Alexandre d'Aspremont

Ibrahim Fayad, Philippe Ciais, Martin Schwartz et al.

Accurate and timely monitoring of forest canopy heights is critical for assessing forest dynamics, biodiversity, carbon sequestration as well as forest degradation and deforestation. Recent advances in deep learning techniques, coupled with the vast amount of spaceborne remote sensing data offer an unprecedented opportunity to map canopy height at high spatial and temporal resolutions. Current techniques for wall-to-wall canopy height mapping correlate remotely sensed 2D information from optical and radar sensors to the vertical structure of trees using LiDAR measurements. While studies using deep learning algorithms have shown promising performances for the accurate mapping of canopy heights, they have limitations due to the type of architectures and loss functions employed. Moreover, mapping canopy heights over tropical forests remains poorly studied, and the accurate height estimation of tall canopies is a challenge due to signal saturation from optical and radar sensors, persistent cloud covers and sometimes the limited penetration capabilities of LiDARs. Here, we map heights at 10 m resolution across the diverse landscape of Ghana with a new vision transformer (ViT) model optimized concurrently with a classification (discrete) and a regression (continuous) loss function. This model achieves better accuracy than previously used convolutional based approaches (ConvNets) optimized with only a continuous loss function. The ViT model results show that our proposed discrete/continuous loss significantly increases the sensitivity for very tall trees (i.e., > 35m), for which other approaches show saturation effects. The height maps generated by the ViT also have better ground sampling distance and better sensitivity to sparse vegetation in comparison to a convolutional model. Our ViT model has a RMSE of 3.12m in comparison to a reference dataset while the ConvNet model has a RMSE of 4.3m.

Fajwel Fogel, Yohann Perron, Nikola Besic et al.

Estimating canopy height and its changes at meter resolution from satellite imagery is a significant challenge in computer vision with critical environmental applications. However, the lack of open-access datasets at this resolution hinders the reproducibility and evaluation of models. We introduce Open-Canopy, the first open-access, country-scale benchmark for very high-resolution (1.5 m) canopy height estimation, covering over 87,000 km$^2$ across France with 1.5 m resolution satellite imagery and aerial LiDAR data. Additionally, we present Open-Canopy-$Δ$, a benchmark for canopy height change detection between images from different years at tree level-a challenging task for current computer vision models. We evaluate state-of-the-art architectures on these benchmarks, highlighting significant challenges and opportunities for improvement. Our datasets and code are publicly available at https://github.com/fajwel/Open-Canopy.

Alexis Groshenry, Clement Giron, Thomas Lauvaux et al.

The new generation of hyperspectral imagers, such as PRISMA, has improved significantly our detection capability of methane (CH4) plumes from space at high spatial resolution (30m). We present here a complete framework to identify CH4 plumes using images from the PRISMA satellite mission and a deep learning model able to detect plumes over large areas. To compensate for the relative scarcity of PRISMA images, we trained our model by transposing high resolution plumes from Sentinel-2 to PRISMA. Our methodology thus avoids computationally expensive synthetic plume generation from Large Eddy Simulations by generating a broad and realistic training database, and paves the way for large-scale detection of methane plumes using future hyperspectral sensors (EnMAP, EMIT, CarbonMapper).

Alexandre d'Aspremont, Cristóbal Guzmán, Clément Lezane

Inspired by regularization techniques in statistics and machine learning, we study complementary composite minimization in the stochastic setting. This problem corresponds to the minimization of the sum of a (weakly) smooth function endowed with a stochastic first-order oracle, and a structured uniformly convex (possibly nonsmooth and non-Lipschitz) regularization term. Despite intensive work on closely related settings, prior to our work no complexity bounds for this problem were known. We close this gap by providing novel excess risk bounds, both in expectation and with high probability. Our algorithms are nearly optimal, which we prove via novel lower complexity bounds for this class of problems. We conclude by providing numerical results comparing our methods to the state of the art.

Martin Schwartz, Fajwel Fogel, Nikola Besic et al.

The recent decline of the European forest carbon sink highlights the need for spatially explicit and frequently updated forest monitoring tools. Yet, existing satellite-based disturbance products remain too coarse to detect changes at the scale of individual trees, typically below 100 m$^{2}$. Here, we introduce FORMSpoT (Forest Mapping with SPOT Time series), a decade-long (2014-2024) nationwide mapping of forest canopy height at 1.5 m resolution, together with annual disturbance polygons (FORMSpoT-$Δ$) covering mainland France. Canopy heights were derived from annual SPOT-6/7 composites using a hierarchical transformer model (PVTv2) trained on high-resolution airborne laser scanning (ALS) data. To enable robust change detection across heterogeneous acquisitions, we developed a dedicated post-processing pipeline combining co-registration and spatio-temporal total variation denoising. Validation against ALS revisits across 19 sites and 5,087 National Forest Inventory plots shows that FORMSpoT-$Δ$ substantially outperforms existing disturbance products. In mountainous forests, where disturbances are small and spatially fragmented, FORMSpoT-$Δ$ achieves an F1-score of 0.44, representing an order of magnitude higher than existing benchmarks. By enabling tree-level monitoring of forest dynamics at national scale, FORMSpoT-$Δ$ provides a unique tool to analyze management practices, detect early signals of forest decline, and better quantify carbon losses from subtle disturbances such as thinning or selective logging. These results underscore the critical importance of sustaining very high-resolution satellite missions like SPOT and open-data initiatives such as DINAMIS for monitoring forests under climate change.

Grégoire Mialon, Dexiong Chen, Alexandre d'Aspremont et al.

We address the problem of learning on sets of features, motivated by the need of performing pooling operations in long biological sequences of varying sizes, with long-range dependencies, and possibly few labeled data. To address this challenging task, we introduce a parametrized representation of fixed size, which embeds and then aggregates elements from a given input set according to the optimal transport plan between the set and a trainable reference. Our approach scales to large datasets and allows end-to-end training of the reference, while also providing a simple unsupervised learning mechanism with small computational cost. Our aggregation technique admits two useful interpretations: it may be seen as a mechanism related to attention layers in neural networks, or it may be seen as a scalable surrogate of a classical optimal transport-based kernel. We experimentally demonstrate the effectiveness of our approach on biological sequences, achieving state-of-the-art results for protein fold recognition and detection of chromatin profiles tasks, and, as a proof of concept, we show promising results for processing natural language sequences. We provide an open-source implementation of our embedding that can be used alone or as a module in larger learning models at https://github.com/claying/OTK.

Christophe Roux, Max Zimmer, Alexandre d'Aspremont et al.

Pruning is a common technique to reduce the compute and storage requirements of Neural Networks. While conventional approaches typically retrain the model to recover pruning-induced performance degradation, state-of-the-art Large Language Model (LLM) pruning methods operate layer-wise, minimizing the per-layer pruning error on a small calibration dataset to avoid full retraining, which is considered computationally prohibitive for LLMs. However, finding the optimal pruning mask is a hard combinatorial problem and solving it to optimality is intractable. Existing methods hence rely on greedy heuristics that ignore the weight interactions in the pruning objective. In this work, we instead consider the convex relaxation of these combinatorial constraints and solve the resulting problem using the Frank-Wolfe (FW) algorithm. Our method drastically reduces the per-layer pruning error, outperforms strong baselines on state-of-the-art GPT architectures, and remains memory-efficient. We provide theoretical justification by showing that, combined with the convergence guarantees of the FW algorithm, we obtain an approximate solution to the original combinatorial problem upon rounding the relaxed solution to integrality.

Ibrahim Fayad, Max Zimmer, Martin Schwartz et al.

Significant efforts have been directed towards adapting self-supervised multimodal learning for Earth observation applications. However, most current methods produce coarse patch-sized embeddings, limiting their effectiveness and integration with other modalities like LiDAR. To close this gap, we present DUNIA, an approach to learn pixel-sized embeddings through cross-modal alignment between images and full-waveform LiDAR data. As the model is trained in a contrastive manner, the embeddings can be directly leveraged in the context of a variety of environmental monitoring tasks in a zero-shot setting. In our experiments, we demonstrate the effectiveness of the embeddings for seven such tasks: canopy height mapping, fractional canopy cover, land cover mapping, tree species identification, plant area index, crop type classification, and per-pixel waveform-based vertical structure mapping. The results show that the embeddings, along with zero-shot classifiers, often outperform specialized supervised models, even in low-data regimes. In the fine-tuning setting, we show strong performances near or better than the state-of-the-art on five out of six tasks.

Thomas Kerdreux, Christophe Roux, Alexandre d'Aspremont et al.

Linear bandit algorithms yield $\tilde{\mathcal{O}}(n\sqrt{T})$ pseudo-regret bounds on compact convex action sets $\mathcal{K}\subset\mathbb{R}^n$ and two types of structural assumptions lead to better pseudo-regret bounds. When $\mathcal{K}$ is the simplex or an $\ell_p$ ball with $p\in]1,2]$, there exist bandits algorithms with $\tilde{\mathcal{O}}(\sqrt{nT})$ pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond $\ell_p$ balls that enjoy pseudo-regret bounds of $\tilde{\mathcal{O}}(\sqrt{nT})$, which answers an open question from [BCB12, §5.5.]. Interestingly, when the action set is uniformly convex but not necessarily strongly convex, we obtain pseudo-regret bounds with a dimension dependency smaller than $\mathcal{O}(\sqrt{n})$. However, this comes at the expense of asymptotic rates in $T$ varying between $\tilde{\mathcal{O}}(\sqrt{T})$ and $\tilde{\mathcal{O}}(T)$.

Thomas Kerdreux, Alexandre d'Aspremont, Sebastian Pokutta

We review various characterizations of uniform convexity and smoothness on norm balls in finite-dimensional spaces and connect results stemming from the geometry of Banach spaces with \textit{scaling inequalities} used in analysing the convergence of optimization methods. In particular, we establish local versions of these conditions to provide sharper insights on a recent body of complexity results in learning theory, online learning, or offline optimization, which rely on the strong convexity of the feasible set. While they have a significant impact on complexity, these strong convexity or uniform convexity properties of feasible sets are not exploited as thoroughly as their functional counterparts, and this work is an effort to correct this imbalance. We conclude with some practical examples in optimization and machine learning where leveraging these conditions and localized assumptions lead to new complexity results.

Alexandre d'Aspremont, Damien Scieur, Adrien Taylor

This monograph covers some recent advances in a range of acceleration techniques frequently used in convex optimization. We first use quadratic optimization problems to introduce two key families of methods, namely momentum and nested optimization schemes. They coincide in the quadratic case to form the Chebyshev method. We discuss momentum methods in detail, starting with the seminal work of Nesterov and structure convergence proofs using a few master templates, such as that for optimized gradient methods, which provide the key benefit of showing how momentum methods optimize convergence guarantees. We further cover proximal acceleration, at the heart of the Catalyst and Accelerated Hybrid Proximal Extragradient frameworks, using similar algorithmic patterns. Common acceleration techniques rely directly on the knowledge of some of the regularity parameters in the problem at hand. We conclude by discussing restart schemes, a set of simple techniques for reaching nearly optimal convergence rates while adapting to unobserved regularity parameters.

Manon Romain, Alexandre d'Aspremont

We develop a Bregman proximal gradient method for structure learning on linear structural causal models. While the problem is non-convex, has high curvature and is in fact NP-hard, Bregman gradient methods allow us to neutralize at least part of the impact of curvature by measuring smoothness against a highly nonlinear kernel. This allows the method to make longer steps and significantly improves convergence. Each iteration requires solving a Bregman proximal step which is convex and efficiently solvable for our particular choice of kernel. We test our method on various synthetic and real data sets.

Mathieu Barré, Clément Giron, Matthieu Mazzolini et al.

Hyperspectral satellite images report greenhouse gas concentrations worldwide on a daily basis. While taking simple averages of these images over time produces a rough estimate of relative emission rates, atmospheric transport means that simple averages fail to pinpoint the source of these emissions. We propose using Wasserstein barycenters coupled with weather data to average gas concentration data sets and better concentrate the mass around significant sources.

Armin Askari, Quentin Rebjock, Alexandre d'Aspremont et al.

We describe a series of algorithms that efficiently implement Gaussian model-X knockoffs to control the false discovery rate on large scale feature selection problems. Identifying the knockoff distribution requires solving a large scale semidefinite program for which we derive several efficient methods. One handles generic covariance matrices, has a complexity scaling as $O(p^3)$ where $p$ is the ambient dimension, while another assumes a rank $k$ factor model on the covariance matrix to reduce this complexity bound to $O(pk^2)$. We also derive efficient procedures to both estimate factor models and sample knockoff covariates with complexity linear in the dimension. We test our methods on problems with $p$ as large as $500,000$.

Alexandre d'Aspremont, Mert Pilanci

We derive global convergence bounds for the Frank Wolfe algorithm when training one hidden layer neural networks. When using the ReLU activation function, and under tractable preconditioning assumptions on the sample data set, the linear minimization oracle used to incrementally form the solution can be solved explicitly as a second order cone program. The classical Frank Wolfe algorithm then converges with rate $O(1/T)$ where $T$ is both the number of neurons and the number of calls to the oracle.

Mathieu Barré, Adrien Taylor, Alexandre d'Aspremont

In smooth strongly convex optimization, knowledge of the strong convexity parameter is critical for obtaining simple methods with accelerated rates. In this work, we study a class of methods, based on Polyak steps, where this knowledge is substituted by that of the optimal value, $f_*$. We first show slightly improved convergence bounds than previously known for the classical case of simple gradient descent with Polyak steps, we then derive an accelerated gradient method with Polyak steps and momentum, along with convergence guarantees.

Grégoire Mialon, Alexandre d'Aspremont, Julien Mairal

We design simple screening tests to automatically discard data samples in empirical risk minimization without losing optimization guarantees. We derive loss functions that produce dual objectives with a sparse solution. We also show how to regularize convex losses to ensure such a dual sparsity-inducing property, and propose a general method to design screening tests for classification or regression based on ellipsoidal approximations of the optimal set. In addition to producing computational gains, our approach also allows us to compress a dataset into a subset of representative points.

Alexandre d'Aspremont, Mihai Cucuringu, Hemant Tyagi

Given a measurement graph $G= (V,E)$ and an unknown signal $r \in \mathbb{R}^n$, we investigate algorithms for recovering $r$ from pairwise measurements of the form $r_i - r_j$; $\{i,j\} \in E$. This problem arises in a variety of applications, such as ranking teams in sports data and time synchronization of distributed networks. Framed in the context of ranking, the task is to recover the ranking of $n$ teams (induced by $r$) given a small subset of noisy pairwise rank offsets. We propose a simple SVD-based algorithmic pipeline for both the problem of time synchronization and ranking. We provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise perturbations with outliers, using results from matrix perturbation and random matrix theory. Our theoretical findings are complemented by a detailed set of numerical experiments on both synthetic and real data, showcasing the competitiveness of our proposed algorithms with other state-of-the-art methods.

François-Pierre Paty, Alexandre d'Aspremont, Marco Cuturi

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transport (OT) can only be used in machine learning if it is substantially regularized. On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth. We propose in this work to draw inspiration from this theory and use regularity as a regularization tool. We give algorithms operating on two discrete measures that can recover nearly optimal transport maps with small distortion, or equivalently, nearly optimal Brenier potentials that are strongly convex and smooth. The problem boils down to solving alternatively a convex QCQP and a discrete OT problem, granting access to the values and gradients of the Brenier potential not only on sampled points, but also out of sample at the cost of solving a simpler QCQP for each evaluation. We propose algorithms to estimate and evaluate transport maps with desired regularity properties, benchmark their statistical performance, apply them to domain adaptation and visualize their action on a color transfer task.

Armin Askari, Alexandre d'Aspremont, Laurent El Ghaoui

Due to its linear complexity, naive Bayes classification remains an attractive supervised learning method, especially in very large-scale settings. We propose a sparse version of naive Bayes, which can be used for feature selection. This leads to a combinatorial maximum-likelihood problem, for which we provide an exact solution in the case of binary data, or a bound in the multinomial case. We prove that our convex relaxation bounds becomes tight as the marginal contribution of additional features decreases, using a priori duality gap bounds dervied from the Shapley-Folkman theorem. We show how to produce primal solutions satisfying these bounds. Both binary and multinomial sparse models are solvable in time almost linear in problem size, representing a very small extra relative cost compared to the classical naive Bayes. Numerical experiments on text data show that the naive Bayes feature selection method is as statistically effective as state-of-the-art feature selection methods such as recursive feature elimination, $l_1$-penalized logistic regression and LASSO, while being orders of magnitude faster.

Anastasia Podosinnikova, Amelia Perry, Alexander Wein et al.

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.

Damien Scieur, Edouard Oyallon, Alexandre d'Aspremont et al.

The Regularized Nonlinear Acceleration (RNA) algorithm is an acceleration method capable of improving the rate of convergence of many optimization schemes such as gradient descend, SAGA or SVRG. Until now, its analysis is limited to convex problems, but empirical observations shows that RNA may be extended to wider settings. In this paper, we investigate further the benefits of RNA when applied to neural networks, in particular for the task of image recognition on CIFAR10 and ImageNet. With very few modifications of exiting frameworks, RNA improves slightly the optimization process of CNNs, after training.

Damien Scieur, Edouard Oyallon, Alexandre d'Aspremont et al.

Regularized nonlinear acceleration (RNA) estimates the minimum of a function by post-processing iterates from an algorithm such as the gradient method. It can be seen as a regularized version of Anderson acceleration, a classical acceleration scheme from numerical analysis. The new scheme provably improves the rate of convergence of fixed step gradient descent, and its empirical performance is comparable to that of quasi-Newton methods. However, RNA cannot accelerate faster multistep algorithms like Nesterov's method and often diverges in this context. Here, we adapt RNA to overcome these issues, so that our scheme can be used on fast algorithms such as gradient methods with momentum. We show optimal complexity bounds for quadratics and asymptotically optimal rates on general convex minimization problems. Moreover, this new scheme works online, i.e., extrapolated solution estimates can be reinjected at each iteration, significantly improving numerical performance over classical accelerated methods.

Thomas Kerdreux, Fabian Pedregosa, Alexandre d'Aspremont

We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm. While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only requires to solve a linear minimization problem over a small \emph{subset} of the original domain. The first algorithm that we propose is a randomized variant of the original FW algorithm and achieves a $\mathcal{O}(1/t)$ sublinear convergence rate as in the deterministic counterpart. The second algorithm is a randomized variant of the Away-step FW algorithm, and again as its deterministic counterpart, reaches linear (i.e., exponential) convergence rate making it the first provably convergent randomized variant of Away-step FW. In both cases, while subsampling reduces the convergence rate by a constant factor, the linear minimization step can be a fraction of the cost of that of the deterministic versions, especially when the data is streamed. We illustrate computational gains of the algorithms on regression problems, involving both $\ell_1$ and latent group lasso penalties.

Vincent Roulet, Fajwel Fogel, Alexandre d'Aspremont et al.

We study supervised learning problems using clustering constraints to impose structure on either features or samples, seeking to help both prediction and interpretation. The problem of clustering features arises naturally in text classification for instance, to reduce dimensionality by grouping words together and identify synonyms. The sample clustering problem on the other hand, applies to multiclass problems where we are allowed to make multiple predictions and the performance of the best answer is recorded. We derive a unified optimization formulation highlighting the common structure of these problems and produce algorithms whose core iteration complexity amounts to a k-means clustering step, which can be approximated efficiently. We extend these results to combine sparsity and clustering constraints, and develop a new projection algorithm on the set of clustered sparse vectors. We prove convergence of our algorithms on random instances, based on a union of subspaces interpretation of the clustering structure. Finally, we test the robustness of our methods on artificial data sets as well as real data extracted from movie reviews.

Fajwel Fogel, Alexandre d'Aspremont, Milan Vojnovic

We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder this matrix and construct a ranking. We first show that this spectral seriation algorithm recovers the true ranking when all pairwise comparisons are observed and consistent with a total order. We then show that ranking reconstruction is still exact when some pairwise comparisons are corrupted or missing, and that seriation based spectral ranking is more robust to noise than classical scoring methods. Finally, we bound the ranking error when only a random subset of the comparions are observed. An additional benefit of the seriation formulation is that it allows us to solve semi-supervised ranking problems. Experiments on both synthetic and real datasets demonstrate that seriation based spectral ranking achieves competitive and in some cases superior performance compared to classical ranking methods.

Noureddine El Karoui, Alexandre d'Aspremont

We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that averaging eigenvectors of randomly subsampled matrices efficiently approximates the true eigenvectors of the original matrix under certain conditions on the incoherence of the spectral decomposition. This incoherence assumption is typically milder than those made in matrix completion and allows eigenvectors to be sparse. We discuss applications to spectral methods in dimensionality reduction and information retrieval.