Publications

2024

• 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.
• On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers
  
  • Durmus Alain
  • Gruffaz Samuel
  , 2024. There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.
• Score diffusion models without early stopping: finite Fisher information is all you need
  
  • Conforti Giovanni
  • Durmus Alain
  • Silveri Marta Gentiloni
  , 2023. Diffusion models are a new class of generative models that revolve around the estimation of the score function associated with a stochastic differential equation. Subsequent to its acquisition, the approximated score function is then harnessed to simulate the corresponding time-reversal process, ultimately enabling the generation of approximate data samples. Despite their evident practical significance these models carry, a notable challenge persists in the form of a lack of comprehensive quantitative results, especially in scenarios involving non-regular scores and estimators. In almost all reported bounds in Kullback Leibler (KL) divergence, it is assumed that either the score function or its approximation is Lipschitz uniformly in time. However, this condition is very restrictive in practice or appears to be difficult to establish. To circumvent this issue, previous works mainly focused on establishing convergence bounds in KL for an early stopped version of the diffusion model and a smoothed version of the data distribution, or assuming that the data distribution is supported on a compact manifold. These explorations have lead to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However, the question remains about the relevance of such early-stopping procedure or compactness conditions. In particular, if there exist a natural and mild condition ensuring explicit and sharp convergence bounds in KL. In this article, we tackle the aforementioned limitations by focusing on score diffusion models with fixed step size stemming from the Ornstein-Ulhenbeck semigroup and its kinetic counterpart. Our study provides a rigorous analysis, yielding simple, improved and sharp convergence bounds in KL applicable to any data distribution with finite Fisher information with respect to the standard Gaussian distribution.
• Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals
  
  • Conforti Giovanni
  • Durmus Alain
  • Greco Giacomo
  , 2024. We show non-asymptotic exponential convergence of Sinkhorn iterates to the Schr\"odinger potentials, solutions of the quadratic Entropic Optimal Transport problem on $\mathbb{R}^d$. Our results hold under mild assumptions on the marginal inputs: in particular, we only assume that they admit an asymptotically positive log-concavity profile, covering as special cases log-concave distributions and bounded smooth perturbations of quadratic potentials. More precisely, we provide exponential $\mathrm{L}^1$ and pointwise convergence of the iterates (resp. their gradient and Hessian) to Schr\"odinger potentials (resp. their gradient and Hessian) for large enough values of the regularization parameter. As a corollary, we establish exponential convergence of Sinkhorn plans with respect to a symmetric relative entropy and in Wasserstein distance of order one. Up to the authors' knowledge, these are the first results which establish exponential convergence of Sinkhorn's algorithm in a general setting without assuming bounded cost functions or compactly supported marginals.
• Monte Carlo guided diffusion for Bayesian linear inverse problems
  
  • Victorino Cardoso Gabriel
  • Janati El Idrissi Yazid
  • Le Corff Sylvain
  • Moulines Eric
  ICLR International Conference on Learning Representations, 2024. Ill-posed linear inverse problems arise frequently in various applications, from computational photography to medical imaging. A recent line of research exploits Bayesian inference with informative priors to handle the ill-posedness of such problems. Amongst such priors, score-based generative models (SGM) have recently been successfully applied to several different inverse problems. In this study, we exploit the particular structure of the prior defined by the SGM to define a sequence of intermediate linear inverse problems. As the noise level decreases, the posteriors of these inverse problems get closer to the target posterior of the original inverse problem. To sample from this sequence of posteriors, we propose the use of Sequential Monte Carlo (SMC) methods. The proposed algorithm, MCGDiff, is shown to be theoretically grounded and we provide numerical simulations showing that it outperforms competing baselines when dealing with ill-posed inverse problems in a Bayesian setting. (10.48550/arXiv.2308.07983)
  DOI : 10.48550/arXiv.2308.07983
• Transverse Brownian Motion for Pareto Front Identification
  
  • Jones Zachary
  • Congedo Pietro Marco
  • Le Maitre Olivier
  , 2024. Including uncertainty sources in multi-objective optimization allows more robust design decisions at the cost of transforming the objective into an expectation. The stochastic multi-gradient algorithm (SMGDA)\cite{mercier} extends the Robbins-Monro approach to the multi-objective case, allowing for the minimization of the expected objectives without having to directly calculate them. However, a bias in the algorithm and the inherent noise in stochastic gradients cause the algorithm to converge to only a subset of the whole Pareto front, limiting its use. We reduce the bias of the stochastic multi-gradient calculation using an exponential smoothing technique and promote the exploration of the Pareto front by adding non-vanishing noise tangential to the front. We prove that this algorithm, Transverse Brownian Motion, generates samples in a concentrated set containing the whole Pareto front. Finally, we estimate the set of Pareto optimal design points using only the sequence generated during optimization while also providing bootstrapped confidence intervals using a nearest-neighbor model calibrated with a novel procedure based on the hypervolume metric. Our proposed method allows for the estimation of the whole of the Pareto front using significantly fewer evaluations of the random quantities of interest when compared to a direct sample-based estimation, which is valuable in the context of costly model evaluations. We illustrate the efficacy of our approach with numerical examples in increasing dimension and discuss how to apply the method to more complex problems.
• Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk
  
  • Alouges François
  • Averseng Martin
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2024, 58 (2), pp.793-831. We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in $\R^3$. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolutions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber $k=0$) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of $k$ and on locally refined meshes. (10.1051/m2an/2023105)
  DOI : 10.1051/m2an/2023105
• Structured Dictionary Learning of Rating Migration Matrices for Credit Risk Modeling
  
  • Allouche Michaël
  • Gobet Emmanuel
  • Lage Clara
  • Mangin Edwin
  Computational Statistics, Springer Verlag, 2024. Rating Migration Matrix is a crux to assess credit risks. Modeling and predicting these matrices are then an issue of great importance for risk managers in any financial institution. As a challenger to usual parametric modeling approaches, we propose a new structured dictionary learning model with auto-regressive regularization that is able to meet key expectations and constraints: small amount of data, fast evolution in time of these matrices, economic interpretability of the calibrated model. To show the model applicability, we present a numerical test with real data. The source code and the data are available at https://github.com/michael-allouche/ dictionary-learning-RMM.git for the sake of reproducibility of our research. (10.1007/s00180-023-01449-y)
  DOI : 10.1007/s00180-023-01449-y
• Three-Dimensional Random Wave Coupling Along a Boundary and an Associated Inverse Problem
  
  • de Hoop Maarten
  • Garnier Josselin
  • Sølna Knut
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2024, 22 (1), pp.39-65. (10.1137/23M1544842)
  DOI : 10.1137/23M1544842
• A deformation theorem for tensor flat chains and applications (complement to "Tensor rectifiable G-flat chains")
  
  • Goldman Michael
  • Merlet Benoît
  , 2024. In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.
• Risk ratio, odds ratio, risk difference... Which causal measure is easier to generalize?
  
  • Colnet Bénédicte
  • Josse Julie
  • Varoquaux Gaël
  • Scornet Erwan
  , 2023. There are many measures to report so-called treatment or causal effect: absolute difference, ratio, odds ratio, number needed to treat, and so on. The choice of a measure, eg absolute versus relative, is often debated because it leads to different appreciations of the same phenomenon; but it also implies different heterogeneity of treatment effect. In addition some measures -- but not all -- have appealing properties such as collapsibility, matching the intuition of a population summary. We review common measures and their pros and cons typically brought forward. Doing so, we clarify notions of collapsibility and treatment effect heterogeneity, unifying different existing definitions. Our main contribution is to propose to reverse the thinking: rather than starting from the measure, we start from a non-parametric generative model of the outcome. Depending on the nature of the outcome, some causal measures disentangle treatment modulations from baseline risk. Therefore, our analysis outlines an understanding what heterogeneity and homogeneity of treatment effect mean, not through the lens of the measure, but through the lens of the covariates. Our goal is the generalization of causal measures. We show that different sets of covariates are needed to generalize an effect to a different target population depending on (i) the causal measure of interest, (ii) the nature of the outcome, and (iii) the generalization's method itself (generalizing either conditional outcome or local effects).
• On using derivatives and multiple kernel methods for clustering and classifying functional data
  
  • Ah-Pine Julien
  • Yao Anne-Françoise
  Neurocomputing, Elsevier, 2024, 621, pp.129231. In this paper, we propose a framework for rich representation of smooth functional data, leveraging a multiview approach that considers functions and their derivatives as complementary sources of information. Additionally, motivated by the non-linear nature of functional data, we advocate for kernel methods as a suitable modeling approach. We extend existing multiple kernel learning techniques for multivariate data to handle functional data. In particular, we introduce a general procedure for linearly combining different kernel functions. We apply this framework to both clustering and classification tasks, extending multiple kernel k-means and multiple kernel SVM methods to Sobolev functions in H q . Our experiments involve both simulated and real-world data, demonstrating the effectiveness of our proposed methods. (10.1016/j.neucom.2024.129231)
  DOI : 10.1016/j.neucom.2024.129231
• A holographic uniqueness theorem
  
  • Novikov Roman
  Proceedings of the Steklov Institute of Mathematics, MAIK Nauka/Interperiodica, 2024, 325, pp.218–223. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in R^3. We consider a ray in this region, such that its direction is different from the propagation direction of the plane wave. We show that the restriction of the radiation solution to this ray is uniquely determined by the intensity of the total solution on an interval of this ray. As a corollary, we also obtain that the restriction of the radiation solution to any plane in the exterior region is uniquely determined by the intensity of the total solution on an open domain in this plane. In particular, these results solve one of old mathematical questions of holography. (10.1134/S0081543824020123)
  DOI : 10.1134/S0081543824020123
• BAYESIAN CALIBRATION WITH ADAPTIVE MODEL DISCREPANCY
  
  • Leoni Nicolas
  • Le Maitre Olivier
  • Rodio Maria-Giovanna
  • Congedo Pietro Marco
  International Journal for Uncertainty Quantification, Begell House Publishers, 2024, 14 (1), pp.19-41. We investigate a computer model calibration technique inspired by the well-known Bayesian framework of Kennedy and O'Hagan (KOH). We tackle the full Bayesian formulation where model parameter and model discrepancy hyperparameters are estimated jointly and reduce the problem dimensionality by introducing a functional relationship that we call the full maximum a posteriori (FMP) method. This method also eliminates the need for a true value of model parameters that caused identifiability issues in the KOH formulation. When the joint posterior is approximated as a mixture of Gaussians, the FMP calibration is proven to avoid some pitfalls of the KOH calibration, namely missing some probability regions and underestimating the posterior variance. We then illustrate two numerical examples where both model error and measurement uncertainty are estimated together. Using the solution to the full Bayesian problem as a reference, we show that the FMP results are accurate and robust, and avoid the need for high-dimensional Markov chains for sampling. (10.1615/Int.J.UncertaintyQuantification.2023046331)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2023046331
• On the numerical approximation of Blaschke–Santaló diagrams using Centroidal Voronoi Tessellations
  
  • Bogosel Beniamin
  • Buttazzo Giuseppe
  • Oudet Edouard
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2024, 58 (1), pp.393-420. Blaschke–Santaló diagrams are images of maps defined on a set of parameters, taking values into an Euclidean space. Typically, the dimension of the source space is high, possibly infinite, while the target space is two or three dimensional. These diagrams help characterize geometrically various inequalities and are of particular interest in the field of shape optimization. We propose a numerical method, based on Centroidal Voronoi Tessellations, which produces sample points in the parameter space that have uniformly distributed images in the Blaschle–Santaló diagram, therefore providing an accurate description of the latter. Compared with the classical Monte Carlo methods, which simply use a large number of images corresponding to random parameters, the method proposed is computationally efficient and precise. Simulations for two and three dimensional diagrams are presented involving examples in algebra and shape optimization. (10.1051/m2an/2023092)
  DOI : 10.1051/m2an/2023092
• Asymptotic estimations of a perturbed symmetric eigenproblem
  
  • Gissler Armand
  • Auger Anne
  • Hansen Nikolaus
  Applied Mathematics Letters, Elsevier, 2024, 150, pp.108951. We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends to infinity, we bound the values of the coordinates of the eigenvectors of the perturbed matrix. Equivalently, in the coordinate system where the initial matrix is diagonal, we bound the rate of convergence of coordinates that tend to zero. (10.1016/j.aml.2023.108951)
  DOI : 10.1016/j.aml.2023.108951
• Activity report ciroquo research & industry consortium
  
  • Blanchet-Scalliet Christophette
  • Helbert Céline
  • Sinoquet Delphine
  • Munoz Zuniga Miguel Munoz
  • Le Riche Rodolphe
  • Rullière Didier
  • Prieur Clémentine
  • Zahm Olivier
  • Garnier Josselin
  • Brockhoff Dimo
  • Roustant Olivier
  • Bachoc François
  • Pronzato Luc
  • Richet Yann
  • Rohmer Jérémy
  • Huguet Frédéric
  • Gaudrie David
  • Marrel Amandine
  • Kerleguer Baptiste
  • Perrillat-Bottonet Thomas
  • Durantin Cedric
  , 2024, pp.1-11. This project emulated the major scientific challenges of the discipline, such as code transposition/calibration/validation, modeling of complex environments and stochastic codes. In each of these scientific fields, particular attention has been paid to large-scale problems. Real-life problems sometimes involve dozens or hundreds of inputs. Methodological advances have been proposed to take account of this additional difficulty. Developments in mathematical research are distinguished from the dominant work in machine learning by the exploration of costly numerical simulations, i.e., by the need to be thrifty in terms of the quantity of data available.
• Neural network time-series classifiers for gravitational-wave searches in single-detector periods
  
  • Trovato A
  • Chassande-Mottin É
  • Bejger M
  • Flamary R
  • Courty N
  Classical and Quantum Gravity, IOP Publishing, 2024, 41 (12), pp.125003. The search for gravitational-wave signals is limited by non-Gaussian transient noises that mimic astrophysical signals. Temporal coincidence between two or more detectors is used to mitigate contamination by these instrumental glitches. However, when a single detector is in operation, coincidence is impossible, and other strategies have to be used. We explore the possibility of using neural network classifiers and present the results obtained with three types of architectures: convolutional neural network, temporal convolutional network, and inception time. The last two architectures are specifically designed to process time-series data. The classifiers are trained on a month of data from the LIGO Livingston detector during the first observing run (O1) to identify data segments that include the signature of a binary black hole merger. Their performances are assessed and compared. We then apply trained classifiers to the remaining three months of O1 data, focusing specifically on single-detector times. The most promising candidate from our search is 2016-01-04 12:24:17 UTC. Although we are not able to constrain the significance of this event to the level conventionally followed in gravitational-wave searches, we show that the signal is compatible with the merger of two black holes with masses $m_1 = 50.7^{+10.4}_{-8.9}\,M_{\odot}$ and $m_2 = 24.4^{+20.2}_{-9.3}\,M_{\odot}$ at the luminosity distance of $d_L = 564^{+812}_{-338}\,\mathrm{Mpc}$. (10.1088/1361-6382/ad40f0)
  DOI : 10.1088/1361-6382/ad40f0
• SCAFFLSA: Taming Heterogeneity in Federated Linear Stochastic Approximation and TD Learning
  
  • Mangold Paul
  • Samsonov Sergey
  • Labbi Safwan
  • Levin Ilya
  • Alami Reda
  • Naumov Alexey
  • Moulines Eric
  , 2024. In this paper, we analyze the sample and communication complexity of the federated linear stochastic approximation (FedLSA) algorithm. We explicitly quantify the effects of local training with agent heterogeneity. We show that the communication complexity of FedLSA scales polynomially with the inverse of the desired accuracy ϵ. To overcome this, we propose SCAFFLSA, a new variant of FedLSA that uses control variates to correct for client drift, and establish its sample and communication complexities. We show that for statistically heterogeneous agents, its communication complexity scales logarithmically with the desired accuracy, similar to Scaffnew [37]. An important finding is that, compared to the existing results for Scaffnew, the sample complexity scales with the inverse of the number of agents, a property referred to as linear speed-up. Achieving this linear speed-up requires completely new theoretical arguments. We apply the proposed method to federated temporal difference learning with linear function approximation and analyze the corresponding complexity improvements.
• A second-order model of small-strain incompatible elasticity
  
  • Amstutz Samuel
  • van Goethem Nicolas
  Mathematics and Mechanics of Solids, SAGE Publications, 2024, 29 (3), pp.503-530. This work deals with the modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations. Our approach relies on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zero-th and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws.
• Inverse source problem for discrete Helmholtz equation
  
  • Novikov Roman
  • Sharma Basant Lal
  Inverse Problems, IOP Publishing, 2024, 40 (10), pp.105005. We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Z^d , d ≥ 1. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided. (10.1088/1361-6420/ad7054)
  DOI : 10.1088/1361-6420/ad7054
• Super-resolution reconstruction from truncated Fourier transform
  
  • Isaev Mikhail
  • Novikov Roman
  • Sabinin Grigory
  , 2024. We present recent theoretical and numerical results on recovering a compactly supported function v on R^d, d ≥ 1, from its Fourier transform Fv given within the ball B_r. We proceed from known results on the prolate spheroidal wave functions and on the Radon transform. The most interesting point of our numerical examples consists in super-resolution, that is, in recovering details beyond the diffraction limit, that is, details of size less than pi/r, where r is the radius of the ball mentioned above. This short review is based on the works Isaev, Novikov (2022 J. Math. Pures Appl. 163 318–333) and Isaev, Novikov, Sabinin (2022 Inverse Problems 38 105002). (10.1007/978-3-031-41665-1_7)
  DOI : 10.1007/978-3-031-41665-1_7
• Hermitian Preconditioning for a class of Non-Hermitian Linear Systems
  
  • Spillane Nicole
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2024, 46 (3). This work considers the convergence of GMRES for non-singular problems. GMRES is interpreted as the GCR method which allows for simple proofs of the convergence estimates. Preconditioning and weighted norms within GMRES are considered. The objective is to provide a way of choosing the preconditioner and GMRES norm that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems). It is proposed to choose a Hermitian preconditioner H and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. This result is illustrated numerically.
• Shower curtain effect and source imaging
  
  • Garnier Josselin
  • Sølna Knut
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2024, 18 (4), pp.993-1023. (10.3934/ipi.2024004)
  DOI : 10.3934/ipi.2024004
• Distance-preserving maps between bounded symmetric domains
  
  • Lemmens Bas
  • Walsh Cormac
  , 2024. We study the rigidity of maps between bounded symmetric domains that preserve the Carathéodory/Kobayashi metric. We show that such maps are only possible when the rank of the codomain is at least as great as that of the domain. When the ranks are equal, and the domain is irreducible, we show that the map is either holomorphic or antiholomorphic. In the holomorphic case, we show that the map is in fact a triple homomorphism, under the additional assumption that the origin is mapped to the origin.