Boolean function analysis is an essential tool in theory of computation. Traditionally, it studies functions on the Boolean cube. Recently, the scope of Boolean function analysis has been extended further, encompassing groups [EFF15b, EFF15a, Pla15, EFF17], association schemes [OW13, Fil16a, Fil16b, FM16, FKMW16, DKK18a, KMS18], error-correcting codes [BGH15], and quantum Boolean functions [MO10]. Boolean function analysis on extended domains has led to progress in learning theory [OW13] and on the unique games conjecture [KMS17, DKK18a, DKK18b, BKS19, KMS18].
Another essential tool in theory of computation is expander graphs. Recently, high-dimensional expanders (HDXs), originally constructed by Lubotzky, Samuels and Vishne [LSV05a, LSV05b], have been used in computer science, with applications to property testing [DK17], lattices [KM18] and list decoding [DHK19]. Just as expander graphs are sparse models of the complete graph, so are high-dimensional expanders sparse models of the complete hypergraph, and hence can be potentially used both for derandomization and to improve constructions of objects such as PCPs.
The goal of this work is to connect these two threads of research, by introducing Boolean function analysis on high-dimensional expanders.
We study Boolean functions on simplicial complexes. A pure -dimensional simplicial complex is a set system consisting of an arbitrary collection of sets of size together with all their subsets. The sets in a simplicial complex are called faces, and it is standard to denote by the faces of whose cardinality is . Our simplicial complexes are weightedon the top-level faces, which induces in a natural way probability distributions on for all : we choose , and then choose an -face uniformly at random. Our main object of study is the space of functions , and in particular, Boolean functions .
1.1 Random-walk based definition of high-dimensional expanders
While much of our work applies to arbitrary complexes, our goal is to study complexes which are high-dimensional expanders. There are several different non-equivalent ways to define high-dimensional expanders, generalizing different properties of expander graphs. One of the main definitions, two-sided link expansion due to Dinur and Kaufman [DK17], extends the spectral definition of expander graphs by requiring two-sided spectral expansion in every link111A related and slightly weaker notion of one-sided spectral expansion appeared in earlier works of Kaufman, Kazhdan and Lubotzky [KKL14] and Evra and Kaufman [EK16].. Dinur and Kaufman [DK17] shows how to construct complexes satisfying this definition from the Ramanujan complexes of Lubotzky, Samuels and Vishne [LSV05a, LSV05b].
We propose a new definition based on high-dimensional random walks on . Denote the real-valued function space on by . There are two natural operators and , which are defined by averaging:
The compositions and are Markov operators of two natural random walks on , the upper random walk and the lower random walk.
The first walk we consider is the upper random walk . Given a face , we choose its neighbour as follows: we pick a random conditioned on and then choose uniformly at random . Note that there is a probability of that . We define the non-lazy upper random walk by choosing conditioned on . We denote the Markov operator of the non-lazy upper walk by .
Similarly, the lower random walk is another random walk on . Here, given a face , we choose a neighbour as follows: we first choose a uniformly at random and then choose a conditioned on .
For instance, if is a graph (a -dimensional simplicial complex), then the non-lazy upper random walk is the usual adjacency walk we define on a weighted graph (i.e. traversing from vertex to vertex by an edge). The (lazy) upper random walk has probability of staying in place, and probability of going to different adjacent vertex. The lower random walk on doesn’t depend on the current vertex: it simply chooses a vertex at random according to the distribution on .
There are several works on these random walks on high-dimensional expanders, which naturally lead to analyzing both real-valued and Boolean-valued functions on , for example [KM18, DK17, KO18]. The most related work is by Kaufman and Oppenheim [KO18], who gave a correspondence between a function and a sequence of functions . This correspondence has the property that
The error in the approximation depends on the one-sided expansion of the complex.
We are now ready to give our definition of a high dimensional expander in terms of these walks.
Definition 1.1 (High-Dimensional Expander).
Let , and let be a -dimensional simplicial complex. We say is a -high dimensional expander (or -HDX) if for all , the non-lazy upper random walk is -similar to the lower random walk in operator norm in the following sense:
In the graph case, this coincides with the definition of a -two-sided spectral expander: recall that the lower walk on is by choosing two vertices independently. Thus
is the second eigenvalue of the adjacency random walk in absolute value. For, we cannot expect the upper random walk to be similar to choosing two independent faces in , since the faces always share a common intersection of elements. Instead, our definition asserts that traversing through a common -face is similar to traversing through a common -face.
We show that this new definition coincides with the aforementioned definition of two-sided link expanders, thus giving these high-dimensional expanders a new characterization. Through this characterization, we decompose real-valued functions in an approximately orthogonal decomposition that respects the upper walk and lower walk operators.
1.2 Decomposition of functions on
We being by recalling the classical decomposition of functions over the Boolean hypercube. Every function on the Boolean cube has a unique representation as a multilinear polynomial. In the case of the Boolean hypercube, it is convenient to view the domain as , in which case the above representation gives the Fourier expansion of the function. The multilinear monomials can be partitioned into “levels” according to their degree, and this corresponds to an orthogonal decomposition of a function into a sum of its homogeneous parts, , a decomposition which is a basic concept in Boolean function analysis.
These concepts have known counterparts for the complete complex, which consists of all subsets of of size at most , where . The facets (top-level faces) of this complex comprise the slice (as it is known to computer scientists) or the Johnson scheme (as it is known to coding theorists), whose spectral theory has been elucidated by Dunkl [Dun76]. For , let if and otherwise (these are the analogs of monomials). Every function on the complete complex has a unique representation as a linear combination of monomials (of various degrees) where the coefficients satisfy the following harmonicity condition: for all and all ,
(If we identify with the product of “variables” , then harmonicity of a multilinear polynomial translates to the condition .) As in the case of the Boolean cube, this unique representation allows us to orthogonally decompose a function into its homogeneous parts (corresponding to the contribution of monomials with fixed ), which plays the same essential part in the complete complex as its counterpart does in the Boolean cube. Moreover, this unique representation allows extending a function from the “slice” to the Boolean cube (which can be viewed as a superset of the “slice”), thus implying further results such as an invariance principle [FKMW16, FM16].
We generalize these concepts for complexes satisfying a technical condition we call properness, which is satisfied by both the complete complex and high-dimensional expanders. We show that the results on unique decomposition for the complete complex hold for arbitrary proper complexes, with a generalized definition of harmonicity which incorporates the distributions . In contrast to the case of the complete complex (and the Boolean cube), in the case of high-dimensional expanders the homogeneous parts are only approximately orthogonal.
The homogeneous components in our decomposition are “approximate eigenfunctions” of the Markov operators defined above, and this allows us to derive an approximate identity relating the total influence (defined through the random walks) to the norms of the components in our decomposition, in complete analogy to the same identity in the Boolean cube (expressing the total influence in terms of the Fourier expansion). All of this is summarized inTheorem 4.6.
1.3 Decomposition of posets
The decomposition we suggest in this paper holds for the more general setting of graded partially ordered sets (posets): A finite graded poset is a poset equipped with a rank function that respects the order, i.e. if then . Additionally, if is minimal with respect to elements that are greater than (i.e. covers ), then . Denoting , we can partition the poset as follows:
We consider graded posets with a unique minimum element .
Every simplicial complex is a graded poset. Another notable example is the Grassmann poset which consists of all subspaces of of dimension at most . The order is the containment relation, and the rank is the dimension minus one, . The Grassmann poset was recently studied in the context of proving the 2-to-1 games conjecture [KMS17, DKK18a, DKK18b, KMS18], where a decomposition of functions of the Grassmann poset was useful. Such a decomposition is a special case of the general decomposition theorem in this paper.
Towards our goal of decomposing functions on graded posets, we generalize the notion of random walks on as follows: A measured poset is a graded poset with a sequence of measures on the different levels , that allow us to define operators similar to the simplicial complex case (for a formal definition see Section 8). The upper random walk defined by the composition is the walk where we choose two consecutive by choosing and then independently. The lower random walk is the walk where we choose two consecutive by choosing and then independently.
Stanley studied a special case of a measured poset that is called a sequentially differential poset [Sta88]. This is a poset where
for all and some constants . There are many interesting examples of sequentially differential posets, such as the Grassmann poset and the complete complex. Definition 1.1 of a high-dimensional expander resembles an approximate version of this equation: in a simplicial complex, one may check that the non-lazy version is . Thus
is equivalent to
which suggests a relaxation of (1) to an expanding poset (eposet).
Definition 1.2 (Expanding Poset (eposet)).
Let , and let . We say is an -expanding poset (or -eposet) if for all :
As we can see, -HDX is also an -eposet, for . In Lemma 8.18 we prove that the converse is also true: every simplicial complex that is an -eposet is an -HDX, under the assumption that the probability is small.
It turns out that eposets are the correct setup to generalize our decomposition of simplicial complexes: in all eposets we can uniquely decompose functions to
where the functions
are “approximate eigenvectors” of. Furthermore, this decomposition is “approximately orthogonal”. Fixing , the error in both approximations is .
1.4 An FKN theorem
Returning to simplicial complexes, as a demonstration of the power of this setup, we generalize the fundamental result of Friedgut, Kalai, and Naor [FKN02] on Boolean functions almost of degree 1. We view this as a first step toward developing a full-fledged theory of Boolean functions on high-dimensional expanders.
An easy exercise shows that a Boolean degree 1 function on the Boolean cube is a dictator, that is, depends on at most one coordinate; we call this the degree one theorem (the easy case of the FKN Theorem with zero-error). The FKN theorem, which is the robust version of this degree one theorem, states that a Boolean function on the Boolean cube which is close to a degree 1 function is in fact close to a dictator, where closeness is measured in .
The degree one theorem holds for the complete complex as well. Recently, the third author [Fil16a] extended the FKN theorem to the complete complex. Surprisingly, the class of approximating functions has to be extended beyond just dictators.
We prove an a degree one theorem for arbitrary proper complexes, and an FKN theorem for high-dimensional expanders. In contrast to the complete complex, Boolean degree 1 functions on arbitrary complexes correspond to independent sets rather than just single points, and this makes the proof of the degree one theorem non-trivial.
Our proof of the FKN theorem for high-dimensional expanders is very different from existing proofs. It follows the same general plan as recent work on the biased Kindler–Safra theorem [DFH19]. The idea is to view a high-dimensional expander as a convex combination of small sub-complexes, each of which is isomorphic to the complete -dimensional complex on vertices. We can then apply the known FKN theorem separately on each of these, and deduce that our function is approximately well-structured on each sub-complex. Finally, we apply the agreement theorem of Dinur and Kaufman [DK17] to show that the same holds on a global level.
1.5 Our results
Our first result is a decomposition for functions on any high-dimensional expander:
Theorem 1.3 (Decomposition theorem for functions on HDX).
Let be a proper -dimensional simplicial complex.222A simplicial complex is proper if the Markov operators of the upper random walks have full rank. All high-dimensional expanders satisfy this property. Every function , for , can be written uniquely as such that:
is a linear combination of the functions for , i.e. when .
Interpreted as a function on , lies in the kernel of the Markov operator of the lower random walk .
If is furthermore a -high dimensional expander, then the above decomposition is an almost orthogonal decomposition in the following sense:
For , .
If then , and in particular .
(For an exact statement in terms of the dependence of error on , see Theorem 4.6).
In Section 8 we give a more general version of this theorem that applies to arbitrary expanding posets.
Our proof of Theorem 1.3 uses the random-walk based definition of high dimensional expander, which appears in Section 4. In Section 5 we show that this definition is equivalent to the earlier notion of a two-sided link expander due to Dinur and Kaufman [DK17], up to a constant factor:
Theorem 1.4 (Equivalence between high-dimensional expander definitions).
Equipped with the decomposition theorem, we prove the following degree one theorem and its robust version, the FKN theorem on high-dimensional expanders.
Definition 1.5 (1-skeleton).
The -skeleton of a simplicial complex is the graph whose vertices are , the -faces of the complex, and whose edges are , the -faces of the complex.
Theorem 1.6 (Degree one theorem on simplicial complexes).
Let be a -dimensional simplicial complex whose -skeleton is connected. If has degree , then is the indicator of either intersecting or not intersecting an independent set of .
Theorem 1.7 (FKN theorem on HDX (informal)).
Let be a -dimensional -HDX. If is -close (in ) to a degree function then there exists a degree function on such that .
We describe our general setup in Section 2. We describe the property of properness and its implications — a unique representation theorem and decomposition of functions into homogeneous parts — in Section 3. We introduce our definition of high-dimensional expanders in Section 4. In Section 5 we show equivalence between our definition and the earlier one of two-sided link expanders. We prove our degree one theorem in Section 6, and our FKN theorem in Section 7.
In Section 8 we define expanding posets, and through them prove that the decomposition in Theorem 3.2 is almost orthogonal. We also show that expanding posets that are simplicial complexes, are in fact high-dimensional expanders. Theorem 4.6 summarizes these results for simplicial complexes.
2 Basic setup
A -dimensional simplicial complex is a non-empty collection of sets of size at most which is closed under taking subsets. We call a set of size an -dimensional face (or -face for short), and denote the collection of all -faces by . A -dimensional simplicial complex is pure if every -face is a subset of some -face. We will only be interested in pure simplicial complexes.
Let be a pure -dimensional simplicial complex. Given a probability distribution on its top-dimensional faces , for each we define a distribution on the -faces using the following experiment: choose a top-dimensional face according to , and remove
points at random. We can couple all of these distributions to a random vectorof which the individual distributions are marginals.
Let be the space of functions on . It is convenient to define , and we also let . We turn to an inner product space by defining and the associated norm .
For , we define the Up operator as follows:333The Up and Down operators differ from the boundary and coboundary operators of algebraic topology, which operate on linear combinations of oriented faces.
where is obtained from by removing a random element. Note that if then .
Similarly, we define the Down operator for as follows:
where is obtained from by conditioning the vector on and taking the th component.
The operators are adjoint to each other. Indeed, if and then
When the domain is understood, we will use instead of . This will be especially useful when considering powers of . For example, if then
Given a face , the function is the indicator function of containing . Our definition of the Up operator guarantees the correctness of the following lemma.
Let . We can think of as a function in for all . Using this convention, .
Direct calculation shows that
and so . ∎
For , the space of harmonic functions on is defined as
We also define . We are interested in decomposing , so let us define for each ,
We can describe , a sub-class of functions of , in more concrete terms.
Every function has a representation of the form
where the coefficients satisfy the following harmonicity condition: for all ,
Furthermore, if is injective on then the representation is unique.
Suppose that . Then for some , which by definition of and the Down operator is equivalent to the condition
for all . In other words, the ’s satisfy the harmonicity condition. It is easy to check that , and so Lemma 2.1 shows that , where
Thus, is a scaling of by a non-zero constant, it follows that the coefficients also satisfy the harmonicity condition.
Now suppose that is injective on , which implies that . The foregoing shows that the dimension of the space of coefficients satisfying the harmonicity conditions is . Since , this shows that the representation is unique. ∎
3 Decomposition of the space and a convenient basis
Our decomposition theorem relies on a crucial property of simplicial complexes, properness.
A -dimensional simplicial complex is proper if (i.e. is positive definite) for all . Equivalently, if it is pure and is trivial for .
We remark that since is PSD, is equivalent to . This is because for any , we would have , implying that .
The complete -dimensional complex on points is proper iff . A pure one-dimensional simplicial complex (i.e., a graph) is proper iff it is not bipartite. Unfortunately, we are not aware of a similar characterization for higher dimensions. However, in Section 5 we show that high-dimensional expanders are proper.
We can now state our decomposition theorem.
If is a proper -dimensional simplicial complex then we have the following decomposition of :
In other words, for every function there is a unique choice of such that the functions satisfy .
We first prove by induction on that every function has a representation , where . This trivially holds when . Suppose now that the claim holds for some , and let . Since is a linear operator, we have , and therefore we can write , where and . Applying induction, we get that , where . Substituting this in completes the proof.
It remains to show that the representation is unique. Since is trivial, for . This shows that . Therefore the operator given by is not only surjective but also injective. In other words, the representation of is unique. ∎
If is a proper -dimensional simplicial complex then every function has a unique representation of the form
where the coefficients satisfy the following harmonicity conditions: for all and all :
Follows directly from Lemma 2.2. ∎
We can now define the degree of a function.
The degree of a function is the maximal cardinality of a face such that in the unique decomposition given by Corollary 3.3.
Thus a function has degree if its decomposition only involves faces whose dimension is less than . The following lemma shows that the functions , for all -dimensional faces , form a basis for the space of all functions of degree at most .
If is a proper -dimensional simplicial complex then the space of functions on of degree at most has the functions as a basis.
The space of functions on of degree at most is spanned, by definition, by the functions for . This space has dimension . Since is proper, for , and so .
Given the above, in order to complete the proof, it suffices to show that for every and , the function can be written as a linear combination of for . This will show that spans the space of functions of degree at most . Since this set contains functions, it forms a basis.
Recall that , where . If contains then it contains exactly many -faces containing , and so
This completes the proof. ∎
We call the “level ” part of , and denote the weight of above level by
We also define and .
4 How to define high-dimensional expansion?
In this section we define a class of simplicial complexes which we call -high-dimensional expanders (or -HDXs). We later show that these simplicial complexes coincide with the high-dimensional expanders defined by Dinur and Kaufman [DK17] via spectral expansion of the links. In addition, we will show the decomposition in Section 3 is almost orthogonal for -HDXs. We will define -HDXs through relations between random walks in different dimensions. It is easy to already state the definition using the operators: a -dimensional simplicial complex is said to be a -HDX if for all levels ,
The operators and induce random walks on the th level of the simplicial complex. Recall that our simplicial complexes come with distributions on the -faces.
Definition 4.1 (The upper random walk ).
Given , we choose the next set as follows:
Choose conditioned on .
Choose uniformly at random such that .
Definition 4.2 (The lower random walk ).
Given , we choose the next set as follows:
Choose uniformly at random such that .
Choose conditioned on .
It is easy to see that the stationary distribution for both these processes is . However, these random walks are not necessarily the same. For example, if , we consider the graph . The upper walk is the -lazy version of the usual adjacency random walk in a graph. The lower random walk is simply choosing two vertices independently, according the distribution . In both walks, the first step and the third step are independent given the second step. In fact, we can view the upper walk (resp. lower walk) as choosing a set (resp. ), and then choosing independently two sets given that they are contained in (resp. given that they contain ).
One property of a random walk is its laziness:
Definition 4.3 (Laziness).
Let be a random walk. The laziness of is
We say that an operator is non-lazy if .
It is easy to see that both walks have some laziness. In the upper walk, the laziness is . We can decompose as
where is the non-lazy version of , i.e. the operator representing the walk when conditioning on . The laziness of the lower version depends on the simplicial complex itself, thus it doesn’t admit a simple decomposition in the general case.
(4) can be written as
A -HDX is a simplicial complex in which the non-lazy upper walk is similar to the lower walk. Thus an equivalent way to state (3) is as follows.
Definition 4.4 (High-dimensional expander).
Let be a simplicial complex, and let . We say that is a -HDX if for all ,
This definition nicely generalizes spectral expansion in graphs, since if is a graph, is the second largest eigenvalue (in absolute value) of the normalized adjacency random walk. In Section 5 we show that this definition is equivalent to the definition of high-dimensional two-sided local spectral expanders that was extensively studied in [DK17, Opp18] and other papers.
If then any -HDX is proper, as shown by the following lemma.
Let be a -dimensional -HDX, for . Then is proper.
To prove this, we directly calculate and show that it is positive when :
and since is a -HDX,
Plugging this in (6), we get
The last part of the sum is non-negative: . Therefore, if then
Hence . ∎
4.1 Almost orthogonality of the decomposition in HDXs
Let be a -dimensional -HDX, where is small enough as a function of . For every function on for , the decomposition of Theorem 3.2 satisfies the following properties:
For , .
, and for all , .
are approximate eigenvectors with eigenvalues in the sense that .
If then .
If then is proper.
The hidden constants in the notations depend only on (and not on the size of ).
5 High-dimensional expanders are two-sided link expanders
In Section 4 we defined -HDXs, see Definition 4.4. Earlier works, such as [EK16, DK17, KO18] for example, gave a different definition of high-dimensional expanders — two-sided link expanders — based on the local link structure. We recall this other definition and prove that the two are equivalent.
Definition 5.1 (Link).
Let be a -dimensional complex with an associated probability distribution on , which induces probability distributions on . For every -dimensional face for , the link of , denoted , is the simplicial complex:
We associate with the weights such that
Definition 5.2 (Underlying graph).
Let . Given , the underlying graph is the weighted graph consisting of the first two levels of the link of . In other words, , where
The weights on the edges are given by
We can also consider directed edges, by choosing a random orientation:
We define the weight of a vertex to
We define an inner product for functions on vertices along the lines of Section 2:
We denote by the adjacency operator of the non-lazy upper-walk on , given by
The corresponding quadratic form is
By definition, fixes constant functions, and is a Markov operator. It is self-adjoint with respect to the inner product above. Thus has eigenvalues , where is the number of vertices. We define . Orthogonality of eigenspaces guarantees that
Definition 5.3 (Two-sided link expander).
Let be a simplicial complex, and let be some constant. We say that is a -two-sided link expander (called -HD expander in [DK17]) if every link of satisfies .
Theorem 5.4 ([Dk17, Lemma 1.5]).
For every and every there exists an explicit infinite family of bounded degree -dimensional complexes which are -two-sided link expanders.
Theorem 5.5 (Equivalence theorem).
Let be a -dimensional simplicial complex.
If is a -two-sided link expander, then is a -HDX.
If is a -HDX then is a -two-sided link expander.
Item 1. Assume that is a -two-sided link expander. We need to show that
for all , where is the non-lazy upper walk. Let be a function on , where . We have