Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile

Kwok, Tsz Chiu; Lau, Lap Chi; Lee, Yin Tat

doi:10.1137/1.9781611974331.ch129

Cited by 8 publications

(16 citation statements)

References 17 publications

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“…Note that the assumption that the robust vertex expansion is large is satisfied by typical average case instances (such as the planted random model). This shows that the evolving set algorithm matches the improved analysis of the spectral partitioning algorithm in [KLL16], partially explaining the success of random walk based algorithms on nonworst-case instances. We refer the reader to [KLL16] for more discussions and motivations for robust vertex expansion.…”

Section: Vertex Expansionmentioning

confidence: 61%

“…Similar results were not known for random walks and evolving sets, as the spectral techniques in [KLL16] are not applicable. These results are proved in this paper by a new analysis of the combinatorial approach of Lovász and Simonovits [LS90].…”

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 64%

“…These results are proved in this paper by a new analysis of the combinatorial approach of Lovász and Simonovits [LS90]. We remark that both the results and the techniques of this paper are different from [KLL16], especially the combinatorial analog of spectral gap, the counterexample for the evolving set algorithm, and the new analysis of Lovász and Simonovits approach using the barchart.…”

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 71%

“…An interesting corollary is a random walk based proof of the improved Cheeger's inequality using vertex expansion proved in [KLL16] with better constant.…”

Section: Vertex Expansionmentioning

confidence: 99%

“…In [KLL16], it was shown that the pagerank algorithm performs better when the robust vertex expansion is large. Similar results were not known for random walks and evolving sets, as the spectral techniques in [KLL16] are not applicable.…”

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 99%

See 4 more Smart Citations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Chan

Kwok

Lau

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Analyzing the mixing time of random walks is a wellstudied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis.1. We define a combinatorial analog of the spectral gap, and use it to prove the convergence of nonlazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph.2. We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process, and also derives an alternative proof of an improved Cheeger's inequality.3. We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of all existing local graph partitioning algorithms in approximating small-set expansion.

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Section: Vertex Expansionmentioning

confidence: 61%

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 64%

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 71%

“…An interesting corollary is a random walk based proof of the improved Cheeger's inequality using vertex expansion proved in [KLL16] with better constant.…”

Section: Vertex Expansionmentioning

confidence: 99%

Section: Mixing Time and Local Graph Partitioningmentioning

confidence: 99%

See 3 more Smart Citations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Chan

Kwok

Lau

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Generalizing the Hypergraph Laplacian via a Diffusion Process with Mediators

Chan

Liang

2018

Lecture Notes in Computer Science

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In a recent breakthrough STOC 2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-beween densities.In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satsifying the Cheeger's inequality.Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.Spectral graph theory, and specifically, the well-known Cheeger's inequality give a relationship between the edge expansion properties of a graph and the eigenvalues of some appropriately defined matrix [1,2]. Loosely speaking, for a given graph, its edge expansion or conductance gives a lower bound on the ratio of the number of edges leaving a subset S of vertices to the sum of vertex degrees in S. It is natural that graph conductance is studied in the context of graph partitioning or clustering [9,17,18], whose goal is to minimize the weight of edges crossing different clusters with respect to intra-cluster edges. The reader can refer to the standard references [5,8] for an introduction to spectral graph theory.Recent Generalization to Hypergraphs. In an edge-weighted hypergraph H = (V, E, w), an edge e ∈ E is a non-empty subset of V . The edges have positive weights indicated by w : E → R + . The weight of each vertex v ∈ V is its weighted degree w v := e∈E:v∈e w e . A subset S of vertices has weight w(S) := v∈S w v , and the edges it cuts is ∂S := {e ∈ E : e intersects both S and V \ S}.The conductance of S ⊆ V is defined as φ(S) := w(∂S) w(S) . The conductance of H is defined as:( 1.1) Until recently, it was an open problem to define a spectral model for hypergraphs. In a breakthrough STOC 2015 paper, Louis [13] considered a continuous diffusion process on hypergraphs (which has been refined in a recent JACM paper [3]), and defined an operator L w f := − df dt , where f ∈ R V is some appropriate vector associated with the diffusion process. As in classical spectral graph theory, L w has non-negative eigenvalues, and the all-ones vector 1 is an eigenvector with eigenvalue 0. Moreover, the operator L w has a second eigenvalue γ 2 , and the Cheeger's inequality can be recovered 1 for hypergraphs:

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Clustering on Laplacian-embedded latent manifolds when clusters overlap

Chrétien

Jagan

Barton

2020

Meas. Sci. Technol.

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The purpose of clustering is to identify groups in a dataset in the hope of revealing some unforeseen latent discrete variable. Clustering however, is known to be one of the most difficult tasks in practice for two main reasons: (i) high dimensionality of data which requires appropriate feature extraction; and (ii) computational complexity of the associated optimisation problems.Spectral clustering was designed as a joint embedding and clustering technique that first embeds the data into a low dimensional space and then delineates between the clusters by considering the sign of the components of (a linearly transformed version of) the second eigenvector of a similarity matrix. Hence, spectral clustering seemingly solves the two main challenges associated with clustering problems, at least when the clusters are well separated.In this paper, we address the question of clustering when clusters overlap. In this regime, one relevant approach to clustering is to consider the modes of the point cloud distribution and, in particular, how the modes of the distribution of the raw data are mapped to the modes of the embedded data via the Laplacian eigenmap.The main contribution of the present paper is to provide a simulation study of the (approximate) mode-preserving property of Laplacian eigenmaps and how relevant the mode chasing approach is to high dimensional clustering. As a consequence of the mode-preserving property of Laplacian eigenmaps, this method is as good as finding modes in the original high dimensional space but with much better computational efficiency. The method is illustrated on simulated data and on satellite data relating to ground movement.

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Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile

Cited by 8 publications

References 17 publications

Random Walks and Evolving Sets: Faster Convergences and Limitations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Generalizing the Hypergraph Laplacian via a Diffusion Process with Mediators

Clustering on Laplacian-embedded latent manifolds when clusters overlap

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