Malik Tiomoko

Vasilii Feofanov, Romain Ilbert, Malik Tiomoko et al.

Foundation models, while highly effective, are often resource-intensive, requiring substantial inference time and memory. This paper addresses the challenge of making these models more accessible with limited computational resources by exploring dimensionality reduction techniques. Our goal is to enable users to run large pre-trained foundation models on standard GPUs without sacrificing performance. We investigate classical methods such as Principal Component Analysis alongside neural network-based adapters, aiming to reduce the dimensionality of multivariate time series data while preserving key features. Our experiments show up to a 10x speedup compared to the baseline model, without performance degradation, and enable up to 4.5x more datasets to fit on a single GPU, paving the way for more user-friendly and scalable foundation models.

Vasilii Feofanov, Malik Tiomoko, Aladin Virmaux

We propose a theoretical framework to analyze semi-supervised classification under the low density separation assumption in a high-dimensional regime. In particular, we introduce QLDS, a linear classification model, where the low density separation assumption is implemented via quadratic margin maximization. The algorithm has an explicit solution with rich theoretical properties, and we show that particular cases of our algorithm are the least-square support vector machine in the supervised case, the spectral clustering in the fully unsupervised regime, and a class of semi-supervised graph-based approaches. As such, QLDS establishes a smooth bridge between these supervised and unsupervised learning methods. Using recent advances in the random matrix theory, we formally derive a theoretical evaluation of the classification error in the asymptotic regime. As an application, we derive a hyperparameter selection policy that finds the best balance between the supervised and the unsupervised terms of our learning criterion. Finally, we provide extensive illustrations of our framework, as well as an experimental study on several benchmarks to demonstrate that QLDS, while being computationally more efficient, improves over cross-validation for hyperparameter selection, indicating a high promise of the usage of random matrix theory for semi-supervised model selection.

Vasilii Feofanov, Songkang Wen, Marius Alonso et al.

In recent years, there has been increasing interest in developing foundation models for time series data that can generalize across diverse downstream tasks. While numerous forecasting-oriented foundation models have been introduced, there is a notable scarcity of models tailored for time series classification. To address this gap, we present Mantis, a new open-source foundation model for time series classification based on the Vision Transformer (ViT) architecture that has been pre-trained using a contrastive learning approach. Our experimental results show that Mantis outperforms existing foundation models both when the backbone is frozen and when fine-tuned, while achieving the lowest calibration error. In addition, we propose several adapters to handle the multivariate setting, reducing memory requirements and modeling channel interdependence.

Ekkehard Schnoor, Jawher Said, Malik Tiomoko et al.

Concept Activation Vectors (CAVs) are a fundamental tool for concept-based explainability in deep learning, yet their practical utility is limited by statistical instability. We analyze the stochastic nature of CAVs and the Testing with CAVs (TCAV) method, deriving the distributions of major CAV classes including PatternCAV, FastCAV, and ridge regression-based CAVs. We then identify a fundamental flaw in the standard TCAV score: its reliance on a discontinuous indicator function induces non-decaying variance in critical regimes. To address this, we introduce $α$-TCAV, a generalized framework that replaces the indicator with a parameterized smooth function, yielding a unified probabilistic formulation that subsumes both TCAV and Multi-TCAV. We characterize the induced distributions of sensitivity scores and different TCAV variants, showing that established state-of-the-art choices lack theoretical justification. We provide principled guidance on tuning the parameter in $α$-TCAV -- either to imitate Multi-TCAV at substantially lower computational cost, or to obtain a calibrated Bayes-optimal probabilistic measure of a concept's influence. Finally, our analysis yields practical recommendations that challenge established routines: most notably, allocating the full sampling budget to a single CAV rather than splitting it across several.

Yessin Moakher, Malik Tiomoko, Cosme Louart et al.

We first study the generalization error of models that use a fixed feature representation (frozen intermediate layers) followed by a trainable readout layer. This setting encompasses a range of architectures, from deep random-feature models to echo-state networks (ESNs) with recurrent dynamics. Working in the high-dimensional regime, we apply Random Matrix Theory to derive a closed-form expression for the asymptotic generalization error. We then apply this analysis to recurrent representations and obtain concise formula that characterize their performance. Surprisingly, we show that a linear ESN is equivalent to ridge regression with an exponentially time-weighted (''memory'') input covariance, revealing a clear inductive bias toward recent inputs. Experiments match predictions: ESNs win in low-sample, short-memory regimes, while ridge prevails with more data or long-range dependencies. Our methodology provides a general framework for analyzing overparameterized models and offers insights into the behavior of deep learning networks.

Youssef Attia El Hili, Albert Thomas, Malik Tiomoko et al.

Model and hyperparameter selection are critical but challenging in machine learning, typically requiring expert intuition or expensive automated search. We investigate whether large language models (LLMs) can act as in-context meta-learners for this task. By converting each dataset into interpretable metadata, we prompt an LLM to recommend both model families and hyperparameters. We study two prompting strategies: (1) a zero-shot mode relying solely on pretrained knowledge, and (2) a meta-informed mode augmented with examples of models and their performance on past tasks. Across synthetic and real-world benchmarks, we show that LLMs can exploit dataset metadata to recommend competitive models and hyperparameters without search, and that improvements from meta-informed prompting demonstrate their capacity for in-context meta-learning. These results highlight a promising new role for LLMs as lightweight, general-purpose assistants for model selection and hyperparameter optimization.

Hamza Cherkaoui, Malik Tiomoko, Mohamed El Amine Seddik et al.

Bootstrap methods have long been a cornerstone of ensemble learning in machine learning. This paper presents a theoretical analysis of bootstrap techniques applied to the Least Square Support Vector Machine (LSSVM) ensemble in the context of large and growing sample sizes and feature dimensionalities. Leveraging tools from Random Matrix Theory, we investigate the performance of this classifier that aggregates decision functions from multiple weak classifiers, each trained on different subsets of the data. We provide insights into the use of bootstrap methods in high-dimensional settings, enhancing our understanding of their impact. Based on these findings, we propose strategies to select the number of subsets and the regularization parameter that maximize the performance of the LSSVM. Empirical experiments on synthetic and real-world datasets validate our theoretical results.

Chiheb Yaakoubi, Cosme Louart, Malik Tiomoko et al.

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hatθ^\top x$ approximately follows the convolution of the (generally non-Gaussian) distribution of $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

Florent Bouchard, Ammar Mian, Malik Tiomoko et al.

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fréchet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine-learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory-based method that estimates Fréchet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.

Malik Tiomoko, Ekkehard Schnoor

Regularized linear regression is central to machine learning, yet its high-dimensional behavior with informative priors remains poorly understood. We provide the first exact asymptotic characterization of training and test risks for maximum a posteriori (MAP) regression with Gaussian priors centered at a domain-informed initialization. Our framework unifies ridge regression, least squares, and prior-informed estimators, and -- using random matrix theory -- yields closed-form risk formulas that expose the bias-variance-prior tradeoff, explain double descent, and quantify prior mismatch. We also identify a closed-form minimizer of test risk, enabling a simple estimator of the optimal regularization parameter. Simulations confirm the theory with high accuracy. By connecting Bayesian priors, classical regularization, and modern asymptotics, our results provide both conceptual clarity and practical guidance for learning with structured prior knowledge.

Ekkehard Schnoor, Malik Tiomoko, Jawher Said et al.

Concept Activation Vectors (CAVs) are a tool from explainable AI, offering a promising approach for understanding how human-understandable concepts are encoded in a model's latent spaces. They are computed from hidden-layer activations of inputs belonging either to a concept class or to non-concept examples. Adopting a probabilistic perspective, the distribution of the (non-)concept inputs induces a distribution over the CAV, making it a random vector in the latent space. This enables us to derive mean and covariance for different types of CAVs, leading to a unified theoretical view. This probabilistic perspective also reveals a potential vulnerability: CAVs can strongly depend on the rather arbitrary non-concept distribution, a factor largely overlooked in prior work. We illustrate this with a simple yet effective adversarial attack, underscoring the need for a more systematic study.

Yessin Moakher, Malik Tiomoko, Cosme Louart et al.

We present a rigorous asymptotic analysis of Echo State Networks (ESNs) in a teacher student setting with a linear teacher with oracle weights. Leveraging random matrix theory, we derive closed form expressions for the asymptotic bias, variance, and mean-squared error (MSE) as functions of the input statistics, the oracle vector, and the ridge regularization parameter. The analysis reveals two key departures from classical ridge regression: (i) ESNs do not exhibit double descent, and (ii) ESNs attain lower MSE when both the number of training samples and the teacher memory length are limited. We further provide an explicit formula for the optimal regularization in the identity input covariance case, and propose an efficient numerical scheme to compute the optimum in the general case. Together, these results offer interpretable theory and practical guidelines for tuning ESNs, helping reconcile recent empirical observations with provable performance guarantees

Malik Tiomoko, Hamza Cherkaoui, Giuseppe Paolo et al.

Time series forecasting models often produce systematic, predictable errors even in critical domains such as energy, finance, and healthcare. We introduce a novel post training adaptive optimization framework that improves forecast accuracy without retraining or architectural changes. Our method automatically applies expressive transformations optimized via reinforcement learning, contextual bandits, or genetic algorithms to correct model outputs in a lightweight and model agnostic way. Theoretically, we prove that affine corrections always reduce the mean squared error; practically, we extend this idea with dynamic action based optimization. The framework also supports an optional human in the loop component: domain experts can guide corrections using natural language, which is parsed into actions by a language model. Across multiple benchmarks (e.g., electricity, weather, traffic), we observe consistent accuracy gains with minimal computational overhead. Our interactive demo shows the framework's real time usability. By combining automated post hoc refinement with interpretable and extensible mechanisms, our approach offers a powerful new direction for practical forecasting systems.

Romain Ilbert, Malik Tiomoko, Cosme Louart et al.

In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

Malik Tiomoko, Romain Couillet, Frédéric Pascal

The article proposes and theoretically analyses a \emph{computationally efficient} multi-task learning (MTL) extension of popular principal component analysis (PCA)-based supervised learning schemes \cite{barshan2011supervised,bair2006prediction}. The analysis reveals that (i) by default learning may dramatically fail by suffering from \emph{negative transfer}, but that (ii) simple counter-measures on data labels avert negative transfer and necessarily result in improved performances. Supporting experiments on synthetic and real data benchmarks show that the proposed method achieves comparable performance with state-of-the-art MTL methods but at a \emph{significantly reduced computational cost}.

Sami Fakhry, Romain Couillet, Malik Tiomoko

This article proposes a distributed multi-task learning (MTL) algorithm based on supervised principal component analysis (SPCA) which is: (i) theoretically optimal for Gaussian mixtures, (ii) computationally cheap and scalable. Supporting experiments on synthetic and real benchmark data demonstrate that significant energy gains can be obtained with no performance loss.

Malik Tiomoko, Romain Couillet, Hafiz Tiomoko

Multi Task Learning (MTL) efficiently leverages useful information contained in multiple related tasks to help improve the generalization performance of all tasks. This article conducts a large dimensional analysis of a simple but, as we shall see, extremely powerful when carefully tuned, Least Square Support Vector Machine (LSSVM) version of MTL, in the regime where the dimension $p$ of the data and their number $n$ grow large at the same rate. Under mild assumptions on the input data, the theoretical analysis of the MTL-LSSVM algorithm first reveals the "sufficient statistics" exploited by the algorithm and their interaction at work. These results demonstrate, as a striking consequence, that the standard approach to MTL-LSSVM is largely suboptimal, can lead to severe effects of negative transfer but that these impairments are easily corrected. These corrections are turned into an improved MTL-LSSVM algorithm which can only benefit from additional data, and the theoretical performance of which is also analyzed. As evidenced and theoretically sustained in numerous recent works, these large dimensional results are robust to broad ranges of data distributions, which our present experiments corroborate. Specifically, the article reports a systematically close behavior between theoretical and empirical performances on popular datasets, which is strongly suggestive of the applicability of the proposed carefully tuned MTL-LSSVM method to real data. This fine-tuning is fully based on the theoretical analysis and does not in particular require any cross validation procedure. Besides, the reported performances on real datasets almost systematically outperform much more elaborate and less intuitive state-of-the-art multi-task and transfer learning methods.

Malik Tiomoko, Romain Couillet

This article proposes a method to consistently estimate functionals $\frac1p\sum_{i=1}^pf(λ_i(C_1C_2))$ of the eigenvalues of the product of two covariance matrices $C_1,C_2\in\mathbb{R}^{p\times p}$ based on the empirical estimates $λ_i(\hat C_1\hat C_2)$ ($\hat C_a=\frac1{n_a}\sum_{i=1}^{n_a} x_i^{(a)}x_i^{(a){\sf T}}$), when the size $p$ and number $n_a$ of the (zero mean) samples $x_i^{(a)}$ are similar. As a corollary, a consistent estimate of the Wasserstein distance (related to the case $f(t)=\sqrt{t}$) between centered Gaussian distributions is derived. The new estimate is shown to largely outperform the classical sample covariance-based `plug-in' estimator. Based on this finding, a practical application to covariance estimation is then devised which demonstrates potentially significant performance gains with respect to state-of-the-art alternatives.

Malik Tiomoko, Florent Bouchard, Guillaume Ginholac et al.

Relying on recent advances in statistical estimation of covariance distances based on random matrix theory, this article proposes an improved covariance and precision matrix estimation for a wide family of metrics. The method is shown to largely outperform the sample covariance matrix estimate and to compete with state-of-the-art methods, while at the same time being computationally simpler. Applications to linear and quadratic discriminant analyses also demonstrate significant gains, therefore suggesting practical interest to statistical machine learning.

Romain Couillet, Malik Tiomoko, Steeve Zozor et al.

Given two sets $x_1^{(1)},\ldots,x_{n_1}^{(1)}$ and $x_1^{(2)},\ldots,x_{n_2}^{(2)}\in\mathbb{R}^p$ (or $\mathbb{C}^p$) of random vectors with zero mean and positive definite covariance matrices $C_1$ and $C_2\in\mathbb{R}^{p\times p}$ (or $\mathbb{C}^{p\times p}$), respectively, this article provides novel estimators for a wide range of distances between $C_1$ and $C_2$ (along with divergences between some zero mean and covariance $C_1$ or $C_2$ probability measures) of the form $\frac1p\sum_{i=1}^n f(λ_i(C_1^{-1}C_2))$ (with $λ_i(X)$ the eigenvalues of matrix $X$). These estimators are derived using recent advances in the field of random matrix theory and are asymptotically consistent as $n_1,n_2,p\to\infty$ with non trivial ratios $p/n_1<1$ and $p/n_2<1$ (the case $p/n_2>1$ is also discussed). A first "generic" estimator, valid for a large set of $f$ functions, is provided under the form of a complex integral. Then, for a selected set of $f$'s of practical interest (namely, $f(t)=t$, $f(t)=\log(t)$, $f(t)=\log(1+st)$ and $f(t)=\log^2(t)$), a closed-form expression is provided. Beside theoretical findings, simulation results suggest an outstanding performance advantage for the proposed estimators when compared to the classical "plug-in" estimator $\frac1p\sum_{i=1}^n f(λ_i(\hat C_1^{-1}\hat C_2))$ (with $\hat C_a=\frac1{n_a}\sum_{i=1}^{n_a}x_i^{(a)}x_i^{(a){\sf T}}$), and this even for very small values of $n_1,n_2,p$.