Making Laplacians Commute: Multimodal Spectral Geometry Using Closest Commuting Operators∗

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. We show that our problem is equivalent to joint diagonalization, however, unlike joint diagonalization, allows to explicitly preserve the properties of Laplacians. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data.