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