  We construct an extension of diffusion geometry to multiple modalities
through joint approximate diagonalization of Laplacian matrices. This 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 manifold learning, retrieval, and clustering demonstrating that the
joint diffusion geometry frequently better captures the inherent structure of
multi-modal data. We also show that many previous attempts to construct
multimodal spectral clustering can be seen as particular cases of joint
approximate diagonalization of the Laplacians.
