Publications

2025

• Personalized Convolutional Dictionary Learning of Physiological Time Series
  
  • Roques Axel
  • Gruffaz Samuel
  • Kim Kyurae
  • Durmus Alain O
  • Oudre Laurent
  , 2025. Human physiological signals tend to exhibit both global and local structures: the former are shared across a population, while the latter reflect inter-individual variability. For instance, kinetic measurements of the gait cycle during locomotion present common characteristics, although idiosyncrasies may be observed due to biomechanical disposition or pathology. To better represent datasets with local-global structure, this work extends Convolutional Dictionary Learning (CDL), a popular method for learning interpretable representations, or dictionaries, of time-series data. In particular, we propose Personalized CDL (PerCDL), in which a local dictionary models local information as a personalized spatiotemporal transformation of a global dictionary. The transformation is learnable and can combine operations such as time warping and rotation. Formal computational and statistical guarantees for PerCDL are provided and its effectiveness on synthetic and real human locomotion data is demonstrated.
• The evolution of partner preferences: Evidence using matrimonial ads from Canada, France, India and the United States
  
  • Lippmann Quentin
  • Surana Khushboo
  Journal of Economic Behavior and Organization, Elsevier, 2025, 233, pp.106950. Using the text from matrimonial ads, we assemble a novel data set to describe the evolution of partner preferences over time and space. Analyzing ads published in Canada, France and India between 1950 and 1995, we show that stated preferences for economic criteria have fallen sharply in favor of personality traits in the two Western countries while they remain the most prevalent in India. Using ads covering various regions from the US and Canada in 1995, we show that personality traits are consistently more demanded than economic criteria. We provide evidence that these results are unlikely to be driven by the composition effects over time, role of parents or changing social norms. We show that the changes over time are particularly strong for women and accompany narrowing gender gaps in labor force participation in Western countries. We discuss the implications for understanding the evolution of assortative mating over time. (10.1016/j.jebo.2025.106950)
  DOI : 10.1016/j.jebo.2025.106950
• Stochastic Dynamics of Incoherent Branched Flow
  
  • Garnier Josselin
  • Picozzi Antonio
  • Torres Theo
  Physical Review Letters, American Physical Society, 2025, 134, pp.223803. Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flow has been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flow by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effects in the development of incoherent branched flow. By considering the paraxial wave equation as a generic representative model, we elaborate a stochastic theory of both coherent and incoherent branched flow. We derive closed-form equations that determine the evolution of the intensity correlation function, as well as the value and the propagation distance of the maximum of the scintillation index, which characterize the dynamical formation of incoherent branched flow. We report accurate numerical simulations that are found in quantitative agreement with the theory without free parameters. Our theory highlights the important impact of coherence and interference on branched flow, thereby providing a framework for exploring branched flow in nonlinear media, in relation with the formation of freak waves in oceans. (10.48550/arXiv.2502.07028)
  DOI : 10.48550/arXiv.2502.07028
• Improving the evaluation of samplers on multi-modal targets
  
  • Grenioux Louis
  • Noble Maxence
  • Gabrié Marylou
  , 2025. Addressing multi-modality constitutes one of the major challenges of sampling. In this reflection paper, we advocate for a more systematic evaluation of samplers towards two sources of difficulty that are mode separation and dimension. For this, we propose a synthetic experimental setting that we illustrate on a selection of samplers, focusing on the challenging criterion of recovery of the mode relative importance. These evaluations are crucial to diagnose the potential of samplers to handle multi-modality and therefore to drive progress in the field.
• A Γ-convergence result for 2D type-I superconductors
  
  • Cosenza Alessandro
  • Goldman Michael
  • Zilio Alessandro
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2025. We consider a 2D non-standard Modica-Mortola type functional. This functional arises from the Ginzburg-Landau theory of type-I superconductors in the case of an infinitely long sample and in the regime of comparable penetration and coherence lengthes. We prove that the functional Γ-converges to the perimeter functional. This result is a first step in understanding how to extend the results of Conti, Goldman, Otto, Serfaty (2018) to the regime of non vanishing Ginzburg-Landau parameter κ.
• Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian
  
  • Abi Jaber Eduardo
  • Attal Elie
  , 2025. <div><p>We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index H decreases and approaches -1/2.</p></div>
• Learned Reference-based Diffusion Sampler for multi-modal distributions
  
  • Noble Maxence
  • Grenioux Louis
  • Gabrié Marylou
  • Durmus Alain Oliviero
  , 2025. Over the past few years, several approaches utilizing score-based diffusion have been proposed to sample from probability distributions, that is without having access to exact samples and relying solely on evaluations of unnormalized densities. The resulting samplers approximate the time-reversal of a noising diffusion process, bridging the target distribution to an easy-to-sample base distribution. In practice, the performance of these methods heavily depends on key hyperparameters that require ground truth samples to be accurately tuned. Our work aims to highlight and address this fundamental issue, focusing in particular on multimodal distributions, which pose significant challenges for existing sampling methods. Building on existing approaches, we introduce Learned Reference-based Diffusion Sampler (LRDS), a methodology specifically designed to leverage prior knowledge on the location of the target modes in order to bypass the obstacle of hyperparameter tuning. LRDS proceeds in two steps by (i) learning a reference diffusion model on samples located in high-density space regions and tailored for multimodality, and (ii) using this reference model to foster the training of a diffusion-based sampler. We experimentally demonstrate that LRDS best exploits prior knowledge on the target distribution compared to competing algorithms on a variety of challenging distributions.
• Watermark anything with localized messages
  
  • Sander Tom
  • Fernandez Pierre
  • Durmus Alain
  • Furon Teddy
  • Douze Matthijs
  , 2025. Image watermarking methods are not tailored to handle small watermarked areas. This restricts applications in real-world scenarios where parts of the image may come from different sources or have been edited. We introduce a deep-learning model for localized image watermarking, dubbed the Watermark Anything Model (WAM). The WAM embedder imperceptibly modifies the input image, while the extractor segments the received image into watermarked and non-watermarked areas and recovers one or several hidden messages from the areas found to be watermarked. The models are jointly trained at low resolution and without perceptual constraints, then post-trained for imperceptibility and multiple watermarks. Experiments show that WAM is competitive with state-of-the art methods in terms of imperceptibility and robustness, especially against inpainting and splicing, even on high-resolution images. Moreover, it offers new capabilities: WAM can locate watermarked areas in spliced images and extract distinct 32-bit messages with less than 1 bit error from multiple small regions -no larger than 10% of the image surface -even for small 256 × 256 images. Training and inference code and model weights are available at github.com/facebookresearch/watermark-anything.
• Heavy-Tailed Diffusion with Denoising Lévy Probabilistic Models
  
  • Shariatian Dario
  • Simsekli Umut
  • Durmus Alain
  , 2025. Exploring noise distributions beyond Gaussian in diffusion models remains an open challenge. While Gaussian-based models succeed within a unified SDE framework, recent studies suggest that heavy-tailed noise distributions, like $α$-stable distributions, may better handle mode collapse and effectively manage datasets exhibiting class imbalance, heavy tails, or prominent outliers. Recently, Yoon et al.\ (NeurIPS 2023), presented the Lévy-Itô model (LIM), directly extending the SDE-based framework to a class of heavy-tailed SDEs, where the injected noise followed an $α$-stable distribution, a rich class of heavy-tailed distributions. However, the LIM framework relies on highly involved mathematical techniques with limited flexibility, potentially hindering broader adoption and further development. In this study, instead of starting from the SDE formulation, we extend the denoising diffusion probabilistic model (DDPM) by replacing the Gaussian noise with $α$-stable noise. By using only elementary proof techniques, the proposed approach, Denoising Lévy Probabilistic Models (DLPM), boils down to vanilla DDPM with minor modifications. As opposed to the Gaussian case, DLPM and LIM yield different training algorithms and different backward processes, leading to distinct sampling algorithms. These fundamental differences translate favorably for DLPM as compared to LIM: our experiments show improvements in coverage of data distribution tails, better robustness to unbalanced datasets, and improved computation times requiring smaller number of backward steps.
• Stochastic numerical approximation for nonlinear Fokker-Planck equations with singular kernels
  
  • Cazacu Nicoleta
  , 2025. This paper studies the convergence rate of the Euler-Maruyama scheme for systems of interacting particles used to approximate solutions of nonlinear Fokker-Planck equations with singular interaction kernels, such as the Keller-Segel model. We derive explicit error estimates in the large-particle limit for two objects: the empirical measure of the interacting particle system and the density distribution of a single particle. Specifically, under certain assumptions on the interaction kernel and initial conditions, we show that the convergence rate of both objects towards solutions of the corresponding nonlinear FokkerPlanck equation depends polynomially on N (the number of particles) and on h (the discretization step). The analysis shows that the scheme converges despite singularities in the drift term. To the best of our knowledge, there are no existing results in the literature of such kind for the singular kernels considered in this work.
• Optimizing Backward Policies in GFlowNets via Trajectory Likelihood Maximization
  
  • Gritsaev Timofei
  • Morozov Nikita
  • Samsonov Sergey
  • Tiapkin Daniil
  , 2025. Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects with probabilities proportional to a given reward function. The key concept behind GFlowNets is the use of two stochastic policies: a forward policy, which incrementally constructs compositional objects, and a backward policy, which sequentially deconstructs them. Recent results show a close relationship between GFlowNet training and entropy-regularized reinforcement learning (RL) problems with a particular reward design. However, this connection applies only in the setting of a fixed backward policy, which might be a significant limitation. As a remedy to this problem, we introduce a simple backward policy optimization algorithm that involves direct maximization of the value function in an entropy-regularized Markov Decision Process (MDP) over intermediate rewards. We provide an extensive experimental evaluation of the proposed approach across various benchmarks in combination with both RL and GFlowNet algorithms and demonstrate its faster convergence and mode discovery in complex environments. (10.48550/arXiv.2410.15474)
  DOI : 10.48550/arXiv.2410.15474
• Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
  
  • Prade Adriano
  , 2025. We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has order $2s$ then the solution is $C^{s-\varepsilon}$ regular for all $\varepsilon \gt 0$ provided the flatness parameter is small enough. The proof relies on an induction argument and its main ingredients are the construction of a suitable barrier and the comparison principle.
• Coupled topology and parametric optimization for electrical machine design with body-fitted meshes
  
  • Gauthey Thomas
  • Allaire Grégoire
  • Bordeu Felipe
  • Hage Hassan Maya
  • Mininger Xavier
  • Ul Rémy
  , 2025.
• Sampling metastable systems using collective variables and Jarzynski-Crooks paths
  
  • Schönle Christoph
  • Gabrié Marylou
  • Lelièvre Tony
  • Stoltz Gabriel
  Journal of Computational Physics, Elsevier, 2025, 527, pp.113806. We consider the problem of sampling a high dimensional multimodal target probability measure. We assume that a good proposal kernel to move only a subset of the degrees of freedoms (also known as collective variables) is known a priori. This proposal kernel can for example be built using normalizing flows. We show how to extend the move from the collective variable space to the full space and how to implement an accept-reject step in order to get a reversible chain with respect to a target probability measure. The accept-reject step does not require to know the marginal of the original measure in the collective variable (namely to know the free energy). The obtained algorithm admits several variants, some of them being very close to methods which have been proposed previously in the literature. We show how the obtained acceptance ratio can be expressed in terms of the work which appears in the Jarzynski-Crooks equality, at least for some variants. Numerical illustrations demonstrate the efficiency of the approach on various simple test cases, and allow us to compare the variants of the algorithm. (10.1016/j.jcp.2025.113806)
  DOI : 10.1016/j.jcp.2025.113806
• Towards History-aware Sensitivity Analysis For Time Series
  
  • Yachouti Mouad
  • Perrin Guillaume
  • Garnier Josselin
  , 2025. Explaining the outcome of dynamic systems is non-trivial due to the temporal nature and correlation of the input variables. In this work, we propose a framework of history-aware sensitivity analysis for stationary time-series to quantify different memory effects and clarify their roles. For this purpose, we decompose the output time series into non-correlated components, namely the instantaneous component and the memory components. The latter are sorted in decreasing order of variance to reflect the importance of the variables. We highlight the compensation phenomena between the resulting components and illustrate them in the case of independent variables in a linear setting. To enable history-aware explanations, variance-based sensitivity indices are derived from the obtained decomposition. We demonstrate the effectiveness of our methodology in providing insights to explain output time-series in both synthetic and real-world cases.
• Identification of moving sources in stochastic flow fields: A bayesian inferential approach with application to marine traffic in the mediterranean sea
  
  • Lakkis Issam
  • Rustom Alexios
  • Hammoud Mohamad Abed El Rahman
  • Issa Leila
  • Knio Omar
  • Le Maitre Olivier
  • Hoteit Ibrahim
  Computational Geosciences, Springer Verlag, 2025, 29 (2), pp.18. A Bayesian inference approach for inferring the source of marine pollution released from a moving source in an uncertain flow field is proposed. A Markov Chain Monte Carlo (MCMC) algorithm is developed and applied for inferring single and multiple release events from vessels moving at known velocity along a predefined path in the Mediterranean Sea. The likelihood is based on a logistic regression cost function that measures the discrepancy between the modeled spill distribution and a binary representation of the observed images. We assess the performance of the proposed methodology using a synthetic release scenario employing realistic ocean currents to drive a stochastic Lagrangian Particle Tracking (LPT) algorithm to generate a probabilistic representation of the spill distribution. The MCMC algorithm employs an adaptive scheme to robustly ensure convergence and well-mixed chains. The proposed Bayesian framework is tested by inferring the location, or injection time, and relative contributions of single and multiple moving sources, contributing to separate and common observation patches, with a focus on various scenarios that demonstrate the efficiency of our sampling algorithm. The performance of the proposed framework was further assessed by comparing the model predictions with the most probable release parameters predicted by a global optimization algorithm. (10.1007/s10596-025-10350-0)
  DOI : 10.1007/s10596-025-10350-0
• Reduced Order Modeling for First Order Hyperbolic Systems with Application to Multiparameter Acoustic Waveform Inversion
  
  • Borcea Liliana
  • Garnier Josselin
  • Mamonov Alexander
  • Zimmerling Jörn
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2025, 18 (2), pp.851-880. Waveform inversion seeks to estimate an inaccessible heterogeneous medium from data gathered by sensors that emit probing signals and measure the generated waves. It is an inverse problem for a second order wave equation or a first order hyperbolic system, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, spatially variable coefficients. The traditional “full waveform inversion” (FWI) formulation estimates the unknown coefficients via minimization of the nonlinear, least squares data fitting objective function. For typical band-limited and high frequency data, this objective function has spurious local minima near and far from the true coefficients. Thus, FWI implemented with gradient based optimization algorithms may fail, even for good initial guesses. Recently, it was shown that it is possible to obtain a better behaved objective function for wave speed estimation, using data driven reduced order models (ROMs) that capture the propagation of pressure waves, governed by the classic second order wave equation. Here we introduce ROMs for vectorial waves, satisfying a general first order hyperbolic system. They are defined via Galerkin projection on the space spanned by the wave snapshots, evaluated on a uniform time grid with appropriately chosen time step. Our ROMs are data driven: They are computed in an efficient and noniterative manner, from the sensor measurements, without knowledge of the medium and the snapshots. The ROM computation applies to any linear waves in lossless and nondispersive media. For the inverse problem we focus attention on acoustic waves in a medium with unknown variable wave speed and density. We show that these can be determined via minimization of an objective function that uses a ROM based approximation of the vectorial wave field inside the inaccessible medium. We assess the performance of our inversion approach with numerical simulations and compare the results to those given by FWI. (10.1137/24M1699784)
  DOI : 10.1137/24M1699784
• Operator Learning for Recommender Systems and Uncertainty Quantification
  
  • Pacreau Grégoire
  , 2025. In this thesis we study estimators for operators and derive non-asymptotic bounds on their precision. We first look at covariance estimation in the presence of outliers and prove that a simple estimator is minimax optimal and outperforms complex state of the art procedures.We then look at multi-task linear bandits, where we assume that the transfer matrix can be decomposed into two factors: a representation matrix shared by all task and a idiosyncratic matrix specific to each task. This decomposition allows for efficient meta-learning, where any new task sharing the same decomposition will only require the learning of the idiosyncractic factor.In a third part we look at non-linear transfer operators. We introduce a novel procedure for learning such operators using deep-learning. This procedure allows for simple MLP architectures to equal or out perform more complex architectures such as normalising flows. Furthermore, our estimator has theoretical garanties on its precision. It also provides confidence intervals without requiring additional training.Finally, we study the learned operators and find ways to infer Granger causality between variables. We extend the study of group-Lasso to linear time series and show that it can be used on non-linear dynamics.
• GEOMETRIC OPTIMIZATION OF A LITHIUM-ION BATTERY WITH THE DOYLE-FULLER-NEWMAN MODEL
  
  • Joly Richard
  • Allaire Grégoire
  • de Loubens Romain
  , 2025. <div><p>This paper studies the geometric optimization of the separator in a lithium-ion battery, following the Doyle-Fuller-Newman model. For a general objective function, we compute its derivative with respect to the interface position by mean of the adjoint method. Our main numerical application is the maximization of the total electric energy during a discharge. Both cases of a fixed final time and a final time depending on the state of charge are examined. Our 2-d numerical implementation is performed in the finite element software FreeFEM with body-fitted meshes. Our main practical conclusion is that optimization over shorter time periods yields more interdigitated designs.</p></div>
• From Glosten-Milgrom to the whole limit order book and applications to financial regulation
  
  • Huang Weibing
  • Pulido Sergio
  • Rosenbaum Mathieu
  • Saliba Pamela
  • Sfendourakis Emmanouil
  , 2025. We build an agent-based model for the order book with three types of market participants: an informed trader, a noise trader and competitive market makers. Using a Glosten-Milgrom like approach, we are able to deduce the whole limit order book (bid-ask spread and volume available at each price) from the interactions between the different agents. More precisely, we obtain a link between efficient price dynamic, proportion of trades due to the noise trader, traded volume, bid-ask spread and equilibrium limit order book state. With this model, we provide a relevant tool for regulators and market platforms. We show for example that it allows us to forecast consequences of a tick size change on the microstructure of an asset. It also enables us to value quantitatively the queue position of a limit order in the book. (10.48550/arXiv.1902.10743)
  DOI : 10.48550/arXiv.1902.10743
• Estimation of extreme risk measures with neural networks
  
  • Allouche Michaël
  • Gobet Emmanuel
  • Girard Stéphane
  , 2025. We propose new parametrizations for neural networks in order to estimate extreme Value-at-Risk and Expected-Shortfall in heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log quantities and their neural network approximation is established. The finite sample performances of the neural network estimator are compared to other bias-reduced extreme-value competitors on both real and simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail.
• Uncertainty quantification in Bayesian inverse problems with neutron and gamma time correlation measurements
  
  • Lartaud Paul
  • Humbert Philippe
  • Garnier Josselin
  Annals of Nuclear Energy, Elsevier Masson, 2025, 213, pp.111123. Neutron noise analysis is a predominant technique for fissile matter identification with passive methods. Quantifying the uncertainties associated with the estimated nuclear parameters is crucial for decision-making. A conservative uncertainty quantification procedure is possible by solving a Bayesian inverse problem with the help of statistical surrogate models but generally leads to large uncertainties due to the surrogate models’ errors. In this work, we develop two methods for robust uncertainty quantification in neutron and gamma noise analysis based on the resolution of Bayesian inverse problems. We show that the uncertainties can be reduced by including information on gamma correlations. The investigation of a joint analysis of the neutron and gamma observations is also conducted with the help of active learning strategies to fine-tune surrogate models. We test our methods on a model of the SILENE reactor core, using simulated and real-world measurements. (10.1016/j.anucene.2024.111123)
  DOI : 10.1016/j.anucene.2024.111123
• statistical modeling for improving decision-making in team sports
  
  • Baouan Ali
  , 2025. In this thesis, we develop statistical frameworks for sports analytics that integrate temporal, spatial, and financial dimensions. The aim is to enhance performance evaluation and strategic decision-making in collective sports, with a focus on the case of football. Our work is organized around four research questions.First, we propose a novel modeling framework based on multivariate Hawkes processes to assess how individual players contribute to the generation of offensive threats. By leveraging the immigration--birth interpretation of Hawkes processes, we introduce new metrics that decompose a player’s influence into direct and indirect contributions, thereby revealing the hierarchical nature of in-game events.Second, we address the representation of spatial information in collective sports by introducing an optimal transport-based embedding of player configurations. By mapping each frame—viewed as a discrete probability measure—into a Euclidean space via successive projections, our method yields a permutation-invariant representation that preserves an interpretable notion of distance through the sliced-Wasserstein metric. This embedding facilitates many learning tasks. In particular, it enables us to measure the degree of similarity between the playing styles of two teams.Third, we develop a probabilistic model to systematically estimate team formations and quantify the rate of role swaps. Modeling observed player positions as a random permutation of latent role positions, our approach enables the identification of a clear team structure. The challenge of the cardinality of the set of permutations is addressed using a sparse selection procedure based on an overlap criterion.Fourth, we combine performance statistics with player-specific attributes using regression techniques, such as Lasso and Random Forests, to predict future market values. This analysis identifies key predictors that drive player valuations, offering actionable insights for talent evaluation and transfer strategies.Overall, the contributions of this thesis advance the state-of-the-art in sports analytics by providing frameworks that capture the intricate dynamics of in-game events, spatial configurations, and economic valuation in modern football. The proposed methodologies are validated on extensive datasets from professional leagues, demonstrating both their practical utility and theoretical robustness.
• A singular infinite dimensional Hamilton-Jacobi-Bellman equation arising from a storage problem
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2025, 35 (03), pp.703-731. In the first part of this paper, we derive an infinite dimensional partial differential equation which describes an economic equilibrium in a model of storage which includes an infinite number of non-atomic agents. This equation has the form of a mean field game master equation. The second part of the paper is devoted to the mathematical study of the Hamilton-Jacobi-Bellman equation from which the previous equation derives. This last equation is both singular and set on a Hilbert space and thus raises new mathematical difficulties. (10.1142/S0218202525500083)
  DOI : 10.1142/S0218202525500083
• Scaffold with Stochastic Gradients: New Analysis with Linear Speed-Up
  
  • Mangold Paul
  • Durmus Alain
  • Dieuleveut Aymeric
  • Moulines Eric
  , 2025. This paper proposes a novel analysis for the Scaffold algorithm, a popular method for dealing with data heterogeneity in federated learning. While its convergence in deterministic settings-where local control variates mitigate client drift-is well established, the impact of stochastic gradient updates on its performance is less understood. To address this problem, we first show that its global parameters and control variates define a Markov chain that converges to a stationary distribution in the Wasserstein distance. Leveraging this result, we prove that Scaffold achieves linear speed-up in the number of clients up to higher-order terms in the step size. Nevertheless, our analysis reveals that Scaffold retains a higher-order bias, similar to FedAvg, that does not decrease as the number of clients increases. This highlights opportunities for developing improved stochastic federated learning algorithms.