Making Laplacians commute
This work addresses the challenge of analyzing multimodal data for researchers in spectral geometry and data analysis, offering an incremental extension of classical methods.
The paper tackled the problem of multimodal spectral geometry by constructing a pair of closest commuting operators to given Laplacians, enabling joint diagonalization and extension of tools like diffusion maps and spectral clustering. The result demonstrated improved capture of inherent structure in multimodal data across applications in dimensionality reduction, shape analysis, and clustering.
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. 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.