number 1 ⩽ ⩽ and denote by T , proceeds as follows: (1) Choose two subsets , ′ ⊆ [ ] uniformly at random conditioned on | ∩ ′ | = .(2) Query [ ], [ ′ ] and check that [ ] and [ ′ ] agree on ∩ ′ .These type of testers have been rst considered and used by Goldreich and Safra [23] in the context of the PCP theorem.They later have been identi ed by Dinur and Reingold [18] as a central component in gap ampli cation.To get some intuition to this test, note that a direct product function clearly passes the test with probability 1.Thus, we say that the tester has perfect completeness.The soundness of the test -namely the probability that a table which is far from a direct product encoding passes the test -is more dicult to analyze.Intuitively, querying at a single location gives the value of a (supposed) on inputs.Thus, if is far from any direct product function, the chance this will be detected should grow with .Formalizing this intuition is more challenging however, and works in the literature are divided into two regimes: the so-called 99% regime, and the 1% regime.To be more precise, suppose the table passes the direct product tester T with probability at least > 0; what can be said about its structure?
In the 99% regime, namely the case where = 1 − is thought of as close to 1, results in the literature [18,20] show that has to be close to a direct product function.More speci cally, for = Θ( ) the result of [20] asserts that there exists : [ ] → {0, 1} such that [ ] = | for 1 − ( ) fraction of the -sets .A structural result of this form is a useful building block in several applications.It can be used to construct constant query PCPs with constant soundness; it also serves as a building block in other results within complexity theory; see for instance [8,13].
The 1% regime, namely the case where = is thought of as a small constant, is more challenging.In this case, the works [14,26] show that has to be correlated with a direct product function.More speci cally, these works show that for (say) = √ if ⩾ 1/ Ω (1) , then there exists : [ ] → {0, 1} such that for at least (1) fraction of the -sets , we have that (1) , where for two strings , ∈ {0, 1} , Δ( , ) = denotes the fractional Hamming distance between them. 2 The motivation for studying this more challenging regime of parameters stems mainly from the perspective of hardness ampli cation (where one wishes to show that if a given task is somewhat hard, then repeating this task -times in parallel gets exponentially harder) as well as from the study of PCPs with small soundness.Indeed, in [26] the authors show that direct product testers similar to the above facilitate soundness ampli cation schemes for PCPs with similar performance to parallel repetition theorems [7,19,25,35,36].Direct product testers in the low soundness regime have additional applications in property testing, as well as in the study of the complexity of satis able constraint satisfaction problems [3][4][5][6].

Size E cient 2-Query Direct Product Testing
In the context of PCPs and hardness ampli cation, one typically thinks of the parameter as very large, and as a large constant number.With this in mind, representing an assignment : [ ] → {0, 1} using its direct product encoding incurs a polynomial blowup in size.Indeed, this type of step is often the only step in the PCP reduction that introduces a polynomial (as opposed to just linear) blow-up in the instance size.In this light, a natural question is whether it is possible to perform hardness ampli cation with a signi cantly smaller blow up in the encoding/instance size.Ecient schemes of this type are often referred to as "derandomized direct product tests", "derandomized hardness ampli cation" or "derandomized parallel repetition theorems".
In [26] a more e cient hardness ampli cation procedure is proposed.Therein, instead of considering all -sets inside [ ], the domain [ ] is thought of as a vector space F and one considers all subspaces of dimension log ( ).It is easy to see that the encoding size then becomes Θ(log ) , making it more e cient.The paper [26] shows that direct product testers analogous to the tester above work in this setting as well; they essentially match all of the results achieved by the Johnson scheme.Building upon [26], Dinur and Meir [16] show how to establish parallel repetition theorems using the more e cient direct product encoding via subspaces.This parallel repetition theorem works for structured instances, which the authors show to still capture the entire class NP.
High dimensional expanders (HDX), which have recently surged in popularity, can be seen as sparse models of the Johnson graph.This leads us to the main problem considered in this paper, due to Dinur and Kaufman [15]: Do high dimensional expanders facilitate direct product testers in the low soundness regime?
The main goal of this paper is to investigate the type of expansion properties that are necessary and su cient for direct product testing with low soundness.It is known that there are HDXs of size ( ) and (1) degree, and if any of these objects facilitates a direct product tester with small soundness, they would essentially be the ultimate form of derandomized direct product testers. 3To state our results, we rst de ne the usual notion of spectral high dimensional expansion, followed by our variant of the well-known notion of coboundary expansion.

High Dimensional Local Spectral Expanders.
A -dimensional complex is composed of (0) = {∅}, a set of vertices (1), which is often identi ed with [ ] and a set of -uniform hyperedges, ( ) ⊆ (1) , for each = 2, . . ., .A -dimensional complex = ( (0), (1), . . ., ( )) is called simplicial if it is downwards closed.Namely, if for every 1 ⩽ ⩽ ⩽ , and every ∈ ( ), if ⊆ has size , then ∈ ( ).The size of a complex is the total number of hyperedges in .The degree of a vertex ∈ (1) is the number of faces in ( ) containing it, and the degree of a complex is the maximum of the degree over all the vertices in (1).
Distributions over the complex.It is convenient to equip a complex with a measure for each one of its levels ( ).For = we consider the measure which is uniform over ( ); for each < , the measure is the push down measure of : to generate a sample according to , sample ∼ and then ⊆ of size uniformly.Abusing notation, we will refer to all of the measures simply as , as the cardinality of the sets in discussion will always be clear from context.The set of measures in the link of is the natural set of measures we get by conditioning on containing .
Equipped with measures over complexes, we may now de ne the notion of spectral HDX.Definition 1.2.A -dimensional simplicial complex is called a one-sided (two-sided) local spectral expander if for every ∈ of size at most − 2, the second eigenvalue (singular value) of the normalized adjacency matrix of the graph underlying the link of is at most .
In this work, we will only be concerned with simplicial complexes that are very strong spectral expanders.With this regard, following the works of [21,32,33] one can show that for every > 0 and every ∈ N there exists an in nite family of -dimensional complexes of linear size that are one-sided or two-sided local expanders (see [15,Lemma 1.5]).
1.2.2 Results in the High Soundness Regime.Dinur and Kaufman [15] were the rst to consider the question of direct product testing over HDX.They showed that a su ciently good high dimensional spectral expander admits a direct product tester in the high soundness regime.The tester they consider is essentially the same as the tester in the Johnson scheme; one thinks of which is much larger than 1 but much smaller than the dimension of the complex .The tester has parameters 1 ⩽ ⩽ /2 and is given oracle access to a table : ( ) → {0, 1} , and proceeds as follows: Agreement-Test 1 ( , , ).
(3) Sample ⊆ , ′ ⊆ of size uniformly.( Henceforth, we refer to this test as the ( , ) direct product tester over .Dinur and Kaufman consider the case where = /2, and proved that for every > 0, provided that is su ciently small, if : ( ) → {0, 1} passes the above test with probability at least 1 − , then there exists : (1) → {0, 1} such A follow-up work by Dikstein and Dinur [9] further re ned this result, and investigated more general structures that support direct product testing in the high soundness regime.
A problem related to direct product testing, called the list agreement testing problem, was considered in the high soundness regime by Gotlib and Kaufman [24].In the list agreement testing problem, each face is assigned a list of = (1) functions, and one performs a local test on these lists to check that they are consistent.With this in mind, the result of Gotlib and Kaufman [24] asserts that under certain structural assumptions on the lists, if the underlying complex has su ciently good coboundary expansion, then one can design a 3-query list agreement tester that is sound.The list agreement problem will play an important role in the current work, and while we do not know how to use the result of Gotlib and Kaufman for our purposes, their work inspired us to look at connections between agreement testing and notions of coboundary expansion.

Main Results
Despite considerable interest, no positive nor negative results are known regarding the question of whether HDX support direct product testers in the low-soundness regime.In fact, the majority of applications of HDX are in the high soundness regime, with the rst construction of 3 -locally testable codes [12] and quantum LDPC codes [22,30,34].At a rst glance, this seems surprising: very good expander graphs give rise to objects in the low-soundness regime, and high dimensional expanders are essentially their higher order analogs.
The main contribution of this work is an explanation to this phenomenon.We show that, to facilitate direct product testers in the low-soundness regime, a high dimensional spectral expander must posses a property that may be seen as a generalization of coboundary expansion [31].On the other hand, we also show that coboundary expansion is su cient to get direct product testers.Thus, to construct constants degree, sparse complexes facilitating direct product testing, one should rst come up with local spectral expanders that are also coboundary expanders.
Below, we state our main results regarding the soundness of the test, which give analysis of the ( , ) tester de ned above assuming expansion properties of the complex .In a concurrent and independent work, Dikstein and Dinur [10] established related results.

Coboundary Expansion.
For convenience, we follow the presentation of coboundary expansion from [11].Suppose we have a function : (2) → F 2 .The function is said to be consistent on the triangle { , , } ∈ (3) if it holds that ({ , }) + ({ , }) + ({ , }) = 0. What can we say about the structure of functions which are consistent with respect to 1 − measure of the triangles?Clearly, if is a function of the form ({ , }) = ( ) + ( ) for some : (1) → F 2 , then it is consistent with respect to all triangles.In the case that is a coboundary expander, the converse is also true: any which is (1 − ) triangle consistent is ( )-close to a function of this form.
More broadly, the notion of coboundary expansion often refers to a property of higher dimensional faces, and to more general groups beyond F 2 .We refrain from de ning these notions precisely and instead turn to our variant of coboundary expansion, which we show governs the soundness of direct product testing.

Unique-Games Coboundary Expansion.
Our notion of coboundary expansion replaces the group F 2 with non-Abelian groups, more speci cally with the permutation groups ; we also need to consider higher dimensional faces.Some de nitions in this spirit have been made, for example in [17,24], and our notion is inspired by theirs.
Definition 1.3.Let be a -dimensional complex and let be an integer such that ⩽ /3.Let : ( ) → be a function that satis es ( , ) = ( , ) −1 for all ( , ) ∈ [ ].We say that is consistent on the triangle ( , , We say that is (1 − )-consistent on triangles if sampling ∼ 3 and then splitting as a triangle ∪ ∪ uniformly where One way to think of this de nition is as a locally consistent instance of Unique-Games.Indeed, a as above speci es a Unique-Games (UG) instance on the graph [ ] whose constraints are locally consistent on triangles.The goal in this UG instance may be thought of assigning elements from [ ] to the vertices of [ ], namely nding an assignment : ( ) → [ ], so as to maximize the fraction of edges ( , ) for which ( ) = ( , ) ( ).
With this de nition in mind, we can now present a simpli ed version of our notion of coboundary expansion.One way to arrive at a locally consistent UG instance as in De nition 1.3 is to rst pick some function : ( ) → and then de ne ( , ) = ( ) ( ) −1 .Thus, a natural question is whether there are other ways to construct locally consistent UG instances on [ ].In simple terms, our simpli ed notion of UG coboundary expansion asserts that this is essentially the only way to arrive at instances of this form.More precisely: Definition 1.4.We say that a -dimensional simplicial complex is an ( , , , ) UG coboundary expander if for all ⩽ and for all functions : [ ] → that are (1 − )-consistent on triangles, there is : We remark that if a complex is an ( , , , ) UG coboundary expander, then given a (1 − )-locally consistent instance of Unique-Games on [ ] for some ⩽ , one may nd an assignment satisfying at least 1 − fraction of the constraints.Indeed, by de nition, given the constraint map we may nd : ( ) → such that ( , ) = ( ) ( ) −1 with probability at least 1 − over the choice of ∪ ∼ 2 .Thus, taking the labeling ( ) = ( )(1), we see that satis es all edges on which ( , The rst result of this paper asserts that a spectral HDX which is a UG coboundary expander admits a direct product tester in the low soundness regime.Theorem 1.5.Suppose that a simplicial complex is a su ciently good spectral and UG coboundary expander.If : ( ) → {0, 1} passes the ( , √ ) direct product test on with probability , then there is : (1) → {0, 1} such that In words, being a UG coboundary expander is a su cient condition for a spectral expander to support a low soundness direct product tester.As far as we know, however, this condition may not be necessary; below, we present a condition which is both necessary and su cient.Nevertheless, we chose to present its simpler to state version, De nition 1.4, as we nd it more appealing, intuitive and resembling non-Abelian variants of the usual notion of coboundary expansion.
Remark 1.6.The usual de nition of coboundary expansion in the literature refers to Abelian groups such as F 2 , see for example [11,24,[27][28][29].In the F 2 setting, coboundary expansion for the base graph can be seen as a UG instance over F 2 , but it is often phrased in topological notions using the boundary and coboundary maps; these de nitions extend well to higher dimensional faces.Coboundary expansion has also been de ned for non-Abelian groups [17,24,29], however, as far as we know, these de nitions coincide with ours only for the case that = 1 in De nition 1.4.3) is also assigned a list ′ ( ) = ( ′ 1 ( ), . . ., ′ ( )).In words, we would like the permutations to be consistent with the lists with respect to concatenations.Towards this end, we introduce a convenient but informal notation to compare strings.Given , ∈ ( ) that are disjoint and strings ( ), ( ) ∈ {0, 1} , we shall think of ( ) as an assignment to the vertices in and of ( ) as an assignment to the vertices in .Thus, the notation ( ) • ( ) will be a string in {0, 1} 2 which encodes the assignment to ∪ provided by the concatenation of the two assignments.More generally, given , disjoint and list assignments ( ), ( ) we de ne Lastly, given a list ( ) as above and ∈ , we de ne ( ) = ( (1) ( ), . . ., ( ) ( )).Definition 1.7.Let : ( ) → ({0, 1} ) , ′ : (3 ) → ({0, 1} 3 ) , and > 0. We say is (1 − )-consistent with the lists and ′ if choosing ∼ 3 and a splitting = ∪ ∪ into a triangle, we have that We say that is (1 − )-strongly triangle consistent if there are lists and ′ such that is (1 − )-consistent with respect to the lists and ′ .
It is easy to see that if is (1 − )-strongly triangle consistent, then is (1 − ( ))-triangle consistent.Thus, the class of triangle consistent functions is more restrictive.With the notion of strong triangle consistency we are now ready to state a weaker variant of De nition 1.4; the only di erence between the two definitions is that in the de nition below, we only require that any strongly triangle consistent assignment admits a global structure.More precisely: Definition 1.8.We say that a -dimensional simplicial complex is a weak ( , , , ) UG coboundary expander if the following condition is satis ed for all ⩽ .Suppose : [ ] → is a (1− )strongly triangle consistent function.Then there exists : ( ) → such that The parameter in De nition 1.8 is often referred to as the level at which UG coboundary expansion holds.With the notion of weak UG coboundary expansion, we can now state a stronger version of Theorem 1.5.Roughly speaking, the following two results asserts that for a su ciently good spectral simplicial complex , the direct product tester over works in the low soundness regime if and only if is a weak UG coboundary expander with su ciently good parameters.
Theorem 1.9.The following results hold for any simplicial complex .
(1) Weak UG-coboundary is Necessary: If a simplicial complex is a su ciently good spectral expander which is not a UG coboundary expander, then there is > 0 such that for sufciently large , there is : ( ) → {0, 1} that passes the ( , √ ) direct product tester with probability and yet for all : (1) → {0, 1} we have that (2) Weak UG-coboundary is Su cient: For all , > 0, if a simplicial complex is a su ciently good spectral expander and a weak UG coboundary expander on level (1), then the direct product test over with respect to su ciently large has soundness .Namely, if : ( ) → {0, 1} passes the ( , √ ) direct product tester with respect to with probability at least , then there is : (1) → {0, 1} such that We refer the reader to the full version for more formal statements and their proofs.We use our necessary result above to conclude that some of the best known sparse spectral expanders -namely some LSV complexes -do not support direct product testers in the low soundness regime precisely because they fail to satisfy coboundary expansion.As the result of Dinur and Kaufman [15] asserts that LSV complexes admit direct product testers in the high soundness regime, we conclude that the low soundness regime is qualitatively di erent.
In the above theorem, the structure for we get is relatively weak though, and only asserts that with signi cant probability over the choice of ∼ , we have that [ ] = ( ) for (1 − ) fraction of ∈ .In the next theorem, we show that if the level on which coboundary expansion holds is linear in , then the conclusion of Theorem 1.9 can be strengthened to say that with signi cant probability over ∼ , it holds that [ ] = ( ) for all but constantly many of ∈ . 4heorem 1.10.If a simplicial complex is a su ciently good spectral expander, and for ∈ N it holds that is a su ciently good weak UG coboundary expander on level Ω( ), then the direct product test over with respect to has soundness .Namely, for all > 0 there is > 0 such that if : ( ) → {0, 1} passes the ( , ) direct product tester with respect to with probability at least , then there is : (1) → {0, 1} such that In the full version, we examine several well known complexes.We show that dense complexes such as the complete and the Grassmann complex are UG coboundary expanders.On the ip side we use well-known theorems that some LSV complexes are not coboundary expanders, to show that they fail to support direct product testers.

PROOF OF THEOREM 1.9: UG COBOUNDARY IS SUFFICIENT
In this section, we prove the "su cient" part of Theorem 1.9, formally stated below.
We begin by setting up some notations that will be helpful throughout the proof.Given a global function : [ ] → {0, 1} and a set ⊆ [ ] we let ( ) denote the assignment to using .For a function : [ ] → {0, 1} and an assignment : ( ) → {0, 1} we let Agr( , ) denote the subset of ( ) where ( ) = ( ) and agr( , ) denote the probability of this event under the measure .Furthermore for ∈ (0, 1) let Agr ( , ) denote the subset of ( ) where ( ) and ( ) agree on (1 − )-fraction of the elements in and agr ( , ) denotes the probability of this event under .

High Level Structure of the Proof
The proof of Theorem 2.1 follows the outline given in the introduction.For convenience we break it into two parts, encapsulated in the following two lemmas.In the rst lemma we implement the rst four steps in the plan and reduce the problem of direct product testing to the problem of "list agreement" testing.In this problem, for each -face in a complex we have a list [ ] of (1) functions, and we test whether these lists are in 1-to-1 correspondence according to the up-down-up walk on the complex.More precisely, the problem is de ned as follows: List-Agreement-Test 1. Input: a list ( ) for each ∈ ( ) and a parameter ∈ (0, 1).
(2) Choose independently random , ′ ⊇ from ( ).With the list agreement problem formally de ned, we can now state the lemma encapsulating the rst few steps in the argument, saying that an assignment that passes the direct product test with probability bounded away from 1 implies a natural list assignment passing the list agreement test with probability close to 1.
Lemma 2.3.For all > 0, for su ciently large ∈ N, ⩾ poly( )2 poly(1/ ) , su ciently small compared to and = ( 68 ), the following holds.Suppose that is a -dimensional simplicial complex which is a -spectral expander, and : ( ) → {0, 1} passes the ( , √ )-agreement-test 1 with probability .Then, there exists 2 −1/ 1200 ⩽ ′ ⩽ 101 and lists ( [ ]) ∈ ( ) satisfying: Armed with the conversion of our assignment lists that pass the list agreement test with probability close to 1, we implement the next three steps in the introduction.Namely, we show that if is a su ciently good UG coboundary expander, then we can use the lists above to de ne a locally consistent instance of Unique-Games on low levels of the complex and apply UG coboundary expansion to deduce the existence of a global solution.
Lemma 2.4.Assume there exists a collection of lists { [ ]} ∈ ( ) that satisfy the premise of Lemma 2.3, and assume that is aspectral expander for < 1/poly( ) and a weak ( (1/ 12 ), , ( √ ), ) UG coboundary expander for = Θ 12 ′ . Then there exists Here, the distance between a function ( ) and a list of functions [ ] is the minimal distance between ( ) and any function in the list.
The proof of Theorem 2.1 now readily follows from the above two lemmas.

Auxiliary Claims
Our proof requires a few basic auxiliary probabilistic claims, which we record here.The rst claim asserts that if the distance between two functions , : [ ] → {0, 1}, then choosing a random subset ⊆ [ ], we have that the distance between | is also very close to .More precisely: Claim 2.5.Suppose ∈ (0, 1), and let , : [ ] → {0, 1} be functions such that Δ( , ) = .Then, for 1  2 ⩽ ⩽ we have that: Proof.Both of the items are immediate consequences of Cherno 's inequality.The arguments are essentially identical, and we give a proof of the rst item only.
Note that conditioned on | ∩ ′ | = , the distributions D 1 and D 2 are identical.Thus, as the probability of this event is 1− ( 2 / √ ) = 1 − (1) in both distributions, it follows that the statistical distance between D 1 and D 2 is (1).Therefore, Denote by D 2 ( ) the distribution on ( , ′ , ) conditioned on sampling , and by the probability that [ ]| = [ ′ ]| if was chosen.By an averaging argument, with probability at least 4 over the choice ∼ √ we have that ⩾ 2 ; we call such good, and denote the set of good 's by B.
We get that Fix a -face satisfying the above event.Thus, picking ⊂ √ and ( , ′ , ) ∼ D 2 ( ) passes the direct product test with probability at least 2 8 .Let this distribution be D 2 ( ).As before, letting the distribution D 1 ( ) be the distribution over ( , ′ , ) ∼ 1 conditioned on sampling , the statistical distance between D 1 ( ) and D 2 ( ) is (1).Therefore we get that, Pr ∼ ( , ) − direct product test passes w.p.
which completes the proof.□ We refer to a -face ∈ ( ) for which the event in Lemma 2.7 holds as good, and thus conclude that 1 − (1) fraction of thefaces are good.Note that the above argument would also work for /2-faces, and thus we similarly de ne the notion of good /2-faces.

Ge ing a List on Each Good Johnson and Generating a Gap.
Fix a good -face , and consider the assignment when restricted to -sets inside .For notational convenience, we denote this restricted assignment by .Thus, the event in Lemma 2.7 translates to saying that the direct product tester over the Johnson scheme passes inside with noticeable probability.Thus, using direct product testing results over the Johnson scheme, we may "explain" this consistency via correlations of with true direct product functions.Towards this end, we use a result due to [14] (see also [26], who state a version that is more convenient for our purposes).
Theorem 2.8.Suppose that passes the ( , √ ) direct product test in with probability .Then there is a function Theorem 2.8 by itself is not enough for us, and we need an idea that is often useful in conjunction with such results: list decoding.We wish to consider all direct product functions that are correlated with and have these as the lists.Alas, there is a technical issue: the number of direct product functions that are correlated with need not be bounded in terms of , the probability that the test passes.To remedy this issue we require the notion of -covers, de ned below.Definition 2.9.Let F ⊆ G be two families of functions from [ ] to {0, 1}.We say that F is an -cover for G if for any ∈ G there exists ∈ F such that Δ( , ) ⩽ .
We are now ready to present a procedure that, given a good -face , generates a short list of functions that "explain" most of the probability that passes the direct product test inside , and which is also short.The procedure takes as input a restriction of the assignment to a face , which below we denote by , and nds one by one direct product functions that are correlated with , following by randomizing at appropriate places.
Algorithm 1.The short list algorithm.
(3) If ∉ 1 then for all , agr ( , ) < .Proof.First note that by Theorem 2.8 we get that there is at least one function with agr ( ) ⩾ ′ , therefore the list is non-empty.Let us start by proving the upper bound on the size.
Proof of (1): At the ℎ iteration we add a function to the list only if agr ( , ˜ ) > which is always at least ′ − ′20 .Let R ⊆ be the -sets that have been randomized in the algorithm so far, so |R| ⩾ ( ′ − ′20 ) .Using the Cherno bound we get that every function : → {0, 1} satis es: Therefore by a union bound we get that with probability 1 − ( 1), for all functions on the above holds, and we condition on this event.Hence, the contribution of R to the agreement of function found in later steps in the procedure is always at most (1).Thus, each newly found function in the process increases the measure of R by at least ′ − ′20 − (1) ⩾ ′ /2.Therefore, with probability 1 − (1) the process terminates after at most 2/ ′ steps, which is thus also an upper bound on the list size 1 .
Proof of (3): If ∉ 1 then the process terminated before step , meaning that the assignment at that time no longer wascorrelated with any direct product function.
Proof of (4): Denote by R the collection of all -sets in which the assignment has been randomized in steps prior to the + 1th iteration, and consider +1 .By Claim 2.6 for all ⩾ , ∈ 1 we get, (1) and so It follows from the above that which is at most + exp(−Ω( log(1/ ′ ))).□ We will now consider the run of the short list algorithm on a -face with various options for parameters, and its relationship with direct product functions on /2 sub-faces.We will especially care about the relationship between the functions in the list of the -face ∈ ( ), and direct product functions on its /2-faces that have large correlation with the assignment .In a sense, we will want to show that these are "the same functions"; ultimately, this is where the local consistency of the lists comes from.
Towards this end, we will run the algorithm above for faces, and denote the outputted list by [ ], For /2 sub-faces of , we will let [ ] be an -cover for functions that have su cient agreement with | .The following lemma summarizes the properties of such runs of the short list algorithm: Lemma 2.11.Let , > 0, = 2 −1/ 1200 , let be su ciently large and let ⩾ poly( ) exp(poly(1/ )).Suppose that passes the ( , √ ) direct product tester inside with probability at least .Then choosing , ∼ ⌊1/ 80 ⌋ uniformly and running Algorithm 1 with parameters and ′ = −100 on and on all /2 sub-faces, with probability 1 − ( 68 ) the algorithm outputs a list [ ] such that: (1) Non-empty, short list: In words, for each function in the list of , projecting it onto a random ⊆ /2 yields a function which is very close to a function in the list of .
(5) Upwards consistent: In words, choosing a random ⊆ , every function in the list [ ] is close to a projection of some function from the list [ ].
Proof of ( 4 5 For general dense -CSPs they incur a exp(2 2 ) dependence in , which comes from the fact that there can be 2 2 constraints in Ψ that can be satis ed by setting a particular set of variables ⊂ [ ] to a xed assignment ∈ {0, 1} .In our setting, there could only be one constraint that gets satis ed by such xing, and therefore we do not incur this triple-exponential dependence on (though this wouldn't matter for us in any case).
Proof of (5): Note that the list 2 has size at most 1/ ′ , hence with probability at least 1 − ( 74 ) over the choice of , we get that + 1 ∉ 1 .This means that we have a gap: ∀ℎ, agr (ℎ, +1 ) < +1 .Condition on being chosen so that this holds; by Lemma 2.13 we get that Pr Fix a where the above holds, let [ ] be an ′ -cover as in the statement of the lemma, and take ∈ [ ]. Assume for contradiction that Δ( | , ) > ′ + Ω(log(1/ ′ ) ) for all ∈ 3 .By the maximality of the independent set 3 , we get that for all ∈ 2 \ 3 , there exists ′ ∈ 3 such that Δ( , ′ ) < ′ .Therefore if is Ω( log(1/ ′ )) + -far from all ∈ 3 , then it is Ω(log(1/ ′ ) )far from all ′ ∈ 2 and in particular from all ∈ 1 for ⩾ , ∈ Let = 2 −1/ 1200 .We sample and integers between 1 and ⌈1/ 80 ⌉ uniformly, set ′ = −100 and run the short list algorithm on each ∈ D with the parameters and .For each , with probability 1 − ( 68 ) (over the choice of , ) we get a list [ ] as in Lemma 2.11.It follows by linearity of expectation and an averaging argument that we may choose and such that we get lists [ ] for at least 1 − ( 68 ) of ∈ D such that [ ] satis es the conditions of Lemma 2.11, and we x such and henceforth.Below, we refer to a good that additionally has a list [ ] satisfying the conditions of Lemma 2.11 as very good, and we note that the probability that is very good is at least 1 − ( 68 ) − (1) = 1 − ( 68 ).For each ∈ ( /2), we x [ ] to be an ′ cover of the collection of functions : → {0, 1} such that agr ( , | ) ⩾ = ′ − ′100 .
The rst three items in the statement of Lemma 2.3 clearly hold by Lemma 2.11, and in the rest of the argument we argue that the list agreement test passes.Towards this end, consider a generation of queries for the list agreement test.Namely, sample ∼ /2 and independently sample , ′ ⊃ .We say a triple ( , , ′ ) is good if: (1) The -faces and ′ are very good.
( .The same holds when is replaced by ′ .Note that since marginally, each one of and ′ is distributed according to , we get that the rst item holds with probability 1 − ( 68 ).Note that the marginal distribution of ( , ) is the same as sampling ∼ , and then ⊆ /2 .Thus, if the rst item holds, then Δ( [ ]) ⩾ −100 ′ , hence by Claim 2.5 we get that the second item holds with probability 1 − (1).Lastly, if the rst item holds, then by Lemma 2.11 we get that the third item holds with