The Laplacian operator and related constructions play a pivotal role in a
wide range of machine learning and dimensionality reduction applications,
which boil down to finding eigenvectors and eigenvalues of a Laplacian
constructed on some high-dimensional manifold. Important examples include
spectral clustering, eigenmaps and diffusion maps, and diffusion metrics
measuring the ``connectivity'' of points on a manifold. These applications
have been considered mostly in the context of uni-modal data, i.e., a
single data space. However, many applications involve observations and
measurements of data done using different modalities.

In this talk, I will show how to construct an extension of diffusion
geometry to multiple modalities through joint approximate diagonalization
of Laplacian matrices. I will provide several synthetic and real examples
of manifold learning, dimensionality reduction, and clustering, demonstrating
that the joint diffusion geometry better captures the inherent structure of
multi-modal data. I will also show several applications in deformable
shape analysis.

(based on joint work with M. Bronstein, D. Eynard, K. Glashoff, and A. Kovnatsky)