News: New article, with Alain Bensoussan, on the RKHSs of Kalman filtering, check it out
News: New article, with Anna Korba and Flavien Léger, on Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM check it out
News: New article, with Stéphane Gaubert, on the tropical analogues of reproducing kernels with applications to value functions, check it out
Since September 2021, I am a post-doctoral researcher at INRIA SIERRA, working with Alessandro Rudi.
I defended my PhD in July 2021. The manuscript can be found here, the slides are here.
I obtained my PhD in July 2021 from PSL MINES ParisTech (Paris), at the CAS laboratory, where I was advised by Nicolas Petit, working on shape/state constraints in optimal control and nonparametric regression through kernel methods. This revolves around optimization problems in infinite dimensions with infinitely many constraints. I am interested in the near future in studying connections between kernels and dynamical constrained/controlled/estimated systems, among which measures, continuity equations (see ICML21) and structured learning problems.
During my PhD, I defined a new positive definite kernel, dubbed the Linear-Quadratic kernel. It is a matrix-valued reproducing kernel, instrumental in Linear-Quadratic optimal control, related to the Gramian of controllability and to the dual Riccati equation (see pdf for an introduction, pdf for the application to state constraints, arxiv with Alain Bensoussan for a general formula extended to PDE control).
I have also developped a general, convex and modular framework to handle shape constraints in RKHSs (see pdf), applied for instance to trajectory reconstruction (see NeuriPS 2020 video for a quick overview in 180s). This is a line of work, originally inspired by non-crossing quantile regression, with Zoltán Szabó. We were particularly motivated by econometrics and finance studies.
I graduated from École polytechnique (X2013) in 2017, then obtained my Master degree (MVA, Mathematics-Vision-Learning) with Highest Honours after an internship with Jean-Philippe Vert (CBIO-Google) on gene network inference (based on single-cell RNA sequencing).
My research and my lyricomania, a passion I share within the association Juvenilia, do not leave me so much time to spare, but I occasionnaly paint.
PhD in Machine Learning, 2018-2021
MINES ParisTech
MS (M2) in Machine Learning, 2016-2017
ENS Paris-Saclay
MS (M1) in Applied Maths, 2013-2016
École polytechnique
-Coming talk in invited session at MTNS22, September 2022
-Gave a talk in invited session at ORCOS-VC22, July 2022
-Gave a talk in invited session at IFAC-CAO, July 2022
-Gave a talk in invited session at EURO22, July 2022
-Gave a talk in invited session at FGP2022, May 2022
-Gave a talk at Séminaire Parisien d'Optimisation, February 2022
-Gave a talk at GdT Contrôle, January 2022
-Got the award of best post-doc presentation at Lifting Inference with Kernel Embeddings LIKE22 Bern, January 2022,
-Gave a talk at PGMO DAYS 2021, December 2021
-I attended CEMRACS 2021, July-August 2021
-Gave a talk at CT 2021, July 2021, video
-Gave a talk at Congrès SMAI 2021, June 2021, slides
-Gave a talk at Learning & Adaptive Systems Group at ETH Zurich (Zurich), February 2021
-Gave a talk at Séminaire du groupe contrôle at SIERRA (INRIA Paris), January 2021, slides
-Gave a talk at Séminaire du CAS at MINES ParisTech (Paris), December 2020, slides, video
-Gave a talk at Séminaire de mathématiques appliquées du CERMICS at ENPC (Marne-la-Vallée), October 2020, slides
-Gave a talk at Séminaire DEVI at ENAC (Toulouse), October 2020, slides
-Presented a poster at SPIGL'20, information geometry summer school (Les Houches), July 2020, poster
-Presented a poster at virtual MLSS 2020 Tübingen, machine learning summer school, July 2020, slides
-Gave a talk at virtual IFAC World Congress, July 2020, slides, video
-Gave a talk at virtual European Control Conference, May 2020, slides, video
PCAF, Alain Bensoussan, The reproducing kernel Hilbert spaces underlying linear SDE Estimation, Kalman filtering and their relation to optimal control, August 2022, arXiv
PCAF, Anna Korba, Flavien Léger, Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM, June 2022, arXiv
PCAF, Alain Bensoussan, Operator-valued Kernels and Control of Infinite dimensional Dynamic Systems, June 2022, CDC 2022, arXiv
PCAF, Stability of solutions for controlled nonlinear systems under perturbation of state constraints, June 2022, IFAC CAO 2022, arXiv
PCAF, Stéphane Gaubert, Tropical reproducing kernels and optimization, May 2022, arXiv
Anna Korba, PCAF, Szymon Majewski and Pierre Ablin, Kernel Stein Discrepancy Descent, ICML 2021 (long oral), July 2021, article, arXiv, code/website, pdf
(Under revision) PCAF and Zoltan Szabo, Handling Hard Affine SDP Shape Constraints in RKHSs, January 2021, [article], arXiv, HAL, pdf
PCAF, Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods, SIAM Journal on Control and Optimization, February 2021, article, arXiv, HAL, pdf, code
PCAF, Interpreting the dual Riccati equation through the LQ reproducing kernel, Comptes Rendus - Mathématique, January 2021, article, arXiv, HAL, pdf
PCAF and Zoltan Szabo, Hard Shape-Constrained Kernel Machines, NeurIPS 2020, December 2020, article, arXiv, HAL, pdf, code
PCAF, Nicolas Petit and Zoltan Szabo, Kernel Regression for Trajectory Reconstruction of Vehicles under Speed and Inter-Vehicular Distance Constraints, Proceedings IFAC WC 2020, July 2020, article, pdf, slides, video
PCAF and Jean-Philippe Vert, Gene regulation inference from single-cell RNA-seq data with linear differential equations and velocity inference, Bioinformatics, June 2020, article, biorXiv, pdf, supp, code
PCAF and Nicolas Petit, Data-driven approximation of differential inclusions and application to detection of transportation modes, Proceedings ECC 2020, May 2020, article, pdf, slides, video
PCAF, Lipschitz regularity of the minimum time function of differential inclusions with state constraints, Systems & Control Letters, April 2020, article, pdf
Title: Estimation and Control under Constraints through Kernel Methods.
Abstract: Pointwise state and shape constraints in control theory and nonparametric estimation are difficult to handle as they often involve convex optimization problem with an infinite number of inequality constraints. Satisfaction of these constraints is critical in many applications, such as path-planning or joint quantile regression. Reproducing kernels are propitious for pointwise evaluations. However representer theorems, which ensure the numerical applicability of kernels, cannot be applied for an infinite number of evaluations. Through constructive algebraic and geometric arguments, we prove that an infinite number of affine real-valued constraints over derivatives of the model can be tightened into a finite number of second-order cone constraints when looking for functions in vector-valued reproducing kernel Hilbert spaces. We show that state-constrained Linear-Quadratic (LQ) optimal control is a shape-constrained regression over the Hilbert space of linearly-controlled trajectories defined by an explicit LQ kernel related to the Riccati matrix. The efficiency of the developed approach is illustrated on various examples from both linear control theory and nonparametric estimation. Finally, we provide some results for general differential inclusions in minimal time problems and identification of the graph of the set-valued map. Most of all we bring to light a novel connection between reproducing kernels and optimal control theory, identifying the Hilbertian kernel of linearly controlled trajectories.
Hired as top civil servant (Corps des IPEF). Specialized in:
Worked on artificial intelligence tailored to the strategies of the technical and scientific network of the French Ministry of Environment. I handed a report shortly after the Villani mission “For a meaningful Artificial Intelligence”. This report focuses on conceptualizing machine learning approaches and details its possible effects in institutions transforming due to the Digital Revolution.