58th Annual IEEE Symposium on Foundations of Computer Science
High dimensional expanders imply agreement expanders
Irit Dinur Tali Kaufman Department of Computer Science and Applied Mathematics Department of Computer Science Weizmann Institute Bar-Ilan University Rehovot, Israel Ramat Gan, Israel Email: [email protected] Email: [email protected]
Abstract—We show that high dimensional expanders imply of μn elements are seen by almost all sets s ∈ X with derandomized direct product tests, with a number of subsets the correct proportion (i.e. each s should see roughly that is linear in the size of the universe. μ|s| elements). We want the subsets in X to be small, Direct product tests belong to a family of tests called agree- ment tests that are important components in PCP constructions and we want the number of subsets to be not too large. and include, for example, low degree tests such as line vs. line 2) “Agreement expansion”: There should be an agree- and plane vs. plane. ment test for X. An agreement test is a distribution For a generic hypergraph, we introduce the notion of over say pairs of subsets such that, roughly speaking, agreement expansion, which captures the usefulness of the if a given collection has high pairwise agreement on hypergraph for an agreement test. We show that explicit bounded degree agreement expanders exist, based on Ramanu- average, then it is close to being consistent with some jan complexes. global string. to Oded Goldreich, with love and admiration, on the occasion We initiate a study of the following general question: of his 60th birthday which collections of subsets X satisfy the two above prop- erties? We formulate this as a type of high dimensional Keywords-high dimensional expanders; agreement tests; di- rect product; direct sum expansion of X which we term agreement expansion, and show a construction of such an X that has only O(n) I. INTRODUCTION subsets. This paper shows that derandomized direct product tests A. Agreement Expansion - definition and main theorem can be obtained from high dimensional expanders. Direct Let [n] be a ground set and let X(d) be a collection of product tests fit into a more general family of tests called subsets of [n] which, for concreteness, can be the set of all agreement tests which include low degree agreement tests d-dimensional faces of a simplicial complex X on n vertices. such as the plane vs. plane [RS97] and line vs. line test A local assignment is a collection f = {fs} of local [AS97], and were first abstracted by Goldreich and Safra in s functions fs ∈{0, 1} , one per subset s ∈ X(d). To be clear, [GS97]. These are important components in the construction fs specifies a 0/1 value for each x ∈ s. It has no information of nearly all probabilistically checkable proofs (PCPs) and about elements x ∈ s so it is “local”. A local assignment is capture a certain local to global behavior. called global if there is a global function g :[n] →{0, 1} PCPs and agreement tests: In all efficient PCP con- such that structions we break a proof into small pieces, use inefficient ∀s ∈ X, fs ≡ g|s PCPs (i.e. PCP encodings that incur a large blowup) to encode each small piece, and then through an agreement We denote by Global = Global(X(d)) the set of global test put the pieces back together. The agreement test is assignments over X(d). needed because given the collection of encoded pieces, there An agreement-check for a pair of subsets s1,s2 checks ∼ is no guarantee that the different pieces come from the same whether their local functions agree, denoted fs1 fs2 . underlying global proof, i.e. that the proofs of each piece can Formally, be “put back together again”. The PCP system must ensure ∼ ⇔∀∈ ∩ this through agreement testing: take two pieces that have fs1 fs2 x s1 s2,fs1 (x)=fs2 (x). some overlap, and check that they agree. For this idea to It is easy to see that any local assignment that is global work we must be able to pass from good pairwise (local) passes all agreement checks. The converse is also true: a agreement to consistency with a single global proof. local assignment that passes all agreement checks must be That is, the scheme should have two features, global. 1) “Sampling property”: the collection of subsets X = An agreement test is specified by giving a distribution D {s ⊂ [n]} should be a good sampler, so that any set over pairs of subsets s1,s2. We define the agreement of a
0272-5428/17 $31.00 © 2017 IEEE 974 DOI 10.1109/FOCS.2017.94 local assignment to be the probability of agreement, Definition I.2 (One-up agreement-expander). A d +1- dimensional complex X is a c-one-up agreement-expander . ∼ agreeD(f) =Pr[fs1 fs2 ] . s1,s2∼D if its underlying graph is connected and if 1 An agreement theorem shows that if f is a local assign- Υ(X, D↑) ≥ c · d ment with agreeD(f) > 1 − ε then f is 1 − O(ε) close { } ≥ to a global assignment. Such a theorem relates two ways In other words, for every f = fs s∈X(d),ifagreeD↑ (f) of measuring the closeness of f to being global: the actual 1 − ε/d, then there is a global g : X(0) →{0, 1} such that distance dist(f,Global) and the distance we observe when Pr[fs = g|s] ≥ 1 − ε/c. looking at the “boundary”, namely the checks that fail. The s . − 1 latter we denote by disagreeD(f) =1 agreeD(f). This D↑ For the one-up distribution the factor d is necessary gives rise to the following definition of agreement expansion as can be seen from its presence also in the complete d- D of X and as a type of “Rayleigh quotient”, dimensional complex on n vertices (whose d-faces are all d +1element subsets of the vertices). We prove, disagreeD(f) Υ(X, D)= inf , (1) f dist(f,Global(X)) Theorem I.3 (Main). There exists a constant c>0 and an explicit infinite family of bounded degree complexes that are where the infimum is over all possible non-global assign- c-agreement expanders, and c-one-up agreement expanders. ments f. A lower bound on Υ implies that when the disagreement is small then the distance to global is also This theorem implies a very strong derandomization of small. This means that the test “works” in that it provides a direct product tests. Previously, the only known agreement good approximation to the actual distance of f from being test with comparable parameters was known for the complete +1 global. We are now ready to provide the formal definition d-dimensional complex [DS14] which has ≈ nd subsets. of an agreement expander, In comparison, the construction here has only Od(n) subsets. There are some known derandomizations of direct product Definition I.1 (Agreement expander). A d-dimensional tests [GS97], [IKW09] (but none have a linear number of complex X is a c-agreement-expander if its underlying subsets) which we discuss later in the introduction. graph1 is connected and if there exists a distribution D such that B. Agreement expansion from high dimensional expanders Υ(X, D) ≥ c. Our main theorem shows that high dimensional expansion implies agreement expansion. We begin by introducing high { } In other words, for every f = fs s∈X(d),if dimensional expanders. High dimensional expansion of simplicial complexes: agreeD(f) ≥ 1 − ε, A d-dimensional simplicial complex X is a hypergraph on then there is g : X(0) →{0, 1} such that n vertices such that for every hyperedge s that belongs to the hypergraph, all of its subsets also belong to the Pr[fs = g|s] ≥ 1 − ε/c. s hypergraph. Hyperedges of a simplicial complex are also called faces, and the dimension of a face is one less than The “in other words” part of the definition is a statement its cardinality. Simplicial complexes are viewed as higher that is the bread and butter of property testing: if a test dimensional analogs of graphs. It is standard to denote the − passes with probability at least 1 ε then the object is vertices of the complex by X(0), the edges by X(1) and in − 1 ε to the property. Thus, proving that a certain pair general X(i) is the collection of i-dimensional faces, which D (X, ) is an agreement expander is equivalent to showing are subsets of cardinality i+1. The following two definitions D that the property Global(X) is testable with as the test are important, distribution. • The graph underlying a complex is simply the graph obtained by keeping only the vertices and the edges of For a d +1 dimensional complex, there is one arguably the complex. most natural distribution D↑ over pairs of subsets in X(d), • The link of a face s ∈ X(i) in the complex, for i< which we shall call the one-up distribution. It is the distribu- d − 1, is itself a complex that is the neighborhood of tion obtained by choosing a random d+1 dimensional face r, s, formally defined as and then two random d-faces in it s1,s2 ⊂ r independently. (The name is explained from the point of view of s1:we Xs = {t \ s | s ⊂ t ∈ X} . move “one-dimension-up” towards r and then to s2). In recent years several distinct notions of high dimen- 1The graph underlying a complex has an edge between u and v whenever sional expansion (of simplicial complexes) have been ex- they belong to a common face. plored. Coboundary expansion, introduced by Linial and
975 Meshulam [LM06] and by Gromov [Gro10], is an ex- to satisfy the property. tension of graph expansion to higher dimensions, from Indeed this property was shown by [KKL14], [EK16] to a cohomological perspective. A relaxation of the notion imply co-systolic expansion which implies the topological of coboundary expansion which is called cosystolic ex- overlapping property. Additionally in [KM17] it was shown pansion was introduced by [EK16]. Cosystolic expansion that for a complex with the λ-skeleton expansion property it was shown [KKL14], [DKW16] to imply the topologi- holds that all its high order random walks converge rapidly cal overlapping property defined by Gromov [Gro10]. In to their stationary distribution. [KM17] a combinatorial “random-walk” type of expansion In this work we continue this approach of trying to capture was defined. This notion is concerned with the convergence a simple combinatorial property of simplicial complexes and speed of high dimensional random walks to the stationary using that in order to understand further properties of the distribution. Our work is most related to the notion stud- complex. We introduce an arguably cleaner variant of the ied in [KM17], since we essentially prove that agreement λ-skeleton expansion which we term λ-HD expansion. expansion is implied by high order random walks with Definition I.4 (λ-HD expander). A d dimensional simplicial optimal convergence rate. The work of [KM17] showed that complex is a λ-HD expander if for every i 976 † C. Technical results on the way to proving the criterion for t-step random walk: Ak,k+t = B B where agreement expansion B = Mk+t−1k+1Mk+t−2k+t−1 ···Mkk+1. Our proof of Theorem I.6 has two main components. First, we analyze high order random walks on a λ-HD expander, Once we have this description of B as a product of the namely walks that move from k-face to k-face if they M’s, the proof of the next theorem follows directly from the belong together to a (k + t)-face (see Section III for precise previous one through a telescoping product of the eigenvalue definitions). We show that these walks are strongly mixing bounds. in the sense that their spectral behavior is just like that of Theorem I.8 (Spectral gap of t-up random walk). Let X the analogous random walks on the complete complex, up be a d-dimensional λ-HD expander. Consider the random to an error term bounded by λ. walk that moves from a k-face s1 to a random k + t face The second component is a proof of agreement expansion r ⊃ s1, and then to a random k-face s2 ⊂ r. Let Ak,k+t be that proceeds by reduction to the agreement expansion of the the transition matrix of this random walk, then the second k+1 complete complex. The reduction crucially uses the strong + largest eigenvalue of Ak,k t is at most t+k+1 + O(tkλ). mixing of the high order random walks: we essentially prove that strong mixing of high order random walks suffices for Recall that we think of λ as arbitrarily small so this means k+1 inheriting the agreement expansion of the complete complex. that the eigenvalue above is nearly t+k+1 . The fact that k+1 1) Optimal high order random walks from decreasing t+k+1 can be arbitrarily small as t increases is crucial. If differences: The key to our proof is an analysis of random t k then λ(Ak,k+t) is small and this implies that the walks that move from k-face to k-face if they belong bipartite graph whose left vertices are the k-faces and whose together to a k +1face. right vertices are the (k + t)-faces is a good sampler. This sampling property drives our proof of agreement expansion. Theorem I.7 (Spectral gap of one-up random walk). Let X We remark that for the argument above to hold it is crucial be a d-dimensional λ-HD expander. For any k 977 obtaining such an incidence graph. In fact, it is interesting to test to a PCP construction is in incorporating the arbitrary compare this to the complete and to the random construction: PCP query structure into the test. In [DM11] this was • The complete construction is a construction as above done by modifying the PCP itself to fit into the agreement [n] [n] expander. for which V = k and W = d . The complete construction has the same spectral gap but |W | = This raises the question of whether a PCP test can be made n to fit into the high dimensional expanders that we study here. d Ω(n). • Every random construction that is obtained by choosing This would potentially allow using the agreement expansion a sparsification parameter p and then leaving alive in a PCP construction. Whether or not this is possible is left edges or vertices with probability p, is easily seen to to future work, but in the meantime, in this work we show for fail to give these properties. For example, if we choose the first time a derandomized direct product test with a mere ∈ [n] n linear number of subsets. We define, for every simplicial to keep each r d with probability O(n/ d ) so as to leave a linear number of subsets in W , the induced complex X, the direct product encoding corresponding to X graph will be highly disconnected. (see Definition IV.2). Our main theorem can be rephrased as a theorem about the two-query testability of this encoding, 4) Reduction to agreement expansion on the complete see Lemma IV.3. complex, using the double sampler: Given a d dimen- The direct product encoding has been used for hardness sional complex X we move to a lower dimensional skele- amplification in settings outside of PCPs, and it is possible ton X(k) consisting of all the k-faces of X, and prove that this derandomization would be useful there as well. agreement expansion for X(k). Our proof capitalizes on the fact that X(k) contains many copies of the complete The direct sum encoding is very related to the direct complex: one for every a ∈ X(d) consisting of all sets product one: for every subset s we replace f by f (x) {s ∈ X(k) | s ⊂ a}. (In fact X(k) can be viewed as a s x∈s s mod 2, i.e. we simply take the XOR of the bits. This “convex combination” of complete complexes). On each gives an encoding from n bits to |X(k)| bits, i.e. when complete sub-complex we can apply the agreement expan- we map a function on the vertices to a “cochain” which sion theorem of [DS14] to deduce that the sets s ⊂ a is a Boolean function on the k-faces. Concretely, we define must usually agree with one function g : a →{0, 1}.We a for every simplicial complex X, the direct sum encoding crucially use the double sampling property as follows, corresponding to X (see Definition IV.1). When X(k) is • Sampling from d-sets to k-sets is used to prove that for bounded-degree this encoding has “constant rate” since it ∈ many d-faces a X(d) we have high agreement inside maps n bits to O(n) bits. We show in Lemma IV.4 that if the complete sub-complex of sets contained in a. X(k) is an agreement expander then this encoding is testable • Sampling from k-sets to points is used to move from with the minimal number of 3 queries. distance ε between the global majority and ga on the Distance amplification code: Note that this encoding level of k sets, to a distance of ε/k on the level of is far from an error correcting code because of its poor points. This shrinkage in distance allows us to deduce relative distance, which is about k , but nevertheless it has ∈ n that fs agrees with the majority for every x s. the interesting distance amplification property: the distance between every two message strings w, w grows roughly k- D. Derandomized direct products and sums fold. This gives the first construction, to the best of our The study of agreement tests continues a line of work knowledge, of a distance amplification code with constant on direct product tests which are combinatorial analogs of rate that is locally testable with a constant number of queries, parallel repetition (a PCP transformation that obtains strong independent of k. gap amplification). Parallel repetition has a high cost in One can view the set {0, 1}n of possible functions on terms of the blow up which is exactly analogous to the the vertices as a code of distance 1/n that is transformed, fact that the complete complex on n vertices has ≈ nk through the direct sum encoding, to a new code whose k-faces. This lead researchers to look for “derandomized distance is Ω(k/n). If we begin with a restricted set of parallel repetition”, and unfortunately this has hit a wall in functions, say a code C ⊂{0, 1}n whose distance is δ, then that there are known limitations to generic derandomization this transformation results in a new code whose distance is [FK95]. Ω(kδ) (as long as δ<1/k), see Lemma IV.5. However, Nevertheless, in the world of direct product tests which even if C is locally testable to begin with, it is not clear are the combinatorial analog of parallel repetition deran- how to retain the local testability of the amplified code. domization is not ruled out and [IKW09] have come up with a derandomization for which they proved an agreement E. More related work testing theorem (i.e., in our terms, agreement expansion). Works on PCP agreement tests: Agreement tests were This construction was later used [DM11] for a bona fide PCP initially studied as a type of low degree test, e.g. the line construction. The difficulty in moving from an agreement vs. line test of [RS92a], [RS92b], [AS97] and the plane vs. 978 plane test of [RS97]. Goldreich and Safra [GS97] were the We remark that although finding a smaller collection of first to consider the more general question of agreement tests subsets is called a derandomization task (and this can be and listed a set of axioms that imply agreement expansion justified because we want to use fewer random bits to choose (in current terminology). They were interested in finding a a random subset in the collection), it is unlike most other smaller collection of subsets on which such a theorem holds derandomization questions studied in the context of pseu- and proved agreement expansion of a certain (derandomized) dorandom generators or extractors. The difference is that in collection of subsets. However, their result employs a weaker standard derandomization a random object with the correct notion of approximate global consistency, namely that fs ≈ size almost surely has the desired property, and the difficulty g|s instead of fs ≡ g|s. Further works [DR06], [DG08], is coming up with an explicit construction that imitates [IKW09] adopted this approximate consistency notion which the random object. Here, in contrast, a random collection is in fact inherent in the small acceptance regime. The only of linearly many subsets, also called the random sparse setting where an agreement test is known to have the (more complex, is very far from having the desired agreement natural) exact global consistency is in the work of [DS14] behavior. This is for a very similar reason to the fact that a on the complete complex. random sparse simplicial complex is not at all a good high- For the approximate global consistency notion, Impagli- dimensional expander. azzo et. al. [IKW09] suggested to look at a collection of Works on high order random walks: Combinatorial subsets that corresponds to affine subspaces inside a high- high order random walks on high dimensional expanders dimensional vector space. The collection has size that is were first defined and analysed by [KM17], who showed polynomial in the size of the ground set which is much that these walks are rapidly mixing. However the second − 1 better than the exponential size of the complete complex, largest eigenvalue bound obtained by [KM17] is 1 O( k2 ) − 1 but still far from linear and certainly at least quadratic. The and not 1 O( k ). This innocent looking difference is quite [IKW09] agreement test theorem holds also in the so-called important since only the optimal gap of 1/k (that we end small acceptance regime, also known as the 1% regime. up showing in Theorem I.7) suffices for proving the strong Extending our results on the bounded-degree complexes to sampling properties that underly our proof. this regime is an intriguing open question. In particular, we First [Fir16] studies a broad collection of high order conjecture the following to be true random walks and shows that their spectral behavior is the same as that of the infinite Affine Building. This could Conjecture I.10 (Derandomization in the 1% regime). For potentially lead to an alternate way of calculating the spectral every δ>0 there exists a d ∈ N and an infinite family gap of these walks: understand them on the infinite building X1,X2,... of sparse d-dimensional complexes, and for and then transfer the results to the finite quotient. However, each X a distribution D over pairs of subsets, such n n this path has so far not been carried out. that for each X = X the following holds. For every n We refer the reader to the work [LLP17] and the refer- f = {f } ∈ ( ),ifagreeD(f) >δthen there is a global s s X d ences therein for a broader discussion of high order random g : X(0) →{0, 1} such that Pr[f ≈ g| ] ≥ Ω(δ). s s walks. Such a result holds for the complete complex [DG08], and the subspaces complex [IKW09]. F. Discussion So far in the PCP literature essentially two constructions High dimensional expanders and PCPs: We believe are known that give non trivial agreement tests. The first, that there is a true connection between high dimensional called the direct product construction, is where X is the expanders and PCPs. These objects posses a mixture of collection of all subsets of size d, i.e. the complete complex. pseudo-randomness and structure that can not be obtained by The second, called the subspaces construction, is where any known random construction. This is in striking contrast the ground set [n] is identified with the points of some to the one dimensional case, where random graphs easily vector space F m and the subsets correspond to all fixed- give nearly optimal expanders. dimensional linear (or affine) sub-spaces of F m. Apart from We think that further exploration of the relations between these two constructions (and some very similar variants) no these two objects could be beneficial. It could well be the other construction is known and certainly not one with linear case that known high dimensional expanders can be used or nearly-linear size that so much as comes close to results to construct better PCPs, either towards the sliding scale cited above. conjecture or towards linear size PCPs and locally testable Recently agreement tests on the subspaces complex codes. Additionally, it can be the case that known PCP (Grassmann) were studied [KMS16], [DKK+16] towards constructions based on composition can be used to obtain proving strong inapproximability results and in particular new constructions of high dimensional expanders that are the so-called 2-to-1 conjecture. This may be taken as further not algebraic. Remark: although some limitations are known indication of the importance of agreement tests inside PCP regarding constructing high dimensional expanders (under constructions. some conditions only number theoretic constructions can 979 be Ramanujan) there is no limitation for constructing a the space L2(U) of functions f : U → R by generic λ-HD high dimensional expander. It would be very . E interesting to construct such an object without using repre- f,f = u[f(u)f (u)] = puf(u)f (u). sentation theory; This could possibly be achieved through u∈U PCP techniques. and similarly on the space L2(V ). 2 Agreement expansion is a kind of approximate coho- • There are two natural linear operators A : L (U) → mology with local coefficients: Functions on a topological L2(V ) and A† : L2(V ) → L2(U) that are associated space are sometimes easier to specify by giving them as a with G. These are the conditional expectation operators collection of local functions, one per small part of the space. given by, It is required of course that the different local functions agree 2 . ∀f ∈ L (U),Af(v) = E | [f(u)], on the intersection of their domains. This is called the sheaf u v ∀ ∈ 2 † . E condition, and corresponds exactly to our notion of agree- g L (V ),Ag(u) = v|u[g(v)], ment. If the collection of local functions satisfies agreement perfectly then it is a global section (or a cohomology with or, in terms of the normalized adjacency matrix, Auv = puv and (A†) = puv . local coefficients). In this language what we are studying is pu vu pv a notion of “approximate sections”. In case U = V and the distribution is symmetric (i.e. The fact that agreement testing has a natural countepart puv = pvu which corresponds to an undirected graph), then in topology (although exact and not approximate), hints to- it is more natural to view G simply as an undirected graph wards promising relations between these seemingly different instead of connecting two copies of V . Indeed there will areas. be only one marginal distribution on the vertices and only one (self-adjoint) operator A = A†, and so this definition G. Organization coincides with that of a Markov operator for undirected non- The rest of the paper contains some preliminaries followed bipartite graphs. by two sections. The analysis of higher order random walks One can check that for every f ∈ L2(U),g ∈ L2(V ) is given in Section III. We skip the proofs of the main E † theorems (Theorem I.3 and Theorem I.6) and of existence Af, g = xy[f(x)g(y)] = f,A g of sparse λ-HD-expanders of all dimensions (Lemma I.5). justifying the notation. Note that the inner product on the These can be found in the full version of the manuscript. right is over the space L2(U) whereas the inner product We conclude with Section IV about application to direct on the left is over the space L2(V ). The following claim sum and direct product. justifies the use of the term Markov operator, 2 2 II. PRELIMINARIES Claim II.2. Let A : L (U) → L (V ) be a Markov operator ∈ 2 2 ≤ 2 A. Markov operators and singular values as defined above. Then for every f L (U), Af f and also A1 = 1. The following is a slight generalization of the theory of spectral decomposition of graphs to the case of bipar- It now makes sense to consider the space of functions 1 tite graphs with nonnegative weights. By normalizing the orthogonal to and upper bound Af / f in this space, weights to sum up to one we can always think of such a Definition II.3 (Second largest singular value). Let A be a bipartite graph as a probability distribution over pairs of Markov operator. Define vertices (u, v) ∈ U × V . Af Throughout the paper we will be working with Markov λ(A)=sup . ⊥1 f operators that are defined via a distribution. We define this f next, We remark that it also holds that Definition II.1 (Markov operator of a bipartite graph). Af, g λ(A)= sup · . (3) Let G =(U, V, E) be a bipartite graph, and assume that f,g⊥1 f g each edge carries a non-negative weight puv such that Clearly this second definition is only larger because one can u,v puv =1. plug in g = Af, observing that if f ⊥ 1 then also Af ⊥ 1. • The probability distribution {puv} induces a marginal For the other direction use Cauchy Schwartz. probability distribution on U and similarly on V given The following definition coincides with the standard def- by inition, but is slightly more general as it pertains to general pu = puv,pv = puv and not necessarily uniform edge distribution, ∈ ∈ v V u U Definition II.4 (λ-expander). A bipartite graph G = All expectations on U, V are with respect to these (U, V, E) is called a λ-expander if λ(A) ≤ λ, where distributions. Moreover, we define an inner product on A : L2(U) → L2(V ) is the associated Markov operator. 980 A non-bipartite graph G =(V,E) is called a λ-expander The one-dimensional skeleton of a complex X is the graph if λ(A) ≤ λ, where A : L2(V ) → L2(V ) is the associated whose vertices are X(0) and whose edges are X(1). The k- Markov operator. dimensional skeleton of a complex X is the k-dimensional complex whose i-faces are X(i) for every 0 ≤ i ≤ k. 1) Concatenation of two Markov operators: Let U, V, W High dimensional expanders: be three vertex sets. Let G =(U, V, E) and G = (V,W,E) each be a bipartite graph with a probability Proposition II.7 (Ramanujan complexes of [LSV05b], distribution P and P on the respective sets of edges E [LSV05a]). For every d ∈ N and every γ>0 there 2 1 O(d ) and E . Assume further that the marginal distribution that is a number c =(γ ) and an infinite sequence of P induces on V is identical to the marginal distribution that explicitly constructible d-dimensional simplicial complexes 1 2 t t t P induces on V . Define the bipartite graph G =(U, W, E) X ,X ,...where X is on nt vertices and |X (d)|≤c·n , with edge distribution P defined by and for each t, X = Xt has the following properties. For each i Xs = {t \ s | s ⊂ t ∈ X} . III. RANDOM WALKS ON SIMPLICIAL COMPLEXES A. Random walks on simplicial complexes More accurately, we will give each top face in Xs a proba- bility proportional to its probability in X (but renormalized Let X be a pure simplicial complex of dimension d, so that the probabilities sum to 1). and let Dd be an arbitrary probability distribution on X(d). 981 We extend Dd to a natural probability distribution D over B. Random walks on X(k) sequences Define random walk distributions on pairs (s1,s2) ∈ X(k) by defining their Markov operators: sd ⊃ sd−1 ⊃ ...⊃ s1 ⊃ s0,si ∈ X(i) Mkk := Mk+1 k Mkk+1 and Simply choose sd ∈ X(d) according to the distribution Dd, and then sd−1 ⊂ sd by removing a random element from sd, Mkk := Mk−1k Mk k−1 and inductively we choose si−1 ⊂ si by removing a random ∈ element from si. Since X is simplicial, si X(i) for every † † i. We observe that from the definition, and since (A A) = A†A both operators are self adjoint. Morever, since Let D be the probability distribution induced in this way i λ(A†A)=λ(A)2 (see Claim II.6) on X(i). It is easy to see that the probability of a face r ∈ X(i) is directly proportional to the weight of top faces 2 λ(Mkk)=λ(Mkk+1) and s ∈ X(d) that contain it. Note that even if Dd happens to be uniform, for i 2 2 ← and Mk t : L (X(k)) → L (X(t)) defined by Plugging k k +1 into the above equation and moving to the adjoint we get ∀ ∈ E s X(t),Mk tg(s):= r|sg(r) 1 2 1 / λ(M +1)=λ(M +1 ) ≤ 1 − + O(kγ) Easily check that k k k k k +2 (5) † (Mtk) = Mk t, which completes the proof. Next, we show how the lemma implies Theorem I.8. namely for every f : X(k) → R and g : X(t) → R Proof of Theorem I.8: E ··· Mk tf,g = (r,s)∼Ptk [g(r)f(s)] = f,Mtkg . Mtk = Mk−1kMk−2k−1 Mtt+1, 982 ∈ E so by relying on Lemma II.5 and plugging in the bound by setting for each vertex i [n], h1(i)= j|ih(i, j) and from (5), also let h0 = Ei[h1(i)]. Define 2 2 δ1 = E [(h1(i)−h0) ] and δ2 = E [(h(i, j)−h1(i)) ] λ(Mtk) ≤ λ(Mk−1k) · λ(Mk−2k−1) ···λ(Mtt+1) i ij k 1/2 1 where all expectations above are with respect to the normal- ≤ 1 − + O(tkγ) ≥ · − ized edge and vertex distribution of G. Then δ2 δ1 (1 λ). =t+1 1 2 IV. DERANDOMIZED DIRECT PRODUCT AND DIRECT SUM t / = + O(tkγ) ENCODINGS k The direct product and direct sum encodings are studied where the last equality is because of a telescoping argument, in various complexity settings espectially since they are and the previous inequality is true as long assuming that very useful for hardness amplification. In the direct sum γ<1/k. encoding, we map a string w ∈{0, 1}n to the string ∈{ }Y (k−1) − [n] DS(w) 0, 1 where Y (k 1) = k is the set C. Decreasing differences and proof of Lemma III.1 of all possible k-element subsets of [n], namely, Y is the Let f : X(k) → R be such that E[f]=0and E[f 2]=1. complete (k − 1)-dimensional complex on n vertices. The ≤ → R encoding is defined by Let us define, for 0 i 983 Definition IV.2 (Direct product encoding with respect to We omit the proof of this lemma, but let us explain the a simplicial complex). A simplicial complex X(k − 1) on main idea. The idea is to rely on the testing result from n vertices gives rise to the following encoding, called the [DDG+15] to show that for typical r ∈ X(k), there is one X(0) r direct product encoding, that maps strings w ∈{0, 1} function hr ∈{0, 1} whose DS encoding agrees with 1−ε to strings DS(w) ∈{0, 1}k×X(k−1) via fraction the sets s ⊂ r, |s| = k. This is enough to prove that the local function {h } ∈ ( ) has agreement at least ∀ ∈ − | r r X d s X(k 1), DPX (w)(s)=w s. agreeD(h) ≥ 1 − O(ε). We apply Theorem I.3 to deduce a global function g that agrees with most of hr and therefore where we view w|s as a k-bit string using some fixed with most of fs. ordering on the vertex set X(0). A. Distance amplification code The crucial point is that if X is a bounded degree complex, namely |X(k−1)| = O(|X(0)|), then the encoding Note that the direct sum (and the direct product encoding) length is linear in the message length, quite a big savings is far from an error correcting code because of its poor k compared to the non-derandomized situation. Agreement relative distance, which is about n , but nevertheless it has the interesting distance amplification property: the distance expansion of X impies quite directly that these encodings can be locally tested with 2 or 3 queries. between every two message strings w, w grows roughly k- fold. This gives the first construction, to the best of our Lemma IV.3 (Derandomized Direct Product - two query knowledge, of a distance amplification code with constant test). Let X(k−1) be a k−1 dimensional simplicial complex rate that is locally testable with a constant number of queries D on n vertices that is an agreement expander. Let be a that is independent of k. D ≥ D distribution for which γ(X, ) Ω(1). Then gives rise One can view the set {0, 1}n of possible functions on to a natural two-query agreement test : the vertices as a code of distance 1/n that is transformed, • Choose (s1,s2) ∼D through the direct sum encoding, to a new code whose • Read the rows of f corresponding to s1,s2 distance is Ω(k/n). If we begin with a restricted set of n • Accept iff for every x ∈ s1∩s2 the corresponding values functions, say a code C ⊂{0, 1} whose distance is δ, then agree: f[s1](x)=f[s2](x). this transformation results in a new code whose distance is Namely, if f =DS(w) for some w then the test succeeds Ω(kδ) (as long as δ<1/k), see Lemma IV.5. However, with probability 1; and if the test succeeds on f with even if C is locally testable to begin with, it is not clear probability 1 − ε then there is some w ∈{0, 1}n such how to retain the local testability of the amplified code. that for at least 1 − O(ε) of the sets s ∈ X(k − 1), We next prove a lemma showing distance amplification of the direct product encoding. This easily implies a similar f[s]=DPX (w)(s). result for the direct sum encoding as well, but we omit the The proof of this lemma is immediate from Theorem I.3. details. Using a reduction from [DDG+15] from direct sum to Lemma IV.5 (Distance Amplification). Let X(d) be a β-HD direct product, and relying on the fact that inside an r-set expander -dimensional simplicial complex on vertices, we are exactly in the setting of the complete complex as d n + and assume . Then for every ≤ and every studied in [DDG 15], we can prove β<1/d 1 984 Finally, we wish to thank Oded Goldreich for his true [Gro10] M. Gromov. 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