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.
In REF , the authors have proposed to find the pair of the closest commuting operators to a given pair of graph Laplacian matrices, which constitute a commuting matrix pair, and thus have the same eigenbases.
Making Laplacians commute
ArXiv
ArXiv
Computer Science Mathematics