Publications

2024

• Neural networks based learning applied to extreme statistics and sampling rare events
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  • Pachebat Jean
  , 2024. Feedforward neural networks based on Rectified linear units (ReLU) cannot efficiently approximate quantile functions which are not bounded, especially in the case of heavy-tailed distributions. This is a major issue for any use in quantile approximation or sampling scheme (such as generative modelling). In this talk we design and compare several new parametrizations for the quantiles (and for the generator of a Generative adversarial network) adapted to this framework, basing on extreme-value theory. We provide theoretical convergence results (as the complexity of the NN increases) and we illustrate the resulting methodologies with experimentations on both simulated data and real data.
• Augmented Quantization: a General Approach to Mixture Models
  
  • Sire Charlie
  • Le Riche Rodolphe
  • Rullière Didier
  • Rohmer Jérémy
  • Pheulpin Lucie
  • Richet Yann
  , 2024. The investigation of mixture models is a key to understand and visualize the distribution of multivariate data. Most mixture models approaches are based on likelihoods, and are not adapted to distribution with finite support or without a well-defined density function. This study proposes the Augmented Quantization method, which is a reformulation of the classical quantization problem but which uses the p-Wasserstein distance. This metric can be computed in very general distribution spaces, in particular with varying supports. The clustering interpretation of quantization is revisited in a more general framework. The performance of Augmented Quantization is first demonstrated through analytical toy problems. Subsequently, it is applied to a practical case study involving river flooding, wherein mixtures of Dirac and Uniform distributions are built in the input space, enabling the identification of the most influential variables.
• Subgraph densities and scaling limits of random graphs with a prescribed modular decomposition
  
  • Lenoir Théo
  , 2024. We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a consequence, we obtain the convergence of a uniform random graph in such classes to a Brownian limit object in the space of graphons. Our proofs rely on combinatorial arguments, computing generating series using the symbolic method and deriving asymptotics using singularity analysis.
• Dense and nondense limits for uniform random intersection graphs
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Féray Valentin
  • Gerin Lucas
  • Pierrot Adeline
  , 2024. We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval [0,1] endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size k (k fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique.
• Model of Cellular Growth under Stress : Emergence of Heterogeneity and Impact of the Environment
  
  • Madrid Canales Ignacio
  , 2024. This thesis focuses on understanding individual-scale cell growth under stress. Starting from the analysis of the data collected by Sebastián Jaramillo and James Broughton under the supervision of Meriem El Karoui, we have developed various models to comprehend the impact of the heterogeneous response to genotoxic stress (SOS response) on the growth of a Escherichia coli populations. We employ measure-values stochastic processes to model the dynamics of these populations.Firstly, we construct a stochastic model based on the "adder" size-control model, extended to incorporate the dynamics of the SOS response and its effect on cell division. The chosen framework is parametric, and the model is fitted by maximum likelihood to individual lineage data obtained in mother machine. This allows us to quantitatively compare inferred parameters in different environments.Next, we explore the ergodic properties of a more general model than the "adder," addressing open questions about its long-time behaviour. We consider a general deterministic flow and a fragmentation kernel that is not necessarily self-similar. We demonstrate the existence of eigenelements. Then, a Doob dollar_h_dollar-transform with the found eigenfunction reduces the problem to the study of a conservative process. Finally, by proving a "petite set" property for the compact sets of the state space, we obtain the exponential convergence.Finally, we consider a bitype age-structured model capturing the phenotypic plasticity observed in the stress response. We study the survival probability of the population and the population growth rate in constant and periodic environments. We evince a trade-off for population establishment, as well as a sensitivity with respect to the model parameters that differs for survival probability and growth rate.We conclude with an independent section, collaborative work initiated during CEMRACS 2022. We investigate numerically the spatial propagation of size-structured populations modeling the collective movement of Myxobacteria clusters via a system of reaction-diffusion equations.
• Random Lévy Looptrees and Lévy Maps
  
  • Kortchemski Igor
  • Marzouk Cyril
  , 2024. Motivated by scaling limits of random planar maps in random geometry, we introduce and study Lévy looptrees and Lévy maps. They are defined using excursions of general Lévy processes with no negative jump and extend the known stable looptrees and stable maps, associated with stable processes. We compute in particular their fractal dimensions in terms of the upper and lower Blumenthal–Getoor exponents of the coding Lévy process. In a second part, we consider excursions of stable processes with a drift and prove that the corresponding looptrees interpolate continuously as the drift varies from $-\infty$ to $\infty$, or as the self-similarity index varies from $1$ to $2$, between a circle and the Brownian tree, whereas the corresponding maps interpolate between the Brownian tree and the Brownian sphere.
• Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics
  
  • Chen Fan
  • Lin Yiqing
  • Ren Zhenjie
  • Wang Songbo
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2024, 29 (none), pp.article no. 17. We study the kinetic mean field Langevin dynamics under the functional convexity assumption of the mean field energy functional. Using hypocoercivity, we first establish the exponential convergence of the mean field dynamics and then show the corresponding N-particle system converges exponentially in a rate uniform in N modulo a small error. Finally we study the short-time regularization effects of the dynamics and prove its uniform-in-time propagation of chaos property in both the Wasserstein and entropic sense. Our results can be applied to the training of two-layer neural networks with momentum and we include the numerical experiments. (10.1214/24-EJP1079)
  DOI : 10.1214/24-EJP1079
• Exact quantization of multistage stochastic linear problems
  
  • Forcier Maël
  • Gaubert Stéphane
  • Leclère Vincent
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2024, 34 (1), pp.533-562. We show that the multistage linear problem (MSLP) with an arbitrary cost distribution is equivalent to a MSLP on a finite scenario tree. We establish this exact quantization result by analyzing the polyhedral structure of MSLPs. In particular, we show that the expected cost-to-go functions are polyhedral and affine on the cells of a chamber complex, which is independent of the cost distribution. This leads to new complexity results, showing that MSLP is fixed-parameter tractable. (10.1137/22M1508005)
  DOI : 10.1137/22M1508005
• The Dynamic Effects of Weather Shocks on Agricultural Production
  
  • Crofils Cédric
  • Gallic Ewen
  • Vermandel Gauthier
  Journal of Environmental Economics and Management, Elsevier, 2024, 130, pp.103078. This paper investigates the dynamic effects of weather shocks on monthly agricultural production in Peru, using a Local Projection framework. An adverse weather shock, measured by an excess of heat or rain, always generates a delayed negative downturn in agricultural production. The magnitude and duration of this downturn depend on several factors, including the type of crop and the timing of the shock. On average, a weather shock—a temperature shock—can cause a monthly decline of 5% to 15% in agricultural production. The response exhibit important heterogeneity across time, crop, and season dimensions, with hysteresis suggesting farmers’ adaptation over time. At the macroeconomic level, weather shocks are recessionary and entail a decline in inflation, agricultural production, exports, exchange rate and GDP. (10.1016/j.jeem.2024.103078)
  DOI : 10.1016/j.jeem.2024.103078
• Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates
  
  • Leluc Rémi
  • Dieuleveut Aymeric
  • Portier François
  • Segers Johan
  • Zhuman Aigerim
  , 2024. The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.
• Accelerating hypersonic reentry simulations using deep learning-based hybridization (with guarantees)
  
  • Novello Paul
  • Poëtte Gaël
  • Lugato David
  • Peluchon Simon
  • Congedo Pietro Marco
  Journal of Computational Physics, Elsevier, 2024, 498, pp.112700. In this paper, we are interested in the acceleration of numerical simulations. We focus on a hypersonic planetary reentry problem whose simulation involves coupling fluid dynamics and chemical reactions. Simulating chemical reactions takes most of the computational time but, on the other hand, cannot be avoided to obtain accurate predictions. We face a trade-off between cost-efficiency and accuracy: the numerical scheme has to be sufficiently efficient to be used in an operational context but accurate enough to predict the phenomenon faithfully. To tackle this trade-off, we design a hybrid numerical scheme coupling a traditional fluid dynamic solver with a neural network approximating the chemical reactions. We rely on their power in terms of accuracy and dimension reduction when applied in a big data context and on their efficiency stemming from their matrixvector structure to achieve important acceleration factors (×10 to ×18.6). This paper aims to explain how we design such cost-effective hybrid numerical schemes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these schemes as references. (10.1016/j.jcp.2023.112700)
  DOI : 10.1016/j.jcp.2023.112700
• An Optimization-Based Discrete Element Model for Dry Granular Flows: Application to Granular Collapse on Erodible Beds
  
  • Martin Hugo A.
  • Mangeney Anne
  • Lefebvre-Lepot Aline
  • Maury Bertrand
  • Maday Yvon
  Journal of Computational Physics, Elsevier, 2024, 498, pp.112665. Erosion processes and the associated static/flowing transition in granular flows are still poorly understood despite their crucial role in natural hazards such as landslides and debris flows. Continuum models do not yet adequately reproduce the observed increase of runout distance of granular flows on erodible beds or the development of waves at the bed/flow interface. Discrete Element Methods, which simulate each grain’s motion and their complex interactions, provide a unique tool to investigate these processes numerically. Among them, Convex Methods (CM), resulting from the convexification of Contact Dynamics methods, benefit from a robust theoretical framework, ensuring the convergence of the numerical solution at every time iteration. They are also intrinsically more stable than classical Molecular Dynamics methods. However, although already implemented in engineering fields, CMs have not yet been tested in the framework of flows on erodible beds. We present here a Convex Optimization Contact Dynamics (COCD) method and prove that it generates a numerical solution verifying Coulomb’s law at each contact and iteration. After its calibration and validation with experiments and another widely used Contact Dynamics method, we show that our simulations accurately reproduce qualitative and even many quantitative characteristics of experimental granular flows on erodible beds, including the increase of runout distance with the thickness of the erodible bed, the spatio-temporal change of the static/flowing interface and the presence of erosion waves behind the flow front. Beyond erosion processes, our study endorses CMs as potential accurate tools for exploring complex granular mechanisms. (10.1016/j.jcp.2023.112665)
  DOI : 10.1016/j.jcp.2023.112665
• Evolution between two competing macrophyte populations along a resource gradient leads to collapse in a bistable lake ecosystem
  
  • Boucenna Sirine
  • Raoul Gael
  • Dakos Vasilis
  Theoretical Population Biology, Elsevier, 2024, 164, pp.23-36. While it is known that shallow lakes ecosystems may experience abrupt shifts (ie tippingpoints) from one state to a contrasting degraded alternative state as a result of gradual envi-ronmental changes, the role of evolutionary processes and the impact of trait variation in thiscontext remain largely unexplored. It is crucial to elucidate how eco-evolutionary feedbacksaffect abrupt ecological transitions in shallow lakes. These feedbacks can significantly alterthe dynamics of aquatic plants competition, community structure, and species diversity, po-tentially affecting the existence of alternative states or either delay or expedite the thresholdsat which these ecological shifts occur. In this paper, we explore the eco-evolutionary dyna-mics of submerged and floating macrophytes in a shallow lake ecosystem under asymmetriccompetition for nutrients and light. We use adaptive dynamics and a structured populationmodel to analyze the evolution of the growth depth of the submerged and floating macro-phytes population, which influences their competitive ability for the two resources. We showhow rapid trait evolution can result in complex dynamics including evolutionary oscillations,extensive diversification and evolutionary suicide. Furthermore, we find that the co-evolutionof the two competitive species can play a stabilizing role, while not significantly affectingthe overall evolutionary dynamics. Overall, this study shows that evolution can have strongeffects in the ecological dynamics of bistable ecosystems. (10.32942/X2532Z)
  DOI : 10.32942/X2532Z
• Weakly supervised covariance matrices alignment through Stiefel matrices estimation for MEG applications
  
  • Collas Antoine
  • Flamary Rémi
  • Gramfort Alexandre
  , 2024. This paper introduces a novel domain adaptation technique for time series data, called Mixing model Stiefel Adaptation (MSA), specifically addressing the challenge of limited labeled signals in the target dataset. Leveraging a domain-dependent mixing model and the optimal transport domain adaptation assumption, we exploit abundant unlabeled data in the target domain to ensure effective prediction by establishing pairwise correspondence with equivalent signal variances between domains. Theoretical foundations are laid for identifying crucial Stiefel matrices, essential for recovering underlying signal variances from a Riemannian representation of observed signal covariances. We propose an integrated cost function that simultaneously learns these matrices, pairwise domain relationships, and a predictor, classifier, or regressor, depending on the task. Applied to neuroscience problems, MSA outperforms recent methods in brain-age regression with task variations using magnetoencephalography (MEG) signals from the Cam-CAN dataset.
• Foreword to the special issue of Comptes Rendus Mécanique in the memory of Roland Glowinski
  
  • Allaire Grégoire
  • Coron Jean-Michel
  • Girault Vivette
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2024, 351 (S1), pp.1-3. (10.5802/crmeca.238)
  DOI : 10.5802/crmeca.238
• Multi-fidelity Bayesian inference of hypersonic flow free-stream conditions and heterogeneous chemistry model parameters
  
  • Capriati Michele
  , 2024. The characterization of the interaction between a spacecraft entering a planetary atmosphere at hypersonic speeds, and its surrounding gas is a challenging task, requiring accurate experiments and high-fidelity simulations. Numerical predictions strongly depend on modeling and experimental uncertainties, as well as numerical errors, which accumulate during the inference process. In this context, uncertainty quantification methods offer a powerful framework to account for several sources of uncertainty. High-fidelity predictions for aerospace applications combine: i) sufficiently complete physical models to account for complex flow features, leading to expensive and hard-to-perform numerical simulations, ii) rigorous uncertainty quantification permitting the use of experimental data to improve the prediction of quantities of interest. Because of the large number of simulations requested to conduct uncertainty quantification studies, efficient low-fidelity representations are appealing to reduce the computational effort. However, the simplifications contained in the low-fidelity models can lead to reduced accuracy, potentially deteriorating the outcome of an inference problem. In this thesis, our objective is to develop tools and methodologies to perform accurate predictions using high-fidelity solvers and state-of-the-art experimental data. This approach involves the use of the US3D CFD solver and an overall UQ framework to solve inference problems, which permits to include mesh error and to employ multi-fidelity strategies. The first contribution of this thesis concerns the production of high-fidelity solutions for each phenomenon of interest with the US3D solver. We coupled US3D to the open-source Mutation++ physicochemical library, which we expanded to incorporate a state-of-the-art ablation model. The second contribution concerns a study about the influence of the mesh error on the convergence of high-fidelity simulations under uncertainty. We constructed an efficient surrogate model by balancing the grid’s numerical errors and the problem-related uncertainties. We applied this methodology to the propagation of model uncertainties to characterize the pressure and heat flux experienced by a re-entry vehicle. Accurate results were obtained with a coarse mesh automatically aligned to the shock for each training point. The third contribution concerns the development of a multi-fidelity formulation to alleviate the computational cost associated with the construction of the surrogate model of the high-fidelity solver and its use in an inference problem. In particular, we defined a methodology to characterize an under-expanded high-enthalpy jet obtained in the von Karman Institute Plasmatron facility, for which no standardized rebuilding procedure for the free-stream conditions existed to date. The description of such a flow required axisymmetric simulations. The analysis allowed us to characterize the flow conditions at the entrance of the nozzle and the nitrogen catalytic recombination coefficient of the probe used to measure the heat flux and pressure at the stagnation point. The characterized uncertainties were then propagated through the numerical solver yielding an uncertainty-based high-fidelity representation of the supersonic flow structure variability. In the last application, we devised a methodology for the calibration and assessment of a finite-rate chemistry gas-surface interaction model for ablation. Specifically, we inferred the rates of the elementary reactions occurring between a carbon surface and a nitrogen gas from both molecular beam-surface scattering and Plasmatron experiments. The analysis highlighted that both experimental data sets are compatible with the same calibrated model. In conclusion, we proposed powerful stochastic tools, encompassing one or more fidelity levels, to infer hypersonic flow free-stream conditions and heterogeneous chemical model parameters.
• Real-time diagnosis aid method and decision-support for medical diagnosis to a user of a medical system
  
  • Allassonnière Stéphanie
  • Besson Rémi
  • Le Pennec Erwan
  , 2024.
• Layer stripping approach to reconstruct shape defects in waveguides using locally resonant frequencies
  
  • Niclas Angéle
  • Seppecher Laurent
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2024, 84 (1), pp.75-96. This article present a new method to reconstruct slowly varying width defects in 2D waveguides using one-side section measurements at locally resonant frequencies. At these frequencies, locally resonant modes propagate in the waveguide up to a "cut-off" position. In this particular point, the local width of the waveguide can be recovered. Given multi-frequency measurements taken on a section of the waveguide, we perform an efficient layer stripping approach to recover shape variations slice by slice. It provides an L infinite-stable method to reconstruct the width of a slowly monotonous varying waveguide. We validate this method on numerical data and discuss its limits. (10.1137/23M1546336)
  DOI : 10.1137/23M1546336
• On the minimal algebraic complexity of the rank-one approximation problem for general inner products
  
  • Kozhasov Khazhgali
  • Muniz Alan
  • Qi Yang
  • Sodomaco Luca
  , 2023. We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products and conjecture that it achieves its minimal value at Frobenius inner product. We prove our conjecture in the case of matrices. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.
• Assessing the extinction risk of the spontaneous flora in urban tree bases
  
  • Louvet Apolline
  • Mantoux Clément
  • Machon Nathalie
  , 2024. As the design of tree bases along streets in cities makes them potential ecological corridors, urban tree bases may be a key contributor to the overall connectivity of the urban ecosystem. However, they are also a highly fragmented environment in which extinctions are frequent. The goal of this study was to assess the plant species ability to survive and spread through urban tree bases. To do so, we developed a Bayesian framework to assess the extinction risk of a plant metapopulation using presence/absence data, assuming that the occupancy dynamics was described by a Hidden Markov Model. The novelty of our approach is to take into account the combined effect of low-distance dispersal and the potential presence of a seed bank on the extinction risk. We introduced a metric of the extinction risk and examined its performance over a wide range of metapopulation parameters. We applied our framework to yearly floristic inventories carried out in 1324 tree bases in Paris, France. While local extinction risks were generally high, the extinction risk at the street scale varied greatly from one species to another. We identified 10 plant species that could survive and spread through urban tree bases, and three biological traits correlated with the extinction risk at the metapopulation scale: the maximal height, and the beginning and end of the flowering period. Our results suggest that some plant species can use urban tree bases as ecological corridors despite high local extinction risks. We also identified several biological traits correlated with the ability to survive in tree bases, some of them being not previously known. Moreover, our findings demonstrate that the approach we introduced to assess the extinction risk has the potential to be extended to more general metapopulations.
• QuadWire: an extended one dimensional model for efficient mechanical simulations of bead-based additive manufacturing processes
  
  • Preumont Laurane
  • Viano Rafaël
  • Weisz-Patrault Daniel
  • Margerit Pierre
  • Allaire Grégoire
  , 2024. This paper presents the basis of a new mechanical model named QuadWire dedicated to efficient simulations of bead-based additive manufacturing processes in which elongated beads are assembled to form 3D parts. The key contribution is to use a multi-particular approach containing 4 particles per material point to develop an extended 1D model capable of capturing complex 3D mechanical states, while significantly reducing computation time with respect to conventional approaches. Indeed, 3D models usually require at least 3 to 4 elements across the bead section, which results in fine discretization along the tangential direction to avoid conditioning issues, and therefore very fine mesh of the entire 3D part. In the QuadWire model, the bead height and thickness are internal dimensions, enabling a significantly coarser mesh along the tangential direction. Thus, despite the QuadWire has 12 degrees of freedom per material point instead of 3 for classical models, the total number of degrees of freedom is reduced by several order of magnitudes for large parts. The proposed model is classically developed within the framework of the principle of virtual power and standard generalized elastic media. Furthermore, the proposed approach includes native and manageable kinematic constraints between successive beads so that the stress state properly evolves during fabrication. Finite element analysis is used for numerical implementation, and the QuadWire stiffness parameters are optimized so that the mechanical response fit conventional 3D approaches. To validate and demonstrate the capabilities of the proposed strategy, the evolution of displacements and stresses in fused deposition modeling of polylactide is simulated.
• Optimal Scaling Results for a Wide Class of Proximal MALA Algorithms
  
  • Crucinio Francesca R.
  • Durmus Alain
  • Jiménez Pablo
  • Roberts Gareth O.
  , 2024. We consider a recently proposed class of MCMC methods which uses proximity maps instead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets. These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found wide application in imaging contexts. The wider stability properties are obtained by building the Moreau-Yoshida envelope for the target of interest, which depends on a parameter $\lambda$. In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.
• Rosenthal-type inequalities for linear statistics of Markov chains
  
  • Durmus Alain
  • Moulines Eric
  • Naumov Alexey
  • Samsonov Sergey
  • Sheshukova Marina
  , 2024. In this paper, we establish novel deviation bounds for additive functionals of geometrically ergodic Markov chains similar to Rosenthal and Bernstein inequalities for sums of independent random variables. We pay special attention to the dependence of our bounds on the mixing time of the corresponding chain. More precisely, we establish explicit bounds that are linked to the constants from the martingale version of the Rosenthal inequality, as well as the constants that characterize the mixing properties of the underlying Markov kernel. Finally, our proof technique is, up to our knowledge, new and based on a recurrent application of the Poisson decomposition.
• Convergence Analysis of Riemannian Stochastic Approximation Schemes
  
  • Durmus Alain
  • Jiménez Pablo
  • Moulines Éric
  • Said Salem
  • Wai Hoi-To
  , 2024. This paper analyzes the convergence for a large class of Riemannian stochastic approximation (SA) schemes, which aim at tackling stochastic optimization problems. In particular, the recursions we study use either the exponential map of the considered manifold (geodesic schemes) or more general retraction functions (retraction schemes) used as a proxy for the exponential map. Such approximations are of great interest since they are low complexity alternatives to geodesic schemes. Under the assumption that the mean field of the SA is correlated with the gradient of a smooth Lyapunov function (possibly non-convex), we show that the above Riemannian SA schemes find an ${\mathcal{O}}(b_\infty + \log n / \sqrt{n})$-stationary point (in expectation) within ${\mathcal{O}}(n)$ iterations, where $b_\infty \geq 0$ is the asymptotic bias. Compared to previous works, the conditions we derive are considerably milder. First, all our analysis are global as we do not assume iterates to be a-priori bounded. Second, we study biased SA schemes. To be more specific, we consider the case where the mean-field function can only be estimated up to a small bias, and/or the case in which the samples are drawn from a controlled Markov chain. Third, the conditions on retractions required to ensure convergence of the related SA schemes are weak and hold for well-known examples. We illustrate our results on three machine learning problems.
• Boost your favorite Markov Chain Monte Carlo sampler using Kac's theorem: the Kick-Kac teleportation algorithm
  
  • Douc Randal
  • Durmus Alain
  • Enfroy Aurélien
  • Olsson Jimmy
  , 2024. The present paper focuses on the problem of sampling from a given target distribution $\pi$ defined on some general state space. To this end, we introduce a novel class of non-reversible Markov chains, each chain being defined on an extended state space and having an invariant probability measure admitting $\pi$ as a marginal distribution. The proposed methodology is inspired by a new formulation of Kac's theorem and allows global and local dynamics to be smoothly combined. Under mild conditions, the corresponding Markov transition kernel can be shown to be irreducible and Harris recurrent. In addition, we establish that geometric ergodicity holds under appropriate conditions on the global and local dynamics. Finally, we illustrate numerically the use of the proposed method and its potential benefits in comparison to existing Markov chain Monte Carlo (MCMC) algorithms.