GSI'25

Keynote speakers

Prof. Nina MIOLANE

Assistant Professor, AI, UC Santa Barbara. Co-Director, AI Center, Bowers Women's Brain Health Initiative. Affiliate, Stanford SLAC

Topological Deep Learning: Unlocking the Structure of Relational Systems

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. By extending beyond traditional graph-based methods, TDL incorporates higher-order relational structures, providing a new lens to tackle challenges in applied sciences and beyond. This talk will introduce the core principles of TDL and provide a comprehensive review of its rapidly growing literature, with a particular focus on neural network architectures and their performance across various domains. I will present open-source implementations that make TDL methods more accessible and practical for real-world applications. All in all, this talk will showcase how TDL models can effectively capture and reason about the complexity of real-world systems, while highlighting the remaining challenges and exciting opportunities for future advancements in the field.

Bibliography

  1. Hajij, M., Papillon, M., Frantzen, F., Agerberg, J., AlJabea, I., Ballester, R., … & Miolane, N. (2024). TopoX: a suite of Python packages for machine learning on topological domains. Journal of Machine Learning Research, 25(374), 1-8.
  2. Papamarkou, T., Birdal, T., Bronstein, M. M., Carlsson, G. E., Curry, J., Gao, Y., … & Zamzmi, G. (2024). Position: Topological Deep Learning is the New Frontier for Relational Learning. In Forty-first International Conference on Machine Learning.
  3. Papillon, M., Bernárdez, G., Battiloro, C., & Miolane, N. (2024). TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks. arXiv preprint arXiv:2410.06530.
  4. Papillon, M., Sanborn, S., Hajij, M., & Miolane, N. (2023). Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks. arXiv preprint arXiv:2304.10031.
  5. Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., … & Schaub, M. T. (2022). Topological deep learning: Going beyond graph data. arXiv preprint arXiv:2206.00606.

Philip J. MORRISON

The University of Texas at Austin , Physics Department

Metriplectic Dynamics: A Geometrical Framework for Thermodynamically Consistent Dynamical Systems

Classical descriptions of matter present many fluid mechanical and kinetic theory dynamical systems. These include, e.g., the Navier-Stokes-Fourier system, the Cahn-Hilliard-Navier-Stokes system for multiphase fluid flow, and various types of collisional kinetic theories for gaseous and plasma modeling.  A desirable feature of such modeling is thermodynamic consistency, i.e., conservation of energy and production of entropy, in agreement with the first and second laws of thermodynamics. Metriplectic dynamics is a kind of dynamical system (finite or infinite) that encapsulates in a geometrical formalism such thermodynamic consistency.  An algorithmic procedure for building such theories is based on the metriplectic 4-bracket, a bracket akin to the Poisson bracket that maps phase space functions to another.  However,  the 4-bracket maps 4 such functions and has algebraic curvature symmetries.  Metriplectic 4-brackets can be constructed using the Kulkarni-Nomizu product or via a pure Lie algebraic formalism based on the Koszul connection.  The formalism algorithmically produces many known and new dynamical systems, and it provides a pathway for constructing structure preserving numerical algorithms.  

Bibliography:

  1. P. J. Morrison and M. Updike, « Inclusive Curvature-Like Framework for Describing Dissipation: Metriplectic 4-Bracket Dynamics,” Physical Review E 109, 045202 (22pp) (2024). 
  2. A. Zaidni, P. J. Morrison, and S. Benjelloun,, « Thermodynamically Consistent Cahn-Hilliard-Navier-Stokes Equations using the Metriplectic Dynamics Formalism,” Physica D 468, 134303 (11pp) (2024).
  3. N. Sato and P. J. Morrison, « A Collision Operator for Describing Dissipation in Noncanonical Phase Space,”Fundamental Plasma Physics 10, 100054 (18pp) (2024)
  4. W.  Barham, P.  J. Morrison, A.  Zaidni, « A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure, » arXiv:2410.11045v2 [physics.comp-ph] 19 Oct 2024.
  5. A. Zaidni, P. J. Morrison, « Metriplectic 4-bracket algorithm for constructing thermodynamically consistent dynamical systems, » arXiv:2501.00159v1 [physics.flu-dyn] 30 Dec 2024.

Mário A.T. FIGUEIREDO

Instituto de Telecomunicações and Instituto Superior Técnico Universidade de Lisboa, Portugal

Extended Variational Learning via Fenchel-Young Losses

Many statistical learning and inference methods, from Bayesian inference to empirical risk minimization, can be unified through a variational perspective that balances empirical risk and prior knowledge. We introduce a new, general class of variational methods based on Fenchel-Young (FY) losses. These losses, derived using Fenchel conjugation (a central tool of convex analysis), generalize the Kullback-Leibler divergence and encompass Bayesian as well as classical variational learning. This FY framework provides generalized notions of free energy, evidence, evidence lower bound, and posterior, while still enabling standard optimization techniques like alternating minimization and gradient backpropagation. This allows learning a broader class of models than previous variational formulations. This talk will review FY losses and then detail this new generalized variational inference approach to machine learning.

Bibliography:

  1. Blondel, A. Martins, V. Niculae, « Learning with Fenchel-Young losses », Journal of Machine Learning Research, vol. 21, pp. 1532-4435, 2020.
  2. Martins, M. Treviso, A. Farinhas, P. Aguiar, M. Figueiredo, M. Blondel, V. Niculae, « Sparse continuous distributions and Fenchel-Young losses », Journal of Machine Learning Research, vol. 23, pp. 1−74, 2022.
  3. A. Martins, A. Farinhas, M. Treviso, V. Niculae, P. Aguiar, M. Figueiredo, « Sparse and Continuous Attention Mechanisms », Neural Information Processing Systems (NeurIPS), 2020
  4. Sklaviadis, S. Agrawal, A. Farinhas, A. Martins, M. Figueiredo, « Fenchel-Young Variational Learning », arXiv:2502.10295, 2025.

Rita FIORESI

FaBiT, University of Bologna, Italy

A Noncommutative perspective of Graph Neural Networks

In this talk we show how the merging of different languages pertaining to K-theory, algebraic geometry, noncommutative geometry and gauge field theory can be fruitfully integrated to give a unified version of discrete differential geometry on graphs.

Bibliography:

  1. E. J. Beggs, S. Majid, Quantum Riemannian Geometry, Springer, 2020.
  2. Bodnar, C., Di Giovanni, F., Chamberlain, B. P., Lio, P., et Bronstein, M. M., Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs, ICLR 2022.
  3. M. Bronstein, J. Bruna, T. Cohen, et P. Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges, ArXiv:2104.13478, 2021.
  4. Theo Braune, Yiying Tong, François Gay-Balmaz, Mathieu Desbrun, A Discrete Exterior Calculus of Bundle-valued Forms, arXiv:2406.05383.
  5. Curry, J. M., Sheaves, Cosheaves and Applications, 2014.
  6. A. Dimakis, Folkert Mueller-Hoissen, Discrete Differential Calculus: Graphs, Topologies and Gauge Theory, J. Math. Phys. 35 (1994), 6703-6735.
  7. R. Fioresi, A. Simonetti, F. Zanchetta, Sheaves on Graphs and Their Differential Calculus, preprint 2025.
  8. J. Friedman, Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity, arXiv:cs/0512008, 2006.
  9. Hansen, J., Ghrist, R., Toward a Spectral Theory of Cellular Sheaves, J. Appl. and Comput. Topology 3, 315–358 (2019).
  10. J. Kolar, P. Michor, J. Slovak, Natural Operations in Differential Geometry, GTM, Springer Verlag, 1993.

Alice Le Brigant

Université Paris 1 Panthéon-Sorbonne

The L^p Fisher-Rao metrics and the alpha-connections

The information geometry of parametric statistical models is built around two central constructions: the Fisher-Rao metric and the family of dual $\alpha$-connections. Both have non-parametric counterparts, which we discuss in this talk. We introduce the $L^p$-Fisher-Rao metrics, a family of Finsler metrics that generalize the Fisher-Rao metric, and study their links with the $\alpha$-connections. We show that their geodesics coincide on the space of smooth densities, for $p=2/(1-\alpha)$, while they differ on the space of probability densities. This gives a new variational interpretation of $\alpha$-geodesics as being energy minimizing curves. Joint work with Martin Bauer, Yuxiu Lu and Cy Maor.

Bibliography:

  1. M. Bauer, A. Le Brigant, Y. Lu and C. Maor. « The Lp-Fisher–Rao metric and Amari-Cencov α-Connections. » Calculus of Variations and Partial Differential Equations 63.2 (2024): 56.

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