Clustering on Multi-Layer Graphs via Subspace Analysis on Grassmann Manifolds (1303.2221v1)

Abstract: Relationships between entities in datasets are often of multiple nature, like geographical distance, social relationships, or common interests among people in a social network, for example. This information can naturally be modeled by a set of weighted and undirected graphs that form a global multilayer graph, where the common vertex set represents the entities and the edges on different layers capture the similarities of the entities in term of the different modalities. In this paper, we address the problem of analyzing multi-layer graphs and propose methods for clustering the vertices by efficiently merging the information provided by the multiple modalities. To this end, we propose to combine the characteristics of individual graph layers using tools from subspace analysis on a Grassmann manifold. The resulting combination can then be viewed as a low dimensional representation of the original data which preserves the most important information from diverse relationships between entities. We use this information in new clustering methods and test our algorithm on several synthetic and real world datasets where we demonstrate superior or competitive performances compared to baseline and state-of-the-art techniques. Our generic framework further extends to numerous analysis and learning problems that involve different types of information on graphs.

• The paper introduces a novel method that maps multi-layer graph structures to Grassmann manifolds, effectively merging subspace representations.
• The paper employs spectral clustering to derive subspaces from individual graph layers and unifies them using a projection distance measure.
• The paper demonstrates robust performance on synthetic and real-world datasets, outperforming traditional multi-view clustering techniques.

Overview of "Clustering on Multi-Layer Graphs via Subspace Analysis on Grassmann Manifolds"

The paper, authored by Xiaowen Dong, Pascal Frossard, Pierre Vandergheynst, and Nikolai Nefedov, addresses the problem of clustering in datasets represented as multi-layer graphs. These graphs naturally arise in situations where entities are interconnected through various modalities, such as social interactions and geographical proximity. This paper proposes a novel method that leverages subspace analysis on Grassmann manifolds to effectively integrate information from multiple graph layers.

Methodological Approach

The core approach proposed involves a process of summarizing each layer of the multi-layer graph using subspace representations. These subspaces are derived using spectral properties of individual graph layers, aligning with traditional spectral clustering techniques. Each layer of the graph is thus mapped to a point on a Grassmann manifold, a geometrical construct that represents collections of subspaces.

The primary contribution of the paper is in the method for merging these diverse subspace representations into a unified low-dimensional subspace. This merging process uses a distance measure derived from the concept of projection distance on the Grassmann manifold. The resultant subspace captures the essential features of the original multi-layer graph, preserving inter-entity relationships across different modalities.

Experimental Results

The authors validate their proposed methodology through extensive experimentation on both synthetic and real-world datasets. Compared to baseline and existing multi-view clustering methods, the proposed approach demonstrates superior or competitive performance, confirming its efficacy. The experimental analysis also showcases how the proposed framework is robust across varying data contexts, offering a consistent and reliable clustering performance.

Theoretical Implications and Future Directions

Theoretically, the use of the Grassmann manifold introduces a robust framework for understanding relationships between various subspaces. This framework not only aids in clustering but extends to a broad set of other data analysis problems involving multi-layer structures. The methodology provides a structured way to integrate disparate views of data, maintaining essential relationships and offering a coherent summarization.

For future research, there is potential in exploring alternative subspace representations beyond those driven by spectral embeddings, such as those informed by modularity or other graph-based measures. Additionally, the framework can be further developed to incorporate a priori information about the graph layers' relative importance, potentially enhancing clustering outcomes.

Overall, the paper offers insightful advancements in the field of multi-view clustering, structured around the novel application of Grassmann manifolds, and sets a foundation for further exploration into sophisticated multi-layer data analysis methodologies.