Finding Cheeger Cuts in Hypergraphs via Heat Equation

09/12/2018 ∙ by Masahiro Ikeda, et al. ∙ 0

Cheeger's inequality states that a tightly connected subset can be extracted from a graph G using an eigenvector of the normalized Laplacian associated with G. More specifically, we can compute a subset with conductance O(√(phi_G)), where ϕ_G is the minimum conductance of a set in G. It has recently been shown that Cheeger's inequality can be extended to hypergraphs. However, as the normalized Laplacian of a hypergraph is no longer a matrix, we can only approximate to its eigenvectors; this causes a loss in the conductance of the obtained subset. To address this problem, we here consider the heat equation on hypergraphs, which is a differential equation exploiting the normalized Laplacian. We show that the heat equation has a unique solution and that we can extract a subset with conductance √(phi_G) from the solution. An analogous result also holds for directed graphs.

READ FULL TEXT VIEW PDF

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

The goal of spectral clustering of graphs is to extract tightly connected communities from a given weighted graph

, where is a weight function, using eigenvectors of matrices associated with

. One of the most fundamental results in this area is Cheeger’s inequality, which relates the second-smallest eigenvalue of the normalized Laplacian of

and the conductance of . Here, the (random-walk) normalized Laplacian of is defined as , where and are the (weighted) adjacency matrix and the (weighted) degree matrix, respectively, of . Further, is a diagonal matrix with the -th element for being the (weighted) degree of . Note that all eigenvalues of are non-negative and the smallest eigenvalue is always zero, as . The conductance of a set is defined as

where is the set of edges between and , and is the volume of . Intuitively, smaller corresponds to more tightly connected . The conductance of is the minimum conductance of a set in ; that is, . Then, Cheeger’s inequality [2, 3] states that

(1)

where is the second-smallest eigenvalue of . The second inequality of (1) is algorithmic in the sense that we can compute a set with conductance of at most , which is called a Cheeger cut, in polynomial time from an eigenvector corresponding to . Moreover, Cheeger’s inequality is tight in the sense that computing a set with conductance is NP-hard [13], assuming the small set expansion hypothesis (SSEH) [12].

Several attempts to extend Cheeger’s inequality to hypergraphs have been made. To explain the known results, we first extend the concepts of conductance and the normalized Laplacian to hypergraphs. Let be a weighted hypergraph, where is a weight function. The (weighted) degree of a vertex is . For a vertex set , the conductance of is

where is the set of hyperedges intersecting both and , and has the same definition as previously. The conductance of is .

The normalized Laplacian of a hypergraph  [4, 17] is multi-valued and no longer linear (see Section 2 for a detailed definition). In the simplest setting that the hypergraph is unweighted and -regular, that is, every vertex has degree

, and the elements of the given vector

are pairwise distinct, the acts as follows: We create an undirected graph on from by adding for each hyperedge an undirected edge , where and , then return .

When holds for and , we can state that and are an eigenvalue and an eigenvector, respectively, of . As with the graph case, all eigenvalues of are non-negative and the first eigenvalue is zero as holds. Moreover, the second-smallest eigenvalue exists. Cheeger’s inequality for hypergraphs [4, 17] states that

(2)

Again, the second inequality is algorithmic. If we can compute an eigenvector corresponding to , we can obtain a Cheeger cut; that is, a set with , in polynomial time. Unlike the undirected graph case, however, only an -approximation algorithm is available for computing  [17]. Further, this approximation ratio is tight under the SSEH [4]. Hence, the following question arises naturally: Can we compute a Cheeger cut without computing and applying Cheeger’s inequality on the corresponding eigenvector?

To answer this question, we consider the following differential equation called the heat equation [4]:

()

where is an initial vector. Intuitively, we gradually diffuse values (or heat) on vertices along hyperedges so that the maximum and minimum values in each hyperedge become closer. We can show that () always has a (unique) solution for 555Previous works [4, 17] only guaranteed that it has a solution for for some . using the theory of monotone operators and evolution equations [11] (see Section 4 for details), and let be the vector at time . In particular, holds. In addition, if , we can show that holds for any , and that converges to when is connected, where (see [4, Theorem 3.4]).

For a vector , let denote the set of all sweep sets with respect to ; that is, sets of the form either or , for some . The following theorem can now be presented.

Theorem 1.

Let be a hypergraph and be a set. Then, we have

where and is a vector for which and for .

The proof of this theorem is given in Section 3. Theorem 1 means that, when is sufficiently large, we can obtain a set such that , thereby avoiding the problem of computing . Although we cannot solve the differential equation () exactly in polynomial time, we can efficiently simulate it by discretizing time using, e.g., the Euler method or the Runge-Kutta method. Indeed these methods have already been used in practice [16]. Alternatively, we can use difference approximation, developed in the theory of monotone operators and evolution equations [11], to obtain the following:

Theorem 2.

Let be a hypergraph and , and let and . Then, we can compute (a concise representation) of a solution such that for every , where , in time polynomial in , , and .

1.1 Directed graphs

We briefly discuss directed graphs here, as we can show an analogue of Theorem 1 for such graphs with almost the same proof.

For a directed graph , the degree of a vertex is and the volume of a set is . Note that we do not distinguish out-going and in-coming edges when calculating degrees. Then, the conductance of a set is defined as

where and are the sets of edges leaving and entering , respectively. Then, the conductance of is . Note that when is a directed acyclic graph.

Yoshida [16] introduced the notion of a Laplacian for directed graphs and derived Cheeger’s inequality for such graphs, which relates and the second-smallest eigenvalue of the normalized Laplacian of . As with the hypergraph case, computing is problematic, and we can apply an analogue of Theorem 1 to obtain a set of small conductance without computing . In this paper, we focus on hypergraphs for simplicity of exposition.

1.2 Proof sketch

Previously, Chung [5] presented an analogue of Theorem 1 for graphs. Here, we review the proof of that analogue, because the proof of Theorem 1 presented in Section 3 is based on that proof.

For the graph case, we consider the following differential equation:

This differential equation has a unique solution . We define a function as

When is connected, converges to irrespective of ; hence, measures the difference between and its unique stationary distribution . For a set , we define as if and otherwise. Then, we can show that

(3)

for every , where is the minimum conductance of a sweep set with respect to the vector . From the closed solution of , we observe that . Then, we have

(by triangle inequality)

Taking the logarithm yields the desired result.

The main obstacle to extending the above argument to hypergraphs is that does not have a closed-form solution as is no longer a linear operator. To overcome this obstacle, we observe that the sequence with exists, such that acts as a linear operator in each interval . Here, is the normalized Laplacian of a graph constructed from the hypergraph and the vector . Then, we can show a counterpart of (3) for each defined as , which is sufficient for our analysis.

Another obstacle is that the triangle inequality applied in the above argument is not a priori true, because may not generally be equivalent to for the hypergraph case. To derive the triangle inequality, we exploit the theory of maximal monotone operators and evolution equations [11] and borrow the concept of difference approximation of the solution.

1.3 Related work

As noted above, an analogue of Theorem 1 for graphs has been presented by Chung [5]. However, as the normalized Laplacian is a matrix for the graph case, that analysis is simpler than that presented herein. Kloster and Gleich [9] have presented a deterministic algorithm that approximately simulates the heat equation for graphs. Hence, they extracted a tightly connected subset by considering a local part of the graph only.

The concept of the Laplacian for hypergraphs has been implicitly employed in semi-supervised learning on hypergraphs in the form

, where  [8, 18]. This concept was then formally presented by Chan et al. [4] at a later time. Subsequently, the Laplacian concept was further generalized to handle submodular transformations [10, 17]; this development encompasses Laplacians for graphs, hypergraphs [4], and directed graphs [16].

Finally, we note that another type of Laplacian for hypergraphs, which essentially replaces each hyperedge with a clique, has been used in the literature [1, 14]. We stress that that Laplacian differs from the Laplacian for hypergraphs studied in this work.

1.4 Organization

The remainder of this paper is organized as follows. In Section 2, we introduce the basic concepts used throughout this paper. In Section 3, we prove Theorem 1. We show that () has a unique solution in Section 4. In Section 5, we prove the triangle inequality discussed in Section 1.2. A proof of Theorem 2 is given in Section 6.

2 Preliminaries

For a vector and a set , let . For a vector and a positive semidefinite matrix , we define and .

Let be a hypergraph. Hereafter, we omit the subscript from symbols such as , , , and when it is clear from context. For , let denote the characteristic vector of ; that is, if and otherwise. When or , we simply write and , respectively. Further, for , we define a vector as if and otherwise. When or , we simply write and , respectively. For a vector , we write to denote a vector with for each .

2.1 Normalized Laplacian for hypergraphs

Here, we formally define the (random-walk) normalized Laplacian for hypergraphs. Let be a hypergraph. For each hyperedge , we define a polytope , where denotes the convex hull of . Then, the Laplacian of is defined as

(4)

and the normalized Laplacian is defined as .

We can write more explicitly, as follows. For each , let and . Let . Then, we arbitrarily define a function such that only if and , and we have . Then, we construct a graph , where for each and for each . Note that for every . Let be the set of graphs constructed in this manner. Hence, we have .

We can understand Laplacian for hypergraphs in terms of submodular functions. Let be the cut function associated with a hyperedge ; that is, if and only if and . It is known that is submodular; that is, holds for every . Then, is the base polytope of and in (4) is chosen so that , where is the Lovász extension of . See [7] for detailed definitions of these concepts.

When is a graph, its Laplacian and the (random-walk) normalized Laplacian are defined as and , respectively. Indeed, this coincides with (4) when we regard as a hypergraph with each hyperedge having size two.

2.2 Heat equation

Let us briefly review some facts regarding the heat equation (). We say that is a solution of () if is absolutely continuous with respect to (hence, is differentiable at almost all ) and , and satisfies for almost all . As discussed in Section 4 below, the heat equation () always has a unique solution. In addition, when is connected, converges to as for any with , as mentioned previously.

Suppose that we begin the heat equation on a hypergraph with an initial vector . Then, there is a time sequence such that a weighted graph exists for each , such that the heat equation on the interval satisfies

where is the normalized Laplacian associated with . Hence, we can write the solution for as

(5)

For , it is easy to see that

Although was originally defined for , we can extend it to any using (5). Note that, when we wish to stress the initial vector, we write , , etc.

3 Proof of Theorem 1

In this section, we prove Theorem 1.

Consider the heat equation (). For each , we define a function as

Note that, what we wish to stress the initial vector , we write . As the following proposition implies, the value of indicates the difference between and the stationary distribution .

Proposition 1.

For any initial vector , , and , we have

Proof.

The second equality is obtained through a direct calculation. ∎

Theorem 1 is obtained by bounding from above and below. To obtain an upper bound, we define

Again, when we wish to stress the initial vector , we write , etc. In the following lemma, we present an upper bound on when the initial vector is for some set .

Lemma 1.

Consider the heat equation for a set . For , let be such that and let and . Then, we have

Next, we consider a lower bound on , when the initial vector is for some set . For simplicity, we write to denote .

Lemma 2.

Consider the heat equation for a set . For , let be such that and let and . Then, we have

The following lemma is useful to relate the heat equation solutions to the different initial vectors.

Lemma 3.

Let be vectors and let . Then, we have

We provide proofs of Lemmas 12, and 3 in Sections 3.23.3, and 5, respectively.

Based on these lemmas, we show the following:

Theorem 3.

Let be a hypergraph, and and . Then, we have

where .

Proof.

Let such that , , and . By Lemma 2, we have

By [4, Lemma 4.12], we have

Hence is the solution to the heat equation . Similarly, is also the solution to the heat equation . Then, Lemma 3 implies that

where the last equality follows from for any scalar and , and is a maximizer of . Let be such that , and . Then, Lemma 1 instantiated with implies

To summarize, we have obtained the following inequality:

Hence, by taking the logarithm, we have

Note that we have ; hence, and . Thus, the claim holds. ∎

The following lemma relates the conductance of a set in a hypergraph and that in a graph in .

Lemma 4.

Let be a hypergraph and be a vector. For any , we have for every . When is a sweep set with respect to , the equality is attained.

Proof.

We have

Thus, we have as . In addition, the equality holds when is a sweep set with respect to , because holds for every hyperedge . ∎

Proof of Theorem 1.

By Lemma 4, we have and . Hence, Theorem 3 implies Theorem 1. ∎

3.1 Useful lemmas

In this section, we derive several inequalities on that will be useful later. Note that the proofs are deferred to Section A. We define as

(6)
Lemma 5.

For any , we have

Lemma 6.

We have

Lemma 7.

For any , we have

3.2 Proof of Lemma 1

We first derive a lower bound on the log derivative of .

Lemma 8.

For any and , we have

Proof.

By Lemma 5, we have