Klaus Glashoff

Artiom Kovnatsky, Klaus Glashoff, Michael M. Bronstein

Numerous problems in machine learning are formulated as optimization with manifold constraints. In this paper, we propose the Manifold alternating directions method of multipliers (MADMM), an extension of the classical ADMM scheme for manifold-constrained non-smooth optimization problems and show its application to several challenging problems in dimensionality reduction, data analysis, and manifold learning.

Michael M. Bronstein, Klaus Glashoff

In this paper, we introduce heat kernel coupling (HKC) as a method of constructing multimodal spectral geometry on weighted graphs of different size without vertex-wise bijective correspondence. We show that Laplacian averaging can be derived as a limit case of HKC, and demonstrate its applications on several problems from the manifold learning and pattern recognition domain.

Michael M. Bronstein, Klaus Glashoff, Terry A. Loring

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.

Davide Eynard, Klaus Glashoff, Michael M. Bronstein et al.

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.