Informatics Institute, University of Amsterdam and Qualcomm Technologies

https://staff.fnwi.uva.nl/m.welling/

ELLIS Board Member (European Laboratory for Learning and Intelligent Systems: https://ellis.eu/)

**Title:** Exploring Quantum Statistics for Machine Learning

**Abstract:** Quantum mechanics represents a rather bizarre theory of statistics that is very different from the ordinary classical statistics that we are used to. In this talk I will explore if there are ways that we can leverage this theory in developing new machine learning tools: can we design better neural networks by thinking about entangled variables? Can we come up with better samplers by viewing them as observations in a quantum system? Can we generalize probability distributions? We hope to develop better algorithms that can be simulated efficiently on classical computers, but we will naturally also consider the possibility of much faster implementations on future quantum computers. Finally, I hope to discuss the role of symmetries in quantum theories.

**Reference:**

Roberto Bondesan, Max Welling, Quantum Deformed Neural Networks, arXiv:2010.11189v1 [quant-ph], 21st October 2020 ; https://arxiv.org/abs/2010.11189

**Jean PETITOT**

Directeur d'Études, Centre d'Analyse et de Mathématiques, Sociales, École des Hautes Études, Paris.

Born in 1944, Jean Petitot is an applied mathematician interested in dynamical modeling in neurocognitive sciences. He is the former director of the CREA (Applied Epistemology Research Center) at the Ecole Polytechnique.

Philisopher of science http://jeanpetitot.com

**Title :** The primary visual cortex as a Cartan engine

Abstract: Cortical visual neurons detect very local geometric cues as retinal positions, local contrasts, local orientations of boundaries, etc. One of the main theoretical problem of low level vision is to understand how these local cues can be integrated so as to generate the global geometry of the images perceived, with all the well-known phenomena studied since Gestalt theory. It is an empirical evidence that the visual brain is able to perform a lot of routines belonging to differential geometry. But how such routines can be neurally implemented ? Neurons are « point-like » processors and it seems impossible to do differential geometry with them. Since the 1990s, methods of "in vivo optical imaging based on activity-dependent intrinsic signals" have made possible to visualize the extremely special connectivity of the primary visual areas, their “functional architectures.” What we called « Neurogeometry » is based on the discovery that these functional architectures implement structures such as the contact structure and the sub-Riemannian geometry of jet spaces of plane curves. For reasons of principle, it is the geometrical reformulation of differential calculus from Pfaff to Lie, Darboux, Frobenius, Cartan and Goursat which turns out to be suitable for neurogeometry.

**References:**

- Agrachev, A., Barilari, D., Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge University Press, 2020.
- Citti, G., Sarti, A., A cortical based model of perceptual completion in the roto-translation space, Journal of Mathematical Imaging and Vision, 24, 3 (2006) 307-326.
- Petitot, J., Neurogéométrie de la vision. Modèles mathématiques et physiques des architectures fonctionnelles, Les Éditions de l'École Polytechnique, Distribution Ellipses, Paris, 2008.
- Petitot, J., “Landmarks for neurogeometry”, Neuromathematics of Vision, (G. Citti, A. Sarti eds), Springer, Berlin, Heidelberg, 1-85,
- Petitot,J., Elements of Neurogeometry. Functional Architectures of Vision, Lecture Notes in Morphogenesis, Springer, 2017.
- Prandi, D., Gauthier, J.-P., A Semidiscrete Version of the Petitot Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition, https://arxiv.org/abs/1704.03069v1, 2017.

**Yvette Kosmann-Schwarzbach**

Professeur des universités honoraire ; former student of the Ecole normale supérieure Sèvres, 1960-1964; aggregation of mathematics, 1963; CNRS research associate, 1964-1969; doctorate in science, Lie derivatives of spinors, University of Paris, 1970 under supervision of André Lichnerowicz; lecturer then professor at the University of Lille (1970-1976 and 1982-1993), at Brooklyn College, New York (1979-1982), at the École polytechnique (1993-2006)

**Title:** Structures of Poisson Geometry: old and new

**Abstract:** How did the brackets that Siméon-Denis Poisson introduce in 1809 evolve into the Poisson geometry of the 1970's? What are Poisson groups and, more generally, Poisson groupoids? In what sense does Dirac geometry generalize Poisson geometry and why is it relevant for applications? I shall sketch the definition of these structures and try to answer these questions.

**References**

- P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company (1987).
- J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, second edition, Springer (1998).
- C. Laurent-Gengoux, A. Pichereau, and P. Vanhaecke, Poisson Structures, Grundlehren der mathematischen Wissenschaften 347, Springer (2013).
- Y. Kosmann-Schwarzbach, Multiplicativity from Lie groups to generalized geometry, in Geometry of Jets and Fields (K. Grabowska et al., eds), Banach Center Publications 110, 2016.
- Special volume of LMP on Poisson Geometry, guest editors, Anton Alekseev, Alberto Cattaneo, Y. Kosmann-Schwarzbach, and Tudor Ratiu, Letters in Mathematical Physics 90, 2009.
- Y. Kosmann-Schwarzbach (éd.), Siméon-Denis Poisson : les Mathématiques au service de la science, Editions de l'Ecole Polytechnique (2013).
- Y. Kosmann-Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, translated by B. E. Schwarzbach, Sources and Studies in the History of Mathematics and Physical Sciences, Springer (2011).

**Michel Broniatowski**

Sorbonne Université, Paris

**Title:** Some insights on statistical divergences and choice of models

**Abstract:** Divergences between probability laws or more generally between measures define inferential criteria, or risk functions. Their estimation makes it possible to deal with the questions of model choice and statistical inference, in connection with the regularity of the models considered; depending on the nature of these models (parametric or semi-parametric), the nature of the criteria and their estimation methods vary. Representations of these divergences as large deviation rates for specific empirical measures allow their estimation in nonparametric or semi parametric models, by making use of information theory results (Sanov's theorem and Gibbs principles), by Monte Carlo methods. The question of the choice of divergence is wide open; an approach linking nonparametric Bayesian statistics and MAP estimators provides elements of understanding of the specificities of the various divergences in the Ali-Silvey-Csiszar-Arimoto class in relation to the specific choices of the prior distributions.

**References:**

- Broniatowski, Michel ; Stummer, Wolfgang. Some universal insights on divergences for statistics, machine learning and artificial intelligence. In Geometric structures of information; Signals Commun. Technol., Springer, Cham, pp. 149.211, 2019
- Broniatowski, Michel. Minimum divergence estimators, Maximum Likelihood and the generalized bootstrap, to appear in "Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems" Entropy, 2020
- Csiszár, Imre ; Gassiat, Elisabeth. MEM pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory 45, no. 7, 2253–2270, 1999
- Liese, Friedrich; Vajda, Igor. On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52, no. 10, 4394–4412, 2006

**Maurice de Gosson**

Professor, Senior Researcher at the University of Vienna https://homepage.univie.ac.at/maurice.de.gosson

Faculty of Mathematics, NuHAG group

**Title:** Gaussian states from a symplectic geometry point of view

**Abstract:** Gaussian states play an ubiquitous role in quantum information theory and in quantum optics because they are easy to manufacture in the laboratory, and have in addition important extremality properties. Of particular interest are their separability properties. Even if major advances have been made in their study in recent years, the topic is still largely open. In this talk we will discuss separability questions for Gaussian states from a rigorous point of view using symplectic geometry, and present some new results and properties.

**References:**

- M. de Gosson, On the Disentanglement of Gaussian Quantum States by Symplectic Rotations. C.R. Acad. Sci. Paris Volume 358, issue 4, 459-462 (2020)
- M. de Gosson, On Density Operators with Gaussian Weyl symbols, In Advances in Microlocal and Time-Frequency Analysis, Springer (2020)
- M. de Gosson, Symplectic Coarse-Grained Classical and Semiclassical Evolution of Subsystems: New Theoretical Aspects, J. Math. Phys. no. 9, 092102 (2020)
- E. Cordero, M. de Gosson, and F. Nicola, On the Positivity of Trace Class Operators, to appear in Advances in Theoretical and Mathematical Physics 23(8), 2061–2091 (2019)
- E. Cordero, M. de Gosson, and F. Nicola, A characterization of modulation spaces by symplectic rotations, to appear in J. Funct. Anal. 278(11), 108474 (2020)

**Giuseppe LONGO**

Centre Cavaillès, CNRS & Ens Paris and School of Medicine, Tufts University, Boston http://www.di.ens.fr/users/longo/

**Title:** Use and abuse of "digital information" in life sciences, is Geometry of Information a way out?

**Abstract:** Since WWII, the war of coding, and the understanding of the structure of the DNA (1953), the latter has been considered as the digital encoding of the Aristotelian Homunculus. Till now DNA is viewed as the "information carrier" of ontogenesis, the main or unique player and pilot of phylogenesis. This heavily affected our understanding of life and reinforced a mechanistic view of organisms and ecosystems, a component of our disruptive attitude towards ecosystemic dynamics. A different insight into DNA as a major constraint to morphogenetic processes brings in a possible "geometry of information" for biology, yet to be invented. One of the challenges is in the need to move from a classical analysis of morphogenesis, in physical terms, to a "heterogenesis" more proper to the historicity of biology.

**References**

- Arezoo Islami, Giuseppe Longo. Marriages of Mathematics and Physics: a challenge for Biology, Invited Paper, in The Necessary Western Conjunction to the Eastern Philosophy of Exploring the Nature of Mind and Life (K. Matsuno et al., eds), Special Issue of Progress in Biophysics and Molecular Biology, Vol 131, Pages 179¬192, December 2017. (DOI) (SpaceTimeIslamiLongo.pdf)
- Giuseppe Longo. How Future Depends on Past Histories and Rare Events in Systems of Life, Foundations of Science, (DOI), 2017 (biolog-observ-history-future.pdf)
- Giuseppe Longo. Information and Causality: Mathematical Reflections on Cancer Biology. In Organisms. Journal of Biological Sciences, vo. 2, n. 1, 2018. (BiologicalConseq-ofCompute.pdf)
- Giuseppe Longo. Information at the Threshold of Interpretation, Science as Human Construction of Sense. In Bertolaso, M., Sterpetti, F. (Eds.) A Critical Reflection on Automated Science – Will Science Remain Human? Springer, Dordrecht, 2019 (Information-Interpretation.pdf)
- Giuseppe Longo, Matteo Mossio. Geocentrism vs genocentrism: theories without metaphors, metaphors without theories. In Interdisciplinary Science Reviews, 45 (3), pp. 380-405, 2020. (Metaphors-geo-genocentrism.pdf)

Informatics Institute, University of Amsterdam and Qualcomm Technologies

https://staff.fnwi.uva.nl/m.welling/

ELLIS Board Member (European Laboratory for Learning and Intelligent Systems: https://ellis.eu/)

**Title:** Exploring Quantum Statistics for Machine Learning

**Abstract:** Quantum mechanics represents a rather bizarre theory of statistics that is very different from the ordinary classical statistics that we are used to. In this talk I will explore if there are ways that we can leverage this theory in developing new machine learning tools: can we design better neural networks by thinking about entangled variables? Can we come up with better samplers by viewing them as observations in a quantum system? Can we generalize probability distributions? We hope to develop better algorithms that can be simulated efficiently on classical computers, but we will naturally also consider the possibility of much faster implementations on future quantum computers. Finally, I hope to discuss the role of symmetries in quantum theories.

**Reference:**

Roberto Bondesan, Max Welling, Quantum Deformed Neural Networks, arXiv:2010.11189v1 [quant-ph], 21st October 2020 ; https://arxiv.org/abs/2010.11189

**Jean PETITOT**

Directeur d'Études, Centre d'Analyse et de Mathématiques, Sociales, École des Hautes Études, Paris.

Born in 1944, Jean Petitot is an applied mathematician interested in dynamical modeling in neurocognitive sciences. He is the former director of the CREA (Applied Epistemology Research Center) at the Ecole Polytechnique.

Philisopher of science http://jeanpetitot.com

**Title :** The primary visual cortex as a Cartan engine

Abstract: Cortical visual neurons detect very local geometric cues as retinal positions, local contrasts, local orientations of boundaries, etc. One of the main theoretical problem of low level vision is to understand how these local cues can be integrated so as to generate the global geometry of the images perceived, with all the well-known phenomena studied since Gestalt theory. It is an empirical evidence that the visual brain is able to perform a lot of routines belonging to differential geometry. But how such routines can be neurally implemented ? Neurons are « point-like » processors and it seems impossible to do differential geometry with them. Since the 1990s, methods of "in vivo optical imaging based on activity-dependent intrinsic signals" have made possible to visualize the extremely special connectivity of the primary visual areas, their “functional architectures.” What we called « Neurogeometry » is based on the discovery that these functional architectures implement structures such as the contact structure and the sub-Riemannian geometry of jet spaces of plane curves. For reasons of principle, it is the geometrical reformulation of differential calculus from Pfaff to Lie, Darboux, Frobenius, Cartan and Goursat which turns out to be suitable for neurogeometry.

**References:**

- Agrachev, A., Barilari, D., Boscain, U., A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge University Press, 2020.
- Citti, G., Sarti, A., A cortical based model of perceptual completion in the roto-translation space, Journal of Mathematical Imaging and Vision, 24, 3 (2006) 307-326.
- Petitot, J., Neurogéométrie de la vision. Modèles mathématiques et physiques des architectures fonctionnelles, Les Éditions de l'École Polytechnique, Distribution Ellipses, Paris, 2008.
- Petitot, J., “Landmarks for neurogeometry”, Neuromathematics of Vision, (G. Citti, A. Sarti eds), Springer, Berlin, Heidelberg, 1-85,
- Petitot,J., Elements of Neurogeometry. Functional Architectures of Vision, Lecture Notes in Morphogenesis, Springer, 2017.
- Prandi, D., Gauthier, J.-P., A Semidiscrete Version of the Petitot Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition, https://arxiv.org/abs/1704.03069v1, 2017.

**Yvette Kosmann-Schwarzbach**

Professeur des universités honoraire ; former student of the Ecole normale supérieure Sèvres, 1960-1964; aggregation of mathematics, 1963; CNRS research associate, 1964-1969; doctorate in science, Lie derivatives of spinors, University of Paris, 1970 under supervision of André Lichnerowicz; lecturer then professor at the University of Lille (1970-1976 and 1982-1993), at Brooklyn College, New York (1979-1982), at the École polytechnique (1993-2006)

**Title:** Structures of Poisson Geometry: old and new

**Abstract:** How did the brackets that Siméon-Denis Poisson introduce in 1809 evolve into the Poisson geometry of the 1970's? What are Poisson groups and, more generally, Poisson groupoids? In what sense does Dirac geometry generalize Poisson geometry and why is it relevant for applications? I shall sketch the definition of these structures and try to answer these questions.

**References**

- P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company (1987).
- J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, second edition, Springer (1998).
- C. Laurent-Gengoux, A. Pichereau, and P. Vanhaecke, Poisson Structures, Grundlehren der mathematischen Wissenschaften 347, Springer (2013).
- Y. Kosmann-Schwarzbach, Multiplicativity from Lie groups to generalized geometry, in Geometry of Jets and Fields (K. Grabowska et al., eds), Banach Center Publications 110, 2016.
- Special volume of LMP on Poisson Geometry, guest editors, Anton Alekseev, Alberto Cattaneo, Y. Kosmann-Schwarzbach, and Tudor Ratiu, Letters in Mathematical Physics 90, 2009.
- Y. Kosmann-Schwarzbach (éd.), Siméon-Denis Poisson : les Mathématiques au service de la science, Editions de l'Ecole Polytechnique (2013).
- Y. Kosmann-Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, translated by B. E. Schwarzbach, Sources and Studies in the History of Mathematics and Physical Sciences, Springer (2011).

**Michel Broniatowski**

Sorbonne Université, Paris

**Title:** Some insights on statistical divergences and choice of models

**Abstract:** Divergences between probability laws or more generally between measures define inferential criteria, or risk functions. Their estimation makes it possible to deal with the questions of model choice and statistical inference, in connection with the regularity of the models considered; depending on the nature of these models (parametric or semi-parametric), the nature of the criteria and their estimation methods vary. Representations of these divergences as large deviation rates for specific empirical measures allow their estimation in nonparametric or semi parametric models, by making use of information theory results (Sanov's theorem and Gibbs principles), by Monte Carlo methods. The question of the choice of divergence is wide open; an approach linking nonparametric Bayesian statistics and MAP estimators provides elements of understanding of the specificities of the various divergences in the Ali-Silvey-Csiszar-Arimoto class in relation to the specific choices of the prior distributions.

**References:**

- Broniatowski, Michel ; Stummer, Wolfgang. Some universal insights on divergences for statistics, machine learning and artificial intelligence. In Geometric structures of information; Signals Commun. Technol., Springer, Cham, pp. 149.211, 2019
- Broniatowski, Michel. Minimum divergence estimators, Maximum Likelihood and the generalized bootstrap, to appear in "Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems" Entropy, 2020
- Csiszár, Imre ; Gassiat, Elisabeth. MEM pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory 45, no. 7, 2253–2270, 1999
- Liese, Friedrich; Vajda, Igor. On divergences and informations in statistics and information theory. IEEE Trans. Inform. Theory 52, no. 10, 4394–4412, 2006

**Maurice de Gosson**

Professor, Senior Researcher at the University of Vienna https://homepage.univie.ac.at/maurice.de.gosson

Faculty of Mathematics, NuHAG group

**Title:** Gaussian states from a symplectic geometry point of view

**Abstract:** Gaussian states play an ubiquitous role in quantum information theory and in quantum optics because they are easy to manufacture in the laboratory, and have in addition important extremality properties. Of particular interest are their separability properties. Even if major advances have been made in their study in recent years, the topic is still largely open. In this talk we will discuss separability questions for Gaussian states from a rigorous point of view using symplectic geometry, and present some new results and properties.

**References:**

- M. de Gosson, On the Disentanglement of Gaussian Quantum States by Symplectic Rotations. C.R. Acad. Sci. Paris Volume 358, issue 4, 459-462 (2020)
- M. de Gosson, On Density Operators with Gaussian Weyl symbols, In Advances in Microlocal and Time-Frequency Analysis, Springer (2020)
- M. de Gosson, Symplectic Coarse-Grained Classical and Semiclassical Evolution of Subsystems: New Theoretical Aspects, J. Math. Phys. no. 9, 092102 (2020)
- E. Cordero, M. de Gosson, and F. Nicola, On the Positivity of Trace Class Operators, to appear in Advances in Theoretical and Mathematical Physics 23(8), 2061–2091 (2019)
- E. Cordero, M. de Gosson, and F. Nicola, A characterization of modulation spaces by symplectic rotations, to appear in J. Funct. Anal. 278(11), 108474 (2020)

**Giuseppe LONGO**

Centre Cavaillès, CNRS & Ens Paris and School of Medicine, Tufts University, Boston http://www.di.ens.fr/users/longo/

**Title:** Use and abuse of "digital information" in life sciences, is Geometry of Information a way out?

**Abstract:** Since WWII, the war of coding, and the understanding of the structure of the DNA (1953), the latter has been considered as the digital encoding of the Aristotelian Homunculus. Till now DNA is viewed as the "information carrier" of ontogenesis, the main or unique player and pilot of phylogenesis. This heavily affected our understanding of life and reinforced a mechanistic view of organisms and ecosystems, a component of our disruptive attitude towards ecosystemic dynamics. A different insight into DNA as a major constraint to morphogenetic processes brings in a possible "geometry of information" for biology, yet to be invented. One of the challenges is in the need to move from a classical analysis of morphogenesis, in physical terms, to a "heterogenesis" more proper to the historicity of biology.

**References**

- Arezoo Islami, Giuseppe Longo. Marriages of Mathematics and Physics: a challenge for Biology, Invited Paper, in The Necessary Western Conjunction to the Eastern Philosophy of Exploring the Nature of Mind and Life (K. Matsuno et al., eds), Special Issue of Progress in Biophysics and Molecular Biology, Vol 131, Pages 179¬192, December 2017. (DOI) (SpaceTimeIslamiLongo.pdf)
- Giuseppe Longo. How Future Depends on Past Histories and Rare Events in Systems of Life, Foundations of Science, (DOI), 2017 (biolog-observ-history-future.pdf)
- Giuseppe Longo. Information and Causality: Mathematical Reflections on Cancer Biology. In Organisms. Journal of Biological Sciences, vo. 2, n. 1, 2018. (BiologicalConseq-ofCompute.pdf)
- Giuseppe Longo. Information at the Threshold of Interpretation, Science as Human Construction of Sense. In Bertolaso, M., Sterpetti, F. (Eds.) A Critical Reflection on Automated Science – Will Science Remain Human? Springer, Dordrecht, 2019 (Information-Interpretation.pdf)
- Giuseppe Longo, Matteo Mossio. Geocentrism vs genocentrism: theories without metaphors, metaphors without theories. In Interdisciplinary Science Reviews, 45 (3), pp. 380-405, 2020. (Metaphors-geo-genocentrism.pdf)