arXiv subjects

Karim Lounici

Publications and source records attributed to Karim Lounici.

At least 19 recordsLinked to original sources

arXiv2026

Langevin-Informed Transfer Learning: Replacing Target Samples by Black-Box Feedback

Many scientific and machine learning systems, from molecular dynamics to diffusion models and beyond, are governed by stochastic dynamics with low-dimensional structure, evolving on slow timescales. However, target trajectories, used to identify and interpret such dynamics, are often inaccessible: only biased or static samples that explore the underlying manifold are available. We introduce Langevin-Informed Transfer Learning (LITL), a framework for recovering target Langevin dynamics from biased source samples using only black-box feedback. LITL learns the leading spectral structure of the target infinitesimal generator and the projected drift through Dirichlet representation learning, enabling kinetic reconstruction in spectral form and slow-manifold gradient field estimation. We further introduce a spherical variant well suited to steering normalized latent representations commonly used in learning systems toward desired objectives. We establish finite-sample guarantees for eigenvalue, eigenfunction, and projected drift estimation in Sobolev norms, thereby ensuring generalization of these quantities and their first-order derivatives. Empirically, LITL recovers physical transition timescales from biased molecular simulations, builds kinetic structure from static samples of generative models, reconstructs spherical symmetries of physical systems, and enables post-hoc latent steering of trained neural networks under black-box feedback. Together, these results position spectral operator learning as a practical framework for recovering stochastic dynamics under distribution shift and unlock applications across machine learning and the physical sciences.

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arXiv2026

Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification

Estimating conditional statistics and learning representations of a population of conditional distributions are central problems in many data-driven applications, including uncertainty quantification and dynamical systems analysis. Conditional mean operators (CMOs), a class of linear operators between function spaces, resolve these objectives by providing access to a broad class of conditional statistics. However, existing methods typically estimate each CMO independently or constrain it to prespecified function spaces, thereby preventing the exploitation of shared structure across related distributions. In this work, we posit that related CMOs share finite-dimensional input and output function spaces, and are specialized for each task with a linear operator mapping these spaces. Based on this hypothesis, we introduce MTL-CMO, a multi-task framework that jointly learns shared function spaces and task-specific operators across multiple datasets. We further introduce T-CMO, a transfer learning method that reuses the shared spaces to estimate, in closed form, the operator of a new conditional distribution. We establish statistical guarantees quantifying the benefits of jointly learning the shared function spaces. Our experiments demonstrate that learning shared function spaces improves uncertainty quantification across a broad range of conditional distributions and, when applied to Langevin and plasma dynamics, yields compact representations of complex dynamics that retain physically meaningful information and enable parameter identification.

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arXiv2026

Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and long-term behavior. However, dynamical operators are typically estimated independently for each system, preventing the discovery of shared structure across related dynamics. To address this limitation, we posit that related dynamical systems lie near a low-dimensional manifold in spectral operator space. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that learns a dictionary of characteristic spectral dynamics whose combinations approximate this manifold and yield compact, interpretable embeddings of individual systems. Beyond representation learning, DOODL enables fast and interpretable operator estimation from short and partially observed trajectories by constraining the estimation to the learned operator manifold. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate that DOODL scales to highly complex multiscale regimes while capturing characteristic spectral structure governing the dynamics rather than merely fitting trajectories, achieving errors one to two orders of magnitude lower than independent operator estimation methods in challenging low-data regimes.

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arXiv2026

Representation Learning for Equivariant Inference with Guarantees

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made empirical advances by incorporating symmetry and geometry priors, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while providing well-calibrated uncertainty estimates.

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arXiv2026

Toward Scalable and Valid Conditional Independence Testing with Spectral Representations

Conditional independence (CI) is central to causal inference, feature selection, and graphical modeling, yet it is untestable in many settings without additional assumptions. Existing CI tests often rely on restrictive structural conditions, limiting their validity. Kernel methods using partial covariance operators offer a more principled approach but suffer from limited adaptivity and scalability. In this work, we explore whether representation learning can help address these limitations. Specifically, we focus on representations derived from the singular value decomposition of partial covariance operators and use them to construct a simple test statistic. We also introduce a bi-level contrastive algorithm to learn these representations. Our theory links representation learning error to test performance and establishes asymptotic validity and power guarantees. Experiments on real and synthetic data suggest that this approach offers a principled and statistically grounded path toward scalable CI testing, bridging kernel-based theory with modern representation learning.

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arXiv2026

Outcome-Aware Spectral Feature Learning for Instrumental Variable Regression

We address the problem of causal effect estimation in the presence of hidden confounders using nonparametric instrumental variable (IV) regression. An established approach is to use estimators based on learned spectral features, that is, features spanning the top singular subspaces of the operator linking treatments to instruments. While powerful, such features are agnostic to the outcome variable. Consequently, the method can fail when the true causal function is poorly represented by these dominant singular functions. To mitigate, we introduce Augmented Spectral Feature Learning, a framework that makes the feature learning process outcome-aware. Our method learns features by minimizing a novel contrastive loss derived from an augmented operator that incorporates information from the outcome. By learning these task-specific features, our approach remains effective even under spectral misalignment. We provide a theoretical analysis of this framework and validate our approach on challenging benchmarks.

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arXiv2026

Toeplitz Based Spectral Methods for Data-driven Dynamical Systems

We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.

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arXiv2026

VertCoHiRF: Decentralized Vertical Clustering Beyond k-means

Vertical Federated Learning (VFL) enables collaborative analysis across parties holding complementary feature views of the same samples, yet existing approaches are largely restricted to distributed variants of $k$-means, requiring centralized coordination or the exchange of feature-dependent numerical statistics, and exhibiting limited robustness under heterogeneous views or adversarial behavior. We introduce VertCoHiRF, a fully decentralized framework for vertical federated clustering based on structural consensus across heterogeneous views, allowing each agent to apply a base clustering method adapted to its local feature space in a peer-to-peer manner. Rather than exchanging feature-dependent statistics or relying on noise injection for privacy, agents cluster their local views independently and reconcile their proposals through identifier-level consensus. Consensus is achieved via decentralized ordinal ranking to select representative medoids, progressively inducing a shared hierarchical clustering across agents. Communication is limited to sample identifiers, cluster labels, and ordinal rankings, providing privacy by design while supporting overlapping feature partitions and heterogeneous local clustering methods, and yielding an interpretable shared Cluster Fusion Hierarchy (CFH) that captures cross-view agreement at multiple resolutions.We analyze communication complexity and robustness, and experiments demonstrate competitive clustering performance in vertical federated settings.

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arXiv2026

CoHiRF: Hierarchical Consensus for Interpretable Clustering Beyond Scalability Limits

We introduce CoHiRF (Consensus Hierarchical Random Features), a hierarchical consensus framework that enables existing clustering methods to operate beyond their usual computational and memory limits. CoHiRF is a meta-algorithm that operates exclusively on the label assignments produced by a base clustering method, without modifying its objective function, optimization procedure, or geometric assumptions. It repeatedly applies the base method to multiple low-dimensional feature views or stochastic realizations, enforces agreement through consensus, and progressively reduces the problem size via representative-based contraction. Across a diverse set of synthetic and real-world experiments involving centroid-based, kernel-based, density-based, and graph-based methods, we show that CoHiRF can improve robustness to high-dimensional noise, enhance stability under stochastic variability, and enable scalability to regimes where the base method alone is infeasible. We also provide an empirical characterization of when hierarchical consensus is beneficial, highlighting the role of reproducible label relations and their compatibility with representative-based contraction. Beyond flat partitions, CoHiRF produces an explicit Cluster Fusion Hierarchy, offering a multi-resolution and interpretable view of the clustering structure. Together, these results position hierarchical consensus as a practical and flexible tool for large-scale clustering, extending the applicability of existing methods without altering their underlying behavior.

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arXiv2026

kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning

kooplearn is a machine-learning library that implements linear, kernel, and deep-learning estimators of dynamical operators and their spectral decompositions. kooplearn can model both discrete-time evolution operators (Koopman/Transfer) and continuous-time infinitesimal generators. By learning these operators, users can analyze dynamical systems via spectral methods, derive data-driven reduced-order models, and forecast future states and observables. kooplearn's interface is compliant with the scikit-learn API, facilitating its integration into existing machine learning and data science workflows. Additionally, kooplearn includes curated benchmark datasets to support experimentation, reproducibility, and the fair comparison of learning algorithms. The software is available at https://github.com/Machine-Learning-Dynamical-Systems/kooplearn.

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arXiv2025

AdaCap: An Adaptive Contrastive Approach for Small-Data Neural Networks

Neural networks struggle on small tabular datasets, where tree-based models remain dominant. We introduce Adaptive Contrastive Approach (AdaCap), a training scheme that combines a permutation-based contrastive loss with a Tikhonov-based closed-form output mapping. Across 85 real-world regression datasets and multiple architectures, AdaCap yields consistent and statistically significant improvements in the small-sample regime, particularly for residual models. A meta-predictor trained on dataset characteristics (size, skewness, noise) accurately anticipates when AdaCap is beneficial. These results show that AdaCap acts as a targeted regularization mechanism, strengthening neural networks precisely where they are most fragile. All results and code are publicly available at https://github.com/BrunoBelucci/adacap.

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arXiv2025

A conversion theorem and minimax optimality for continuum contextual bandits

We study the contextual continuum bandits problem, where the learner sequentially receives a side information vector and has to choose an action in a convex set, minimizing a function associated with the context. The goal is to minimize all the underlying functions for the received contexts, leading to the contextual notion of regret, which is stronger than the standard static regret. Assuming that the objective functions are $γ$-Hölder with respect to the contexts, $0<γ\le 1,$ we demonstrate that any algorithm achieving a sub-linear static regret can be extended to achieve a sub-linear contextual regret. We prove a static-to-contextual regret conversion theorem that provides an upper bound for the contextual regret of the output algorithm as a function of the static regret of the input algorithm. We further study the implications of this general result for three fundamental cases of dependency of the objective function on the action variable: (a) Lipschitz bandits, (b) convex bandits, (c) strongly convex and smooth bandits. For Lipschitz bandits and $γ=1,$ combining our results with the lower bound of Slivkins (2014), we prove that the minimax optimal contextual regret for the noise-free adversarial setting is achieved. Then, we prove that in the presence of noise, the contextual regret rate as a function of the number of queries is the same for convex bandits as it is for strongly convex and smooth bandits. Lastly, we present a minimax lower bound, implying two key facts. First, obtaining a sub-linear contextual regret may be impossible over functions that are not continuous with respect to the context. Second, for convex bandits and strongly convex and smooth bandits, the algorithms that we propose achieve, up to a logarithmic factor, the minimax optimal rate of contextual regret as a function of the number of queries.

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arXiv2025

A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems

The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors and defining a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fréchet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.

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arXiv2025

Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups

Markov processes serve as a universal model for many real-world random processes. This paper presents a data-driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector-valued regression and Hilbert-Schmidt operator analysis. Existing techniques, including physics-informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time-lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time-lag variations, ensuring accurate eigenvalue learning even for small time-lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments.

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arXiv2025

Neural Conditional Probability for Uncertainty Quantification

We introduce Neural Conditional Probability (NCP), an operator-theoretic approach to learning conditional distributions with a focus on statistical inference tasks. NCP can be used to build conditional confidence regions and extract key statistics such as conditional quantiles, mean, and covariance. It offers streamlined learning via a single unconditional training phase, allowing efficient inference without the need for retraining even when conditioning changes. By leveraging the approximation capabilities of neural networks, NCP efficiently handles a wide variety of complex probability distributions. We provide theoretical guarantees that ensure both optimization consistency and statistical accuracy. In experiments, we show that NCP with a 2-hidden-layer network matches or outperforms leading methods. This demonstrates that a a minimalistic architecture with a theoretically grounded loss can achieve competitive results, even in the face of more complex architectures.

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arXiv2024

Multi-Source and Test-Time Domain Adaptation on Multivariate Signals using Spatio-Temporal Monge Alignment

Machine learning applications on signals such as computer vision or biomedical data often face significant challenges due to the variability that exists across hardware devices or session recordings. This variability poses a Domain Adaptation (DA) problem, as training and testing data distributions often differ. In this work, we propose Spatio-Temporal Monge Alignment (STMA) to mitigate these variabilities. This Optimal Transport (OT) based method adapts the cross-power spectrum density (cross-PSD) of multivariate signals by mapping them to the Wasserstein barycenter of source domains (multi-source DA). Predictions for new domains can be done with a filtering without the need for retraining a model with source data (test-time DA). We also study and discuss two special cases of the method, Temporal Monge Alignment (TMA) and Spatial Monge Alignment (SMA). Non-asymptotic concentration bounds are derived for the mappings estimation, which reveals a bias-plus-variance error structure with a variance decay rate of $\mathcal{O}(n_\ell^{-1/2})$ with $n_\ell$ the signal length. This theoretical guarantee demonstrates the efficiency of the proposed computational schema. Numerical experiments on multivariate biosignals and image data show that STMA leads to significant and consistent performance gains between datasets acquired with very different settings. Notably, STMA is a pre-processing step complementary to state-of-the-art deep learning methods.

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arXiv2024

Learning the Infinitesimal Generator of Stochastic Diffusion Processes

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes. Our approach integrates physical priors through an energy-based risk metric in both full and partial knowledge settings. We evaluate the statistical performance of a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the partial knowledge setting. Notably, our approach provides learning bounds independent of the state space dimension and ensures non-spurious spectral estimation. Additionally, we elucidate how the distortion between the intrinsic energy-induced metric of the stochastic diffusion and the RKHS metric used for generator estimation impacts the spectral learning bounds.

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arXiv2024

Learning invariant representations of time-homogeneous stochastic dynamical systems

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

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