Consider the following random walks [KM17, DK17, KO18, DDFH18] defined111All the definitions in the introduction will be formally defined again in a more general setting in Section 2. on a simplicial complex . Initially, the random walk starts from an arbitrary face of dimension in .
Down-Up Walk: In each step , we choose a uniform random element and delete from , and set to be a uniform random face of dimension in that contains . This is called the -th down-up walk of , and its transition matrix is denoted by .
Up-Down Walk: In each step , we choose a uniform random face of dimension in that contains , and choose a uniform random element and set . This is called the -th up-down walk of , and its transition matrix is denoted by .
The stationary distribution of these random walks is the uniform distribution on the faces of dimensionin the simplicial complex . The question of interest is the mixing time of these random walks, i.e. the number of steps required for the distribution of to be close to the uniform distribution.
A graph is a simplicial complex of dimension . The transition matrix of the lazy random walk on a graph is . Fundamental results in spectral graph theory state that (i) the mixing time of the lazy random walk is small, if and only if (ii) the second eigenvalue of is small, if and only if (iii) the graph is an expander graph. See [HLW06, WLP09] for surveys on this topic.
Since the theory of expander graphs has many applications, there are various motivations in generalizing these results for graphs to simplicial complexes. Several definitions of high-dimensional expanders have been studied in the literature (e.g. [LM06, Gro10, PRT16, DKW16, KM17, Opp18]), and these results have found interesting applications in discrete geometry, complexity theory, coding theory, and property testing (e.g. [LM06, MW09, FGL11, KL14, EK16, KM16, KKL16, DK17, DHK19]).
Local Spectral Expanders
In this paper, we consider the definition of -local-spectral expanders developed in [KM17, DK17, KO18, Opp18, DDFH18] for the study of random walks on simplicial complexes. The local structures of a simplicial complex are described by its links. The link of a face is defined as the simplicial complex . The graph of the link is defined as follows: (i) each vertex in corresponds to a singleton in , (ii) two vertices have an edge in if and only if is contained in some face of , (iii) the weight of an edge is proportional to the number of maximal faces in that contains .
Informally, a simplicial complex is a -local-spectral expander if is an expander graph for every . In the following, we say is a pure simplicial complex if every maximal face of is of the same dimension, and we call this the dimension of . [-local-spectral expanders [Opp18, KO18]] A -dimensional pure simplicial complex is a -local-spectral expander if for every face of dimension up to , where denotes the second largest eigenvalue of the random walk matrix of
(where the transition probabilities are proportional to the edge weights).
Kaufman and Oppenheim [KO18] proved that the -th down-up walk and the -th up-down walk have a non-trivial spectral gap as long as the simplicial complex is a -local-spectral expander for .
[[KO18]] Let be a pure -dimensional simplicial complex. Suppose is a -local-spectral expander. Then, for every ,
Section 1 states that the spectral gap of is at least , which implies by a standard argument (see Section 2.5) that the mixing time of these walks is at most where is the size of the ground set of . For example, if , then the mixing time of is at most .
Section 1 can also be used to bound the spectral gap of certain “longer” random walks on simplicial complexes (see Section 1.2.5 and Section 1.2.5). Dinur and Kaufman [DK17] use these results with the Ramanujan complexes of [LSV05] to construct efficient agreement testers, which have applications to PCP constructions. Recently, these ideas have also found applications in coding theory [DHK19].
Oppenheim’s Trickling Down Theorem
Kaufman-Oppenheim Section 1 provides a way to bound the mixing time of the down-up walks and up-down walks. To apply the theorem, however, one needs to check that for every face of dimension at most . This is not an easy task. There are exponentially many graphs to check, and these graphs are defined implicitly where computing the edge weights involve non-trivial counting problems. A very useful result by Oppenheim [Opp18] makes this task easier, by relating the second eigenvalue of the graph of a lower-dimensional link to that of a higher-dimensional link.
[[Opp18]] Let be a pure -dimensional simplicial complex. Suppose for every face of dimension , and is connected for every face of dimension . Then, for every face of dimension , it holds that
Applying this theorem inductively, we can reduce the problem of bounding for every to bounding for only those faces of highest dimension.
[[Opp18]] Let be a pure -dimensional simplicial complex. Suppose for every face of dimension , and is connected for every face . Then, for every , and for every face of dimension , it holds that
Section 1 is useful for two reasons: First, note that the weight of every edge in for face of dimension is either zero or one, which makes the task of bounding its second eigenvalue more tractable. Second, if one can prove that for every face of dimension and is connected for every face , then one can conclude that for every face and hence the simplicial complex is a -local-spectral expander. So, the reduction of Oppenheim is basically lossless in the regime where Kaufman-Oppenheim’s Section 1 applies.
Analyzing Mixing Times of Markov Chains
Recently, Anari, Liu, Oveis Gharan, and Vinzant [ALOV19] found a striking application of Section 1 and Section 1 in proving the matroid expansion conjecture of Mihail and Vazirani [MV87], answering a long standing open question in Markov chain Monte Carlo methods.
To illustrate their result, consider the special case of sampling a random spanning tree from a graph . Let be the simplicial complex where the ground set is and each acyclic subgraph of is a face of . Then is a pure -dimensional simplicial complex, where and the spanning trees of are the maximal faces of . Note that in is exactly the natural Markov chain on the spanning trees of , where in each step we delete a uniformly random edge from the current spanning tree and add a uniformly random edge so that is a spanning tree. So, the problem of proving the Markov chain on spanning trees is fast mixing is equivalent to upper bounding of the simplicial complex .
Using the nice structures of matroids, Anari, Liu, Oveis Gharan, and Vinzant [ALOV19] showed that the graph is a complete multi-partite graph for every face of dimension , and this implies that for every face of dimension . Thus, it follows from Oppenheim’s Section 1 that for every face .222The result that every matroid complex is a -local-spectral expander was also proved by Huh and Wang [HW17], using techniques from Hodge theory for matroids [AHK18] instead of Oppenheim’s theorem. Then Kaufman-Oppenheim’s Section 1 implies that , and thus the mixing time of the Markov chain of sampling matroid bases is at most . This provides the first FPRAS for counting the number of matroid bases, and also proves that the basis exchange graph of a matroid is an expander graph.
The proof of the matroid expansion conjecture shows that the techniques developed in higher order random walks provide a new simplicial complex approach to analyze mixing times of Markov chains. It is thus natural to investigate whether this approach can be extended to other problems. Here we would like to discuss some limitations of the current techniques. It can be shown that only if is a complete multi-partite graph [God] and more generally a -local-expander is a weighted matroid complex [BH19], and so the same analysis as in [ALOV19] only works for matroids. Note that Kaufman-Oppenheim Section 1 only applies when for every face up to dimension . For many problems that we have considered, it does not hold that even when restricted to faces of dimension .
1.1 Main Result
The main motivation of this work is to extend this simplicial complex approach to analyze mixing times of more general Markov chains. Our main result is the following improved eigenvalue bound for higher order random walks.
Let be a pure -dimensional simplicial complex. Define
For any ,
The following are some remarks about Section 1.1.
A basic result is that a simplicial complex is gallery connected (i.e. ) if is connected (i.e. ) for every face of dimension up to . Section 1.1 provides a quantitative generalization of this result.
The refinement of having a different bound for links of different dimension is very useful for analyzing Markov chains. We will see some applications in Section 4.
Let be a pure -dimensional simplicial complex. For any , suppose and is connected for every face up to dimension , then
This provides a convenient way to bound the mixing time of Markov chains. Recall that the edge weights in for face of dimension are either zero or one, and so it is easier to bound their second eigenvalue. Section 1.1 states that as long as we can prove for these unweighted graphs in the highest dimension, then we can conclude that is fast mixing.
We present several applications of Section 1.1 and Section 1.1, in analyzing mixing times of Markov chains (Section 1.2.1, Section 1.2.2, Section 1.2.3), in analyzing constructions of high-dimensional expanders (Section 1.2.4), and in analyzing longer random walks (Section 1.2.5).
1.2.1 Sampling Independent Sets of Fixed Size
One of the most natural simplicial complexes to consider is the independent set complex of a graph [Mes01, AB06]. Let be a graph. The independent set complex has the vertex set as the ground set, and a subset is a face in if and only if is an independent set in with .
We are interested in bounding for this simplicial complex . The -th down-up walk corresponds to a natural Markov chain on sampling independent sets of size . Initially, the random walk starts from an arbitrary independent set of size . In each step , we choose a uniform random vertex and delete it from , and we choose a uniform random vertex so that is still an independent set of size and set . This Markov chain is known to mix in polynomial time for where is the maximum degree of , by using the path coupling technique [BD97, MU05]. We prove a more refined result using the simplicial complex approach.
 Let be a graph with maximum degree . Let be the -th down-up walk on the simplicial complex . Let be the adjacency matrix of .
It is well-known that for a graph with maximum degree , and so Theorem 1.2.1 recovers the previous result that the Markov chain is fast mixing if . There are various graph classes with smaller than , and Section 1.2.1 allows us to sample larger independent sets. For example, it is known that for planar graphs and more generally for graphs with bounded arboricity [Hay06], and also for random graphs and more generally for two-sided expander graphs [HLW06].
1.2.2 Sampling Common Independent Sets in Two Partition Matroids
A matroid on the ground set with the set of independent sets is a combinatorial object satisfying the following properties:
(containment property) if and , then ,
(extension property) if such that then there is some such that .
A partition matroid is the special case where the ground set is partitioned into disjoint blocks with parameters for , and a subset is in if and only if for .
The intersection of two matroids and over the same ground set
can be used to formulate various interesting combinatorial optimization problems[Sch03]. We are interested in the problem of sampling a uniform random common independent set of size , i.e. a random subset with .
Matroids naturally correspond to simplicial complexes. Let be the matroid intersection complex with ground set , where a subset is a face in if and only if and . The -th down-up walk of this complex corresponds to a natural Markov chain on sampling common independent sets of and of size . We show that this Markov chain is fast mixing for up to one third the size of a maximum common independent set, when and are partition matroids and there are no two elements belonging to the same block in both matroids (i.e. there are no two elements such that and are in the same block in and also in the same block in ).
 Let and be two given partition matroids with a common independent set of size and no two elements belonging to the same block in both matroids. If , then
where is the -th down-up walk on the matroid intersection complex .
The proof of Section 1.2.2 reveals an interesting property of the links of the simplicial complex . For any face of dimension , we show that the graph is the complement of the line graph of a bipartite graph. We note that this holds for any two matroids, not just for partition matroids. By the additional assumptions that the two matroids are partition matroids and there are no two elements in the same block in both matroids, the graph is the line graph of a simple bipartite graph. Using the fact that the adjacency matrix of the line graph of a simple graph has minimum eigenvalue at least , we prove that as long as . We can then use Section 1.1 to conclude Section 1.2.2.
1.2.3 Sampling Independent Sets from Hardcore Distributions
Very recently, Anari, Liu, and Oveis Gharan [ALO] use Section 1.1 to prove a strong result about sampling independent sets from the hardcore distribution. Given a graph and a parameter , the problem is to sample an independent set with probability where is the partition function. An important work of Weitz [Wei06]
gave a deterministic fully polynomial time approximation scheme to estimatefor up to the “uniqueness threshold”, but the exponent of the runtime depends on the maximum degree of . It is conjectured that the natural Markov chain for sampling independent sets mixes in polynomial time up to the uniqueness threshold. Anari, Liu, and Oveis Gharan prove this conjecture and obtain a polynomial time algorithm to estimate up to the uniqueness threshold for any graph (even with unbounded maximum degree). They consider a pure -dimensional simplicial complex for sampling independent sets, and prove that for by using the techniques from correlation decay. Then it follows from Section 1.1 that the Markov chain is fast mixing. Note that it is crucial to have a different bound for links of different dimension in Section 1.1, so even when it is still possible to conclude fast mixing.
1.2.4 Combinatorial Constructions of High Dimensional Expanders
Recently, Liu, Mohanty, and Yang [LMY19]
presented an interesting combinatorial construction of a sparse simplicial complex where all higher order random walks have a constant spectral gap. Their construction is by taking a certain tensor product of a graphon vertices and a small -dimensional complete simplicial complex on vertices.
[[LMY19]] Let be a -regular triangle free graph on vertices. There is an explicit family of simplicial complexes, satisfying the following properties:
is a pure -dimensional simplicial complex with maximal faces.
The spectral gap of the graphs of dimensional links of the complex satisfies
where is the second largest eigenvalue of the normalized adjacency matrix of .
For any ,
The main technical part of their proof is in establishing Item (3) in Section 1.2.4. They use the special structures of their construction and the decomposition technique from [JST04] to bound the spectral gap of the higher order random walks. The authors ask the question whether the spectral property in Item (2) alone is enough to prove the fast mixing result in Item (3). Note that Kaufman-Oppenheim’s Section 1 does not apply in this regime.
1.2.5 Longer Random Walks and Other Applications
Consider the following generalization of the up-down walk where we take “longer” steps. Initially, the random walk starts from an arbitrary face of dimension in . In each step , we sample a uniformly random face of dimension that contains , and set to be a uniformly random subset of of dimension . We call this the -th up-down walk through the -th dimension, and denote its transition matrix by . The -th up-down walk defined before is the special case . Dinur and Kaufman [DK17] derived the following result about from the result about the ordinary up-down walks.
[[DK17]] Let be a -dimensional pure simplicial complex. If is a -local-spectral expander, then for any ,
Let be a -dimensional pure simplicial complex. If is a -local-spectral expander, then for any ,
In particular, if for some , then
Section 1.2.5 has found applications in agreement testing and coding theory [DK17, DHK19, AJQ20]. We believe that Section 1.2.5 can be of independent interest because of those applications. One potential application would be in constructing double samplers from Ramanujan complexes under a weaker expansion assumption [DK17].
1.3 Related Work
Higher Order Random Walks and Applications
Our work follows a sequence of works [KM17, DK17, Opp18, KO18, DDFH18] which use the spectral properties of the links of simplical complexes to analyze higher order random walks. Higher order random walks on simplicial complexes were first introduced by Kaufman and Mass [KM17]. They formulated related but more combinatorial notions of skeleton expansion and colorful expansion to establish fast mixing of higher order random walks. Dinur and Kaufman [DK17] introduced the definition of two-sided -local-spectral expanders, which is similar to Section 1 but requires all but the first eigenvalue to have absolute value at most (i.e. it also controls the negative eigenvalues). They used this stronger assumption to prove a similar theorem as in Section 1, and applied it to construct efficient agreement tester with applications to PCP constructions. The one-sided -local-expander in Section 1 was first studied by Oppenheim [Opp18], where he proved Section 1. Then, Kaufman and Oppenheim [KO18] strengthened the result in [DK17] and prove Section 1.
Dikstein, Dinur, Filmus and Harsha [DDFH18]
studied an alternative definition of high dimensional expanders, based on the operator norm of the difference between the (non-lazy) up-down and down-up operators. Using this definition, they show that it is possible to approximately characterize all the eigenvalues and eigenvectors of higher order random walks. Their techniques were used in[AJT19] to analyze the “swap walks” on high dimensional expanders, with applications in designing good approximation algorithms for solving constraint satisfaction problems on high-dimensional expanders. Independently, the same “swap walks” were also studied by [DD19] under the name “complement walks”, where applications in agreement testing were given.
The results in higher order random walks have also found applications in coding theory. The double samplers in [DK17] are used in [DHK19] to design an efficient algorithm to decode direct product codes over high dimensional expanders. The swap walks in [AJT19] are used in [AJQ20] to recover the same result and also to design an efficient algorithm to decode direct sum codes over high dimensional expanders.
Analyzing Mixing Times of Markov Chains
Mixing time of Markov chains is an extensively studied topic with various applications (see e.g. [WLP09, MT05]). There are several well-developed approaches to bound the mixing time of a Markov chain. Perhaps the most widely used approach is the coupling method (e.g. [Ald83, BD97]), which has applications in sampling graph colorings (e.g. [Jer95, Vig00]) and many other problems (see [WLP09]). The canonical path (or more generally multicommodity flow) method developed in [JS89, Sin92, Sin93] was used in the important problem of sampling perfect matchings in bipartite graphs [JS89, JSV04] and other problems including sampling matroid bases [FM92]. Geometric methods are used in the important problem of sampling a random point in a convex body [DFK91, LV06]. Analytical methods such as (modified) log-Sobolev inequalities and Nash inequalities [DSC96, BT06] are useful in proving sharp bounds on mixing time, e.g. a recent paper [CGM19] used a modified log-Sobolev inequality to prove optimal mixing time of the natural Markov chain on sampling matroid bases.
The simplicial complex approach studied in this paper is quite different from the above approaches. It is linear algebraic and designed to bound the second eigenvalue directly using ideas from simplicial complexes. On the other hand, the coupling method is probabilistic and designed to compare two random processes, while the canonical path method and the geometric method are designed to bound the underlying expansion of the graph or the geometric object. The analytical methods are more diffcult to apply and are not as widely applicable, but when they work they could be used to prove very sharp results.
2.1 Linear Algebra
Vectors and Inner-Products
Bold faces will be used for scalar functions, i.e. . The notation
will be reserved for the all-one vector in; the subscript may be omitted when the vector space is clear from the context.
Throughout this text, we use
to denote various probability distributions, i.e.and for . Given , we use the notations and to denote the inner-product and the norm with respect to the distribution , i.e.
We reserve for the standard inner-product. Given , we write for its -norm, and for its -norm.
Matrices and Eigenvalues
Serif faces will be used for matrices, i.e. . Let be an edge-weighted undirected graph with a weight on each edge . The adjacency matrix of is denoted by with for and for . The diagonal degree matrix of is denoted by with for . The random walk matrix of is denoted by . Note that
is a row-stochastic matrix where every row sums to one. Throughout this text, we will useto denote row-stochastic operators, where .
The adjoint of the operator , with respect to the inner-products defined by and on and , is the unique operator such that
If and , the operator is called self-adjoint if . Note that a real symmetric matrix is self-adjoint with respect to the standard inner-product.
If is a row-stochastic self-adjoint operator (with respect to the stationary distribution ), then the Markov chain described by is called reversible. The random walk operator of an edge-weighted undirected graph is described by the self-adjoint row-stochastic operator (with respect to the stationary distribution ) and is a reversible Markov chain.
Let be a self-adjoint operator with respect to the inner-product defined by . It is a fundamental result in linear algebra that has only real eigenvalues, and an orthonormal set of eigenvectors with respect to the inner-product defined by , i.e. for eigenvectors . We write for the -th largest eigenvalue of so that , and write for the smallest eigenvalue of , i.e. . The largest eigenvalue of a self-adjoint matrix with respect to the measure obeys the variational formula
Given an arbitrary operator we will write for the
-th largest singular value ofso that . It is well known that the singular values of a real operator coincide with the eigenvalues of the self-adjoint operator .
A self-adjoint operator with respect to inner-product defined by is called positive semi-definite, denoted by , if it satisfies for all . The condition is equivalent to the condition that . For self-adjoint operators and with respect to the same inner-product defined by , we will write if
This is equivalent to being positive-semidefinite, i.e. . If is just the standard inner-product, we will drop the subscript .
Let and . Then, the non-zero spectrum of coincides with that of with the same multiplicity.
Let be two self-adjoint matrices with respect to the inner-product defined by satisfying . Then, for all .
[Cauchy Interlacing Theorem] Let be a symmetric matrix and be a principal submatrix of . Let and . For any ,
[Weyl Interlacing Theorem] Let be two symmetric matrices. For any ,
2.2 Simplicial Complexes
A simplicial complex is a collection of subsets that is downward closed, i.e. if and then . The elements in are called faces/simplices of . The dimension of a face is defined as , e.g. an edge is of dimension , a vertex/singleton is of dimension , the empty set is of dimension . The collection of faces of dimension is denoted by . The dimension of a simplicial complex is defined as the maximum dimension of its faces. A -dimensional simplicial complex is called pure if every maximal face is of dimension . All simplicial complexes considered in this paper are pure.
Weighted Simplicial Complexes
A simplicial complex can be equipped with a weighted function which assigns a positive weight to each face of . We follow the formalism of [DDFH18] where the weight function is a probability distribution on the faces of the same dimension. Let be a -dimensional simplicial complex. Given a probability distribution on , we can inductively obtain probability distributions on all by considering the marginal distributions, i.e.
Equivalently, we can understand as the probability distribution of the following random process: Sample a random face using the probability distribution , and then sample a uniform random subset of in . The pair will be referred as a weighted simplicial complex. We write simply as when is the uniform distribution.
Links and Graphs
Let be a pure -dimensional weighted simplicial complex. The link of a face is the simplicial complex defined as
The probability distributions on can naturally be used to define the probability distributions on using conditional probability. Suppose . The probability distribution for is defined as
where the latter equality is obtained by applying Eq. 1 repeatedly.
Given a link , the graph is defined as the -skeleton of . More explicitly, each singleton in is a vertex in , each pair in is an edge in , and the weight of in is equal to . A simple observation is that if is a pure -dimensional simplicial complex and is the uniform distribution on , then for any the weighting on the edges of is uniform. We will use this observation in Section 4.
2.3 Local Spectral Expanders
Random Walk Matrices
The definition of local spectral expanders will be based on the random walk matrix of . Let be the adjacency matrix of . Let be the diagonal degree matrix where where the last equality is by Eq. 1. The random walk matrix of is defined as , with
The distribution is the stationary distribution of , as
The matrix is self-adjoint with respect to the inner-product defined by , as
So, have only real eigenvalues, and an orthonormal basis of eigenvectors with respect to the inner-product defined by . The largest eigenvalue of is , as and is row-stochastic.
Given a vector , we will be interested in writing it as , so that for some scalar and . It follows that . We write as the operator to map to , so that
|(projector to constant functions)|
Local Spectral Expanders and Oppenheim’s Theorem
Let be a pure -dimensional weighted simplicial complex. Define
where is the second largest eigenvalue of the operator . We say is a -local-spectral expander if for .
Oppenheim’s Theorem relates the second eigenvalue of the graph of a lower-dimensional link to that of a higher-dimensional link. It works for any weighted simplicial complex with a “balanced” weight function , where for any and any it holds that
for some constant that only depends on . Note that the weight function in Eq. 1 satisfies this condition with .
[Oppenheim’s Theorem] Let be a pure -dimenisonal weighted simplicial complex where satisfies Eq. 1. For any , if is connected for every , then
An inductive argument proves the following corollary.
[Oppenheim’s Corollary] Let be a pure -dimenisonal weighted simplicial complex where satisfies Eq. 1. If is connected for every and every , then
2.4 Higher Order Random Walks
Up and Down Operators
Let be a pure -dimensional weighted simplicial complex. In the following definitions, , , , , and .
The -th up operator is defined as
The -st down operator is defined as
And it follows that the adjoint of with respect to the inner-products defined by and is , i.e. for all .
We have stayed consistent with the notations in [DDFH18], and named and up and down operators with their right-action on functions (or vectors) in mind. However, in terms of random walks, describes a random down-movement from to , whereas describes a random up-movement from to , since the action of the probability distribution is from the left.
Down-Up Walk, Up-Down Walk, and Non-Lazy Up-Down Walk
We use the up and down operators to define three random walk operators on . The -th down-up walk and the -th up-down walk are defined as
|(down-up walk, up-down walk)|
As , it is easy to observe that these operators are positive semi-definite. One useful property of and is that they have the same non-zero spectrum with the same multiplicity by Section 2.1, and in particular