Victor Kawasaki-Borruat

Victor Kawasaki-Borruat, Clara Grotehans, Pierre Vandergheynst et al.

High-dimensional data are often modeled as lying near a low-dimensional manifold. We study how to construct diffusion processes on this data manifold in the implicit setting. That is, using only point cloud samples and without access to charts, projections, or other geometric primitives. Our main contribution is a data-driven SDE that captures intrinsic diffusion on the underlying manifold while being defined in ambient space. The construction relies on estimating the diffusion's infinitesimal generator and its carré-du-champ (CDC) from a proximity graph built from the data. The generator and CDC together encode the local stochastic and geometric structure of the intended diffusion. We show that, as the number of samples grows, the induced process converges in law on the space of probability paths to its smooth manifold counterpart. We call this construction Implicit Manifold-valued Diffusions (IMDs), and furthermore present a numerical simulation procedure using Euler-Maruyama integration. This gives a rigorous basis for practical implementations of diffusion dynamics on data manifolds, and opens new directions for manifold-aware sampling, exploration, and generative modeling.

Pierre-Gabriel Berlureau, Ali Hariri, Victor Kawasaki-Borruat et al.

Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing. Existing remedies such as graph transformers or rewiring typically incur high computational cost or require altering the graph structure. We introduce a Bakry-Emery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential, inducing task-dependent propagation dynamics without modifying topology. This operator has a well-behaved spectral decomposition and acts as a drop-in replacement for standard Laplacians in spectral GNNs. Building on this insight, we develop mu-ChebNet, a spectral architecture that jointly learns the potential and Chebyshev filters, effectively bridging message-passing adaptivity and spectral efficiency. Our theoretical analysis shows how the potential modulates the spectrum, enabling control of key graph properties. Empirically, mu-ChebNet delivers consistent gains on synthetic long-range reasoning tasks, as well as real-world benchmarks, while offering an interpretable routing field that reveals how information flows through the graph. This establishes the Bakry-Emery Laplacian as a principled and efficient foundation for adaptive spectral graph learning.