Keywords
 A3.4.1. Supervised learning
 A3.4.4. Optimization and learning
 A3.4.6. Neural networks
 A3.4.7. Kernel methods
 A3.4.8. Deep learning
 A3.5. Social networks
 A3.5.1. Analysis of large graphs
 A5.3.2. Sparse modeling and image representation
 A5.9. Signal processing
 A5.9.4. Signal processing over graphs
 A5.9.5. Sparsityaware processing
 A5.9.6. Optimization tools
 A6.3.1. Inverse problems
 A8.2. Optimization
 A8.6. Information theory
 A8.12. Optimal transport
 B2.6. Biological and medical imaging
 B6.6. Embedded systems
 B7.2.1. Smart vehicles
 B9.5.1. Computer science
 B9.5.2. Mathematics
 B9.5.6. Data science
 B9.10. Privacy
1 Team members, visitors, external collaborators
Research Scientists
 Remi Gribonval [Team leader, INRIA, Senior Researcher, HDR]
 Paulo Goncalves [INRIA, Senior Researcher, HDR]
 Mathurin Massias [INRIA, Researcher]
 Titouan Vayer [INRIA, Researcher, from Sep 2022]
Faculty Members
 Marion Foare [CPE LYON, Associate Professor]
 Elisa Riccietti [ENS DE LYON, Associate Professor, Chaire Inria]
PostDoctoral Fellows
 Ayoub Belhadji [ENS DE LYON]
 Luc Giffon [ENS DE LYON, until Jun 2022]
 Rémi Vaudaine [ENS DE LYON]
PhD Students
 Antoine Gonon [ENS DE LYON]
 Clément Lalanne [ENS DE LYON]
 Guillaume Lauga [INRIA ]
 QuocTung Le [ENS DE LYON]
 Can Pouliquen [ENS DE LYON, from Nov 2022]
 Léon Zheng [ENS DE LYON]
Technical Staff
 Badr Moufad [ENS DE LYON, Engineer, from Apr 2022]
Administrative Assistant
 Solene Audoux [INRIA]
Visiting Scientist
 Sibylle Marcotte [ENS RENNES, from May 2022 until May 2022]
External Collaborator
 Márton Karsai [UNIV CEU, HDR]
2 Overall objectives
Building on a culture at the interface of signal modeling, mathematical optimization and statistical machine learning, the global objective of DANTE (and of its followup team Ockham which formal creation was process was formally started in 2022) is to develop computationally efficient and mathematically founded methods and models to process highdimensional data. Our ambition is to develop frugal signal processing and machine learning methods able to exploit structured models, intrinsically associated to resourceefficient implementations, and endowed with solid statistical guarantees.
Challenge 1: Developing frugal methods with robust expressivity.
The idea of frugal approaches means algorithms relying on a controlled use of computing resources, but also methods whose expressivity and flexibility provably relies on the versatile notion of sparsity. This is expected to avoid the current pitfalls of costly overparameterizations and to robustify the approaches with respect to adversarial examples and overfitting. More specifically, it is essential to contribute to the understanding of methods based on neural networks, in order to improve their performance and most of all, their efficiency in resourcelimited environments.
Challenge 2: Integrating models in learning algorithms.
To make statistical machine learning both more frugal and more interpretable, it is important to develop techniques able to exploit not only highdimensional data but also models in various forms when available. When some partial knowledge is available about some phenomena related to the processed data, e.g. under the form of a physical model such as a partial differential equation, or as a graph capturing local or nonlocal correlations, the goal is to use this knowledge as an inspiration to adapt machine learning algorithms. The main challenge is to flexibly articulate a priori knowledge and datadriven information, in order to achieve a controlled extrapolation of predicted phenomena much beyond the particular type of data on which they were observed, and even in applications where training data is scarce.
Challenge 3: Guarantees on interpretability, explainability, and privacy.
The notion of sparsity and its structured avatars –notably via graphs– is known to play a fundamental role in ensuring the identifiability of decompositions in latent spaces, for example for highdimensional inverse problems in signal processing. The team's ambition is to deploy these ideas to ensure not only frugality but also some level of explainability of decisions and an interpretability of learned parameters, which is an important societal stake for the acceptability of “algorithmic decisions”. Learning in smalldimensional latent spaces is also a way to spare computing resources and, by limiting the public exposure of data, it is expected to enable tunable and quantifiable tradeoffs between the utility of the developed methods and their ability to preserve privacy.
3 Research program
This project is resolutely at the interface of signal modeling, mathematical optimization and statistical machine learning, and concentrates on scientific objectives that are both ambitious –as they are difficult and subject to a strong international competition– and realistic thanks to the richness and complementarity of skills they mobilize in the team.
Sparsity constitutes a backbone for this project, not only as a target to ensure resourceefficiency and privacy, but also as prior knowledge to be exploited to ensure the identifiability of parameters and the interpretability of results. Graphs are its necessary alter ego, to flexibly model and exploit relations between variables, signals, and phenomena, whether these relations are known a priori or to be inferred from data. Lastly, advanced largescale optimization is a key tool to handle in a statistically controlled and algorithmically efficient way the dynamic and incremental aspects of learning in varying environments.
The scientific activity of the project is articulated around the three axes described below. A common endeavor to these three axes consists in designing structured lowdimensional models, algorithms of bounded complexity to adjust these models to data through learning mechanisms, and a control of the performance of these algorithms to exploit these models on tasks ranging from lowlevel signal processing to the extraction of highlevel information.
3.1 Axis 1: Sparsity for highdimensional learning.
As now widely documented, the fact that a signal admits a sparse representation in some signal dictionary 51 is an enabling factor not only to address a variety of inverse problems with highdimensional signals and images, such as denoising, deconvolution, or declipping, but also to speedup or decrease the cost of the acquisition of analog signals in certain scenarios compatible with compressive sensing 52, 45. The flexibility of the models, which can incorporate learned dictionaries 63, as well as structured and/or lowrank variants of the nowclassical sparse modeling paradigm 57, has been a key factor of the success of these approaches. Another important factor is the existence of algorithms of bounded complexity with provable performance, often associated to convex regularization and proximal strategies 43, 48, allowing to identify latent sparse signal representations from lowdimensional indirect observations.
While being now wellmastered (and in the core field of expertise of the team), these tools are typically constrained to relatively rigid settings where the unknown is described either as a sparse vector or a lowrank matrix or tensor in high (but finite) dimension. Moreover, the algorithms hardly scale to the dimensions needed to handle inverse problems arising from the discretization of physical models (e.g., for 3D wavefield reconstruction). A major challenge is to establish a comprehensive algorithmic and theoretical toolset to handle continuous notions of sparsity 46, which have been identified as a way to potentially circumvent these bottlenecks. The other main challenge is to extend the sparse modeling paradigm to resourceefficient and interpretable statistical machine learning. The methodological and conceptual output of this axis provides tools for Axes 2 and 3, which in return fuel the questions investigated in this axis.

1.1 Versatile and efficient sparse modeling. The goal is to propose flexible and resourceefficient sparse models, possibly leveraging classical notions of dictionaries and structured factorization, but also the notion of sparsity in continuous domains (e.g. for sketched clustering, mixture model estimation, or image superresolution), lowrank tensor representations, and neural networks with sparse connection patterns.
Besides the empirical validation of these models and of the related algorithms on a diversity of targeted applications, the challenge is to determine conditions under which their success can be mathematically controlled, and to determine the fundamental tradeoffs between the expressivity of these models and their complexity.
 1.2 Sparse optimization. The main objectives are: a) to define cost functions and regularization penalties that integrate not only the targeted learning tasks, but also a priori knowledge, for example under the form of conservation laws or as relation graphs, cf Axis 2; b) to design efficient and scalable algorithms 4, 8 to optimize these cost functions in a controlled manner in a largescale setting. To ensure the resourceefficiency of these algorithms, while avoiding pitfalls related to the discretization of highdimensional problems (aka curse of dimensionality), we investigate the notion of “continuous” sparsity (i.e., with sparse measures), of hierarchies (along the ideas of multilevel methods), and of reduced precision (cf also Axis 3). The nonconvexity and nonsmoothness of the problems are key challenges 2, and the exploitation of proximal algorithms and/or convexifications in the space of Borelian measures are privileged approaches.
 1.3 Identifiability of latent sparse representations. To provide solid guarantees on the interpretability of sparse models obtained via learning, one needs to ensure the identifiability of the latent variables associated to their parameters. This is particularly important when these parameters bear some meaning due to the underlying physics. Viceversa, physical knowledge can guide the choice of which latent parameters to estimate. By leveraging the team's knowhow obtained in the field of inverse problems, compressive sensing and source separation in signal processing, we aim at establishing theoretical guarantees on the uniqueness (modulo some equivalence classes to be characterized) of the solutions of the considered optimization problems, on their stability in the presence of random or adversarial noise, and on the convergence and stability of the algorithms.
3.2 Axis 2: Learning on graphs and learning of graphs.
Graphs provide synthetic and sparse representations of the interactions between potentially highdimensional data, whether in terms of proximity, statistical correlation, functional similarity, or simple affinities. One central task in this domain is how to infer such discrete structures, from the observations, in a way that best accounts for the ties between data, without becoming too complex due to spurious relationships. The graphical lasso 53 is among the most popular and successful algorithm to build a sparse representation of the relations between time series (observed at each node) and that unveils relevant patterns of the data. Recent works (e.g. 58) strived to emphasize the clustered structure of the data by imposing spectral constraints to the Laplacian of the sought graphs, with the aim to improve the performance of spectral approaches to unsupervised classification. In this direction, several challenges remain, such as for instance the transposition of the framework to graphbased semisupervised learning 1, where natural models are stochastic block models rather than strictly multicomponent graphs (e.g. Gaussian mixtures models). As it is done in 69, the standard ${l}_{1}$norm penalization term of graphical lasso could be questioned in this case. On another level, when lowrank (precision) matrices and / or when preservation of privacy are important stakes, one could be inspired by the sketching techniques developed in 55 and 47 to work out a sketched graphical lasso. There exists other situations where the graph is known a priori and does not need to be inferred from the data. This is for instance the case when the data naturally lie on a graph (e.g. social networks or geographical graphs) and so, one has to combine this data structure with the attributes (or measures) carried by the nodes or the edges of these graphs. Graph signal processing (GSP) 619, which underwent methodological developments at a very rapid pace in recent years, is precisely an approach to jointly exploit algebraically these structures and attributes, either by filtering them, by reorganizing them, or by reducing them to principal components. However, as it tends to be more and more the case, data collection processes yield very large data sets with high dimensional graphs. In contrast to standard digital signal processing that relies on regular graph structures (cycle graph or cartesian grid) treating complex structured data in a global form is not an easily scalable task 54. Hence, the notion of distributed GSP 49, 50 has naturally emerged. Yet, very little has been done on graph signals supported on dynamical graphs that undergo vertices/edges editions.
 2.1 Learning of graphs. When the graphical structure of the data is not known a priori, one needs to explore how to build it or to infer it. In the case of partially known graphs, this raises several questions in terms of relevance with respect to sparse learning. For example, a challenge is to determine which edges should be kept, whether they should be oriented, and how attributes on the graph could be taken into account (in particular when considering timeseries on graphs) to better infer the nature and structure of the unobserved interactions. We strive to adapt known approaches such as the graphical lasso to estimate the covariance under a sparsity constraint (integrating also temporal priors), and investigate diffusion approaches to study the identifiability of the graphs. In connection with Axis 1.2, a particular challenge is to incorporate a priori knowledge coming from physical models that offer concise and interpretable descriptions of the data and their interactions.

2.2 Distributed and adaptive learning on graphs. The availability of a known graph structure underlying training data offers many opportunities to develop distributed approaches, open perspectives where graph signal processing and machine learning can mutually fertilize each other.
Some classifiers can be formalized as solutions of a constrained optimization problem, and an important objective is then to reduce their global complexity by developing distributed versions of these algorithms. Compared to costly centralized solutions, distributing the operations by restricting them to local node neighborhoods will enable solutions that are both more frugal and more privacyfriendly. In the case of dynamic graphs, the idea is to get inspiration from adaptive processing techniques to make the algorithms able to track the temporal evolution of data, either in terms of structural evolution or of temporal variations of the attributes. This aspect finds a natural continuation in the objectives of Axis 3.
3.3 Axis 3: Dynamic and frugal learning.
With the resurgence of neural networks approaches in machine learning, training times of the order of days, weeks, or even months are common. Mainstream research in deep learning somehow applies it to an increasingly large class of problems and uses the general wisdom to improve the models prediction accuracy by “stacking more layers”, making the approach ever more resourcehungry. Underpinning theory on which resources are needed for a network architecture to achieve a given accuracy is still in its infancy. Efficient scaling of such techniques to massive sample sizes or dimensions in a resourcerestricted environment remains a challenge and is a particularly active field of academic and industrial R&D, with recent interest in techniques such as sketching, dimension reduction, and approximate optimization.
A central challenge is to develop novel approximate techniques with reduced computational and memory imprint. For certain unsupervised learning tasks such as PCA, unsupervised clustering, or parametric density estimation, random features (e.g. random Fourier features 59) allow to compute aggregated sketches guaranteed to preserve the information needed to learn, and no more: this has led to the compressive learning framework, which is endowed with statistical learning guarantees 55 as well as privacy preservation guarantees 47. A sketch can be seen as an embedding of the empirical probability distribution of the dataset with a particular form of kernel mean embedding 62. Yet, designing random features given a learning task remains something of an art, and a major challenge is to design provably good endtoend sketching pipelines with controlled complexity for supervised classification, structured matrix factorization, and deep learning.
Another crucial direction is the use of dynamical learning methods, capable of exploiting wisely multiple representations at different scales of the problem at hand. For instance, many low and mixedprecision variants of gradientbased methods have been recently proposed 67, 66, which are however based on a static reduced precision policy, while a dynamic approach can lead to much improved energyefficiency. Also, despite their massive success, gradientbased training methods still possess many weaknesses (low convergence rate, dependence on the tuning of the learning parameters, vanishing and exploding gradients) and the use of dynamical information promises to allow for the development of alternative methods, such as secondorder or multilevel methods, which are as scalable as firstorder methods but with faster convergence guarantees 60, 68.
The overall objective in this axis is to adapt in a controlled manner the information that is extracted from datasets or data streams and to dynamically use such information in learning, in order to optimize the tradeoffs between statistical significance, resourceefficiency, privacypreservation and integration of a priori knowledge.
 3.1 Compressive and privacypreserving learning. The goal is to compress training datasets as soon as possible in the processing workflow, before even starting to learn. In the spirit of compressive sensing, this is desirable not only to ensure the frugal use of ressources (memory and computation), but also to preserve privacy by limiting the diffusion of raw datasets and controlling the information that could actually be extracted from the targeted compressed representations, called sketches, obtained by wellchosen nonlinear random projections. We aim to build on a compressive learning framework developed by the team with the viewpoint that sketches provide an embedding of the data distribution, which should preserve some metrics, either associated to the specific learning task or to more generic optimal transport formulations. Besides ensuring the identifiability of the taskspecific information from a sketch (cf Axis 1.3), an objective is to efficiently extract this information from a sketch, for example via algorithms related to avatars of continuous sparsity as studied in Axis 1.2. A particular challenge, connected with Axis 2.1 when inferring dynamic graphs from correlation of nonstationary times series, and with Axis 3.2 below, is to dynamically adapt the sketching mechanism to the analyzed data stream.
 3.2 Sequential sparse learning. Whether aiming at dynamically learning on data streams (cf. Axes 2.1 and 2.2), at integrating a priori physical knowledge when learning, or at ensuring domain adaptation for transfer learning, the objective is to achieve a statistically nearoptimal update of a model from a sequence of observations whose content can also dynamically vary. When considering timeseries on graphs, to preserve resourceefficiency and increase robustness, the algorithms further need to update the current models by dynamically integrating the data stream.
 3.3 Dynamicprecision learning. The goal is to propose new optimization algorithms to overcome the cost of solving large scale problems in learning, by dynamically adapting the precision of the data. The main idea is to exploit multiple representations at different scales of the problem at hand. We explore in particular two different directions to build the scales of problems: a) exploiting ideas coming from multilevel optimization to propose dynamical hierarchical approaches exploiting representations of the problem of progressively reduced dimension; b) leveraging the recent advances in hardware and the possibility of representing data at multiple precision levels provided by them. We aim at improving over stateoftheart training strategies by investigating the design of scalable multilevel and mixedprecision secondorder optimization and quantization methods, possibly derivativefree.
4 Application domains
The primary objectives of this project, which is rooted in Signal Processing and Machine Learning methodology, are to develop flexible methods, endowed with solid mathematical foundations and efficient algorithmic implementations, that can be adapted to numerous application domains. We are nevertheless convinced that such methods are best developed in strong and regular connection with concrete applications, which are not only necessary to validate the approaches but also to fuel the methodological investigations with relevant and fruitful ideas. The following application domains are primarily investigated in partnership with research groups with the relevant expertise.
4.1 Frugal AI on embedded devices
There is a strong need to drastically compress signal processing and machine learning models (typically, but not only, deep neural networks) to fit them on embedded devices. For example, on autonomous vehicles, due to strong constraints (reliability, energy consumption, production costs), the memory and computing resources of dedicated highend imageanalysis hardware are two orders of magnitude more limited than what is typically required to run stateoftheart deep network models in realtime. The research conducted in the DANTE project finds direct applications in these areas, including: compressing deep neural networks to obtain lowbandwidth videocodecs that can run on smartphones with limited memory resources; sketched learning and sparse networks for autonomous vehicles; or sketching algorithms tailored to exploit optical processing units for energy efficient largescale learning.
4.2 Imaging in physics and medicine
Many problems in imaging involve the reconstruction of large scale data from limited and noisecorrupted measurements. In this context, the research conducted in DANTE pays a special attention to modeling domain knowledge such as physical constraints or prior medical knowledge. This finds applications from physics to medical imaging, including: multiphase flow image characterization; near infrared polarization imaging in circumstellar imaging; compressive sensing for joint segmentation and highresolution 3D MRI imaging; or graph signal processing for radio astronomy imaging with the Square Kilometer Array (SKA).
4.3 Interactions with computational social sciences
Based on collaborations with the relevant experts the team also regularly investigates applications in computational social science. For example, modeling infection disease epidemics requires efficient methods to reduce the complexity of large networked datasets while preserving the ability to feed effective and realistic datadriven models of spreading phenomena. In another area, estimating the vote transfer matrices between two elections is an illposed problem that requires the design of adapted regularization schemes together with the associated optimization algorithms.
5 Social and environmental responsibility
5.1 Contribution to the monitoring of the Covid19 pandemic
Robust prediction of the spatiotemporal evolution of the reproduction number $R\left(t\right)$ of the Covid19 pandemic from open data (SantéPubliqueFrance and the European Center for Disease Prevention).
Following our past work 42, where an algorithm exploiting sparsity and convex optimization was developed, and dynamic maps were proposed, we identified robustness to outliers as a critical issue.
This is addressed using convex regularization in a journal paper published this year 14.
6 Highlights of the year
7 New software and platforms
In an effort towards reproducible research, the default policy of the team is to release opensource code (typically python or matlab) associated to research papers that report experiments 36, 37, 38, 39, 40, 41. When applicable and possible, more engineered software is developed and maintained over several years to provide more robust and consistent implementations of selected results.
7.1 New software
7.1.1 FAuST

Keywords:
Learning, Sparsity, Fast transform, Multilayer sparse factorisation

Scientific Description:
FAuST allows to approximate a given dense matrix by a product of sparse matrices, with considerable potential gains in terms of storage and speedup for matrixvector multiplications.

Functional Description:
FAUST is a C++ toolbox designed to decompose a given dense matrix into a product of sparse matrices in order to reduce its computational complexity (both for storage and manipulation).
Faust includes Matlab and Python wrappers and scripts to reproduce the experimental results of the following papers:  Le Magoarou L. and Gribonval R,. "Flexible multilayer sparse approximations of matrices and applications", Journal of Selected Topics in Signal Processing, 2016.  Le Magoarou L., Gribonval R., Tremblay N. "Approximate fast graph Fourier transforms via multilayer sparse", IEEE Transactions on Signal and Information Processing over Networks, 2018  QuocTung Le, Rémi Gribonval. Structured Support Exploration For Multilayer Sparse Matrix Factorization. ICASSP 2021 – IEEE International Conference on Acoustics, Speech and Signal Processing, Jun 2021, Toronto, Ontario, Canada. pp.15.  Sibylle Marcotte, Amélie Barbe, Rémi Gribonval, Titouan Vayer, Marc Sebban, et al.. Fast Multiscale Diffusion on Graphs. 2021.

Release Contributions:
Faust 1.x contains Matlab routines to reproduce experiments of the PANAMA team on learned fast transforms.
Faust 2.x contains a C++ implementation with preliminary Matlab / Python wrappers.
Faust 3.x includes Python and Matlab wrappers around a C++ core with GPU acceleration, new algorithms.

News of the Year:
In 2022, major efforts were put to optimize code efficiency (in particular for socalled butterly structures), and an anaconda package was made available. New Faust implementations of toeplitz, circulant, dct and dst matrices and more were made available.
In 2021, new algorithms bringing improved precision and/or accelerations were incorporated into Faust, GPU support was completed together with a systematic optimization of the code (including the ability to run it in float instead of double precision), and PIP packages were made available to ease the installation of faust.
In 2020, major efforts were put into finalizing Python wrappers, producing tutorials using Jupyter notebooks and Matlab livescripts, as well as substantial refactoring of the code to optimize its efficiency and exploit GPUs.
In april 2018, a Software Development Initiative (ADT REVELATION) started in for the maturation of FAuST. A first step was to complete and robustify Matlab wrappers, to code Python wrappers with the same functionality, and to setup a continuous integration process. A second step was to simplify the parameterization of the main algorithms. The roadmap for next year includes showcasing examples and optimizing computational efficiency.
In 2017, new Matlab code for fast approximate Fourier Graph Transforms have been included. based on the approach described in the papers:
Luc Le Magoarou, Rémi Gribonval, "Are There Approximate Fast Fourier Transforms On Graphs?", ICASSP 2016 .
Luc Le Magoarou, Rémi Gribonval, Nicolas Tremblay, "Approximate fast graph Fourier transforms via multilayer sparse approximations", IEEE Transactions on Signal and Information Processing over Networks,2017.
 URL:
 Publications:

Contact:
Remi Gribonval

Participants:
Luc Le Magoarou, Nicolas Tremblay, Remi Gribonval, Nicolas Bellot, Adrien Leman, Hakim HadjDjilani
7.1.2 Celer

Keywords:
Mathematical Optimization, Machine learning, Sparsity

Functional Description:
celer is a Python package that solves Lassolike problems and provides estimators that under the popular scikitlearn API. Thanks to a tailored implementation, celer provides a fast solver that tackles largescale datasets with millions of features up to 100 times faster than scikitlearn. It handles Lasso, ElasticNet, Group Lasso, Multitask Lasso and Sparse Logistic regression, and comes with  automated parallel crossvalidation  support of sparse and dense data  optional feature centering and normalization  unpenalized intercept fitting
celer also provides easytouse estimators as it is designed under the scikitlearn API.

News of the Year:
In 2022 we added a fast solver based on coordinate descent for the Elastic Net problem.
 URL:
 Publications:

Contact:
Mathurin Massias

Participants:
Badr Moufad, Alexandre Gramfort
7.1.3 skglm

Keywords:
Optimization, Machine learning, Sparsity

Functional Description:
skglm is a Python package that offers fast estimators for Generalized Linear Models (GLMs) that are compatible with scikitlearn. It is highly flexible and supports a wide range of GLMs. Its main feature is flexibility: you can implement virtually any estimator as a combination of datafit and penalty.
Thanks to this flexible design, skglm supports many missing models in scikitlearn while ensuring high performance. There are several reasons to opt for skglm:
 SUpport for many fast solvers able to tackle large datasets, either dense or sparse, with millions of features up to 100 times faster than scikitlearn  Userfriendly API than enables composing custom estimators with any combination of existing datafits and penalties  Flexible design that makes it simple and easy to implement new datafits and penalties, a matter of few lines of code  Estimators fully compatible with the scikitlearn API and dropin replacements of its GLM estimators
skglm is integrated into scikitlearn via the scikitlearncontrib organization.

News of the Year:
2022: first release
 URL:
 Publication:

Contact:
Mathurin Massias

Participants:
Mathurin Massias, Badr Moufad
7.1.4 Benchopt

Keywords:
Mathematical Optimization, Benchmarking, Reproducibility

Functional Description:
BenchOpt is a package to simplify, make more transparent and more reproducible the comparisons of optimization algorithms. It is written in Python but it is available with many programming languages. So far it has been tested with Python, R, Julia and compiled binaries written in C/C++ available via a terminal command. If it can be installed via conda, it should just work!
BenchOpt is used through a simple command line and ultimately running and replicating an optimization benchmark should be as easy a cloning a repo and launching the computation with a single command line. For now, BenchOpt features benchmarks for around 10 convex optimization problems and we are working on expanding this to feature more complex optimization problems. We are also developing a website to display the benchmark results easily.

Release Contributions:
https://github.com/benchopt/benchopt/releases/tag/1.3.0
 Publication:

Contact:
Thomas Moreau

Participants:
Thomas Moreau, Alexandre Gramfort, Mathurin Massias, Badr Moufad
8 New results
8.1 Integrating Structured Models in Machine Learning and Signal Processing
8.1.1 Optimal Transport and Machine Learning on Graphs
Participants: Titouan Vayer.
Collaborations with Cédric VincentCuaz (PhD student, MAASAI, Université Côte d'Azur), Rémi Flamary (CMAP, Ecole Polytechnique), Marco Corneli (MAASAI, Université Côte d'Azur) and Nicolas Courty (IRISA, Université Bretagne Sud).
The GromovWasserstein (GW) distance is derived from optimal transport (OT) theory. The interest of OT lies both in its ability to provide relationships, connections, between sets of points and distances between probability distributions. By modeling graphs as probability distributions GW has become an important tool in many ML tasks involving structured data. In a previous work 65 we proposed an efficient graph dictionary learning algorithm based on GW that allows to describe graphs as a simple composition of smaller graphs (atoms of the dictionary) and we showed that these representations are particularly efficient for tasks such as change detection for structured data and clustering of graphs. In 24 we proposed an alternative approach whose goal is to learn a single graph of large size whose subgraphs will best match (according to the GW criterion) the graphs of the dataset. This approach has the merit of being much more efficient to compute and more interpretable. We also validate this method for supervised learning tasks such as classification of multiple graphs 28.
In another line of works 25, we build upon the flexibility of the optimal transport framework and GW distance to define a novel graph neural network (GNN) architecture for graphs classification. More precisely, we propose a novel graph representation as GW distances to some learnable graph templates. We postulate that the vector of GW distances to a set of template graphs has a strong discriminative power, which is then fed to a nonlinear classifier for final predictions. Distance embedding can be seen as a new layer, and can leverage on existing message passing techniques to promote sensible feature representations (and are learnt in an endtoend fashion by differentiating through this layer). We empirically validate our claim on several synthetic and real life graph classification datasets, where our method is competitive or surpasses kernel and GNN stateoftheart approaches.
8.1.2 Diffused Wasserstein Distance for Optimal Transport between Attributed Graphs
Participants: Paulo Gonçalves, Rémi Gribonval, Titouan Vayer.
This work is a collaboration with Pierre Borgnat (CNRS) from the the Physics Lab of ENS de Lyon, Marc Sebban, Professor at the LabHC of University Jean Monet, and Sibylle Marcotte (student at ENS de Rennes).
Within the Ph.D work of A. Barbe (20182021), we introduced the Diffusion Wasserstein distance, a generalization of the standard Wasserstein to undirected and connected graphs where nodes are described by feature vectors. The last advance on this subject was to reduce the computational cost of the diffusion Wasserstein distance, by proposing a Chebyshev approximation of the diffusion operator applied to the features vectors. In the course of this work, we were also able to tighten the theoretical approximation bounds, which in turn allowed to significantly improve estimates of the polynomial order for a prescribed error. This work led to a joint publication 21.
8.1.3 Structured Time Series Modeling
Participants: Titouan Vayer.
Collaborations with Romain Tavenard (IRISA, Université de Rennes 2), Laetitia Chapel (IRISA, Université Bretagne Sud), Rémi Flamary (CMAP, Ecole Polytechnique) and Nicolas Courty (IRISA, Université Bretagne Sud).
Multivariate time series are ubiquitous objects in signal processing, yet defining a distance or similarity between two such objects can be very difficult as soon as the temporal dynamics and the representation of the time series, i.e. the nature of the observed quantities, differ from one another. In the article 16, we propose a novel distance accounting for both feature space and temporal variabilities by learning a latent global transformation of the feature space together with a temporal alignment, cast as a joint optimization problem. The versatility of our framework allows for several variants depending on the structure of the time series at stake. Among other contributions, we define a differentiable loss for time series and present two algorithms for the computation of time series barycenters under this new geometry. We illustrate the interest of our approach on both simulated and real world data and show the robustness of our approach compared to stateoftheart methods.
8.2 Sparse deep neural networks : theory and algorithms
8.2.1 Mathematics of deep learning: approximation theory, scaleinvariance, and regularization
Participants: Rémi Gribonval, Antoine Gonon, Elisa Riccietti, Mathurin Massias.
Collaborations with Facebook AI Research, Paris, with Nicolas Brisebarre (ARIC team, ENS de Lyon), and with Yann Traonmilin (IMB, Bordeaux) and Samuel Vaiter (JAD, Dijon)
Neural networks with the ReLU activation function are described by weights and bias parameters, and realized a piecewise linear continuous function. Natural scalings and permutations operations on the parameters leave the realization unchanged, leading to equivalence classes of parameters that yield the same realization. These considerations in turn lead to the notion of identifiability – the ability to recover (the equivalence class of) parameters from the sole knowledge of the realization of the corresponding network. We studied this problem in depth throught the lens of a new embedding of ReLU neural network parameters of any depth. The proposed embedding is invariant to scalings and provides a locally linear parameterization of the realization of the network. Leveraging these two key properties, we derived some conditions under which a deep ReLU network is indeed locally identifiable from the knowledge of the realization on a finite set of samples. We studied the shallow case in more depth, establishing necessary and sufficient conditions for the network to be identifiable from the knowledge of its realization on some appropriate bounded domain. These results have been published this year 15.
Motivated by the importance of quantizing networks besides pruning them to achieve sparsity, we studied the expressivity of quantized deep networks from an approximation theoretic perspective 30. Our objective is to define and compare the corresponding approximation classes 7 with the unquantized ones. We also characterize the error of nearestneighbour uniform quantization of ReLU networks and we investigate when ReLU networks can be expected, or not, to have better approximation properties than other classical approximation families.
Another important challenge in deep learning is to promote sparsity during the learning phase using a regularizer. In the classical setting of linear inverse problems, it is well known that the ${\ell}^{1}$ norm is a convex regularizer lending itself to efficient optimization and endowed with stable recovery guarantees. A particular challenge is to understand to what extent using an ${\ell}^{1}$ penalty in this context is also wellfounded theoretically, and to possibly design alternate regularizers if possible.
On the one hand, we started investigating the properties of minimizers of the ${\ell}^{1}$ norm in deep learning problems. On the other hand, we considered the abstract problem of recovering elements of a lowdimensional model set from underdetermined linear measurements. Considering the minimization of a convex regularizer subject to a data fit constraint, we explored the notion of a "best" convex regularizer given a model set. This was formalized as a regularizer that maximizes a compliance measure with respect to the model. Several notions of compliance were studied and analytical expressions were obtained for compliance measures based on the bestknown recovery guarantees with the restricted isometry property. This lead to a formal proof of the optimality of the ${\ell}^{1}$norm for sparse recovery and of the nuclear norm for lowrank matrix recovery for these compliance measures. We also investigated the construction of an optimal convex regularizer using the example of sparsity in levels 34.
8.2.2 Algorithms for quantized networks
Participants: Rémi Gribonval, Elisa Riccietti.
Collaboration with Facebook AI Research Paris, Silviu Filip (IRISA, Rennes), Theo Mary (LIP6, Paris)
From a more computational perspective, we pursued the study of efficient optimization algorithms to solve problems involving quantized networks.
As a first step towards a better understanding of nonlinear quantized networks, we started from the linear case and investigated the problem of optimally quantizing low rank matrices. We showed that exploiting scaling invariances inherent to the optimization problem, much more accurate quantizations can be obtained than by a simple round to nearest strategy. We proposed an optimal solution algorithm with polynomial complexity in the dimension of the problem and exponential complexity in the number of bits.
Within the framework of the Ph.D. of Paul Estano, we studied the design of gradientbased training methods for neural networks, capable of exploiting multiple quantization levels. The proposed methods are supported by an error analysis, which suggests a good rule to switch among the available quantization levels, yielding a procedure that provides the same accuracy of classical training strategies but with a lower energy consumption.
8.2.3 Deep sparse factorizations: hardness, algorithms and identifiability
Participants: Rémi Gribonval, Elisa Ricietti, Marion Foare, Léon Zheng, QuocTung Le.
Collaboration with Valeo AI, Paris; Valérie Castin (M1 internship with DANTE)
Matrix factorization with sparsity constraints plays an important role in many machine learning and signal processing problems such as dictionary learning, data visualization, dimension reduction.
From a theoretical perspective, we pursued the study started last year on the hardness and uniqueness properties of sparse matrix factorization. Three papers have been published on this subject. First, in 13 we show that, even with only two factors and a fixed, known support, optimizing the coefficients of the sparse factors can be an NPhard problem. Second, we study the landscape of the corresponding optimization problem and exhibite "easy" instances where the problem can be solved to global optimality with an algorithm demonstrated to be orders of magnitude faster than classical gradient based methods. Then, in 17 we investigate the essential uniqueness of sparse matrix factorizations in a multilayer setting 17. More details on the case with two factors can be found in the technical report 70of last year. Third, in 20 we combine these results with a focus on socalled butterfly supports to achieve a multilayer sparse factorization algorithm able to learn fast transforms essentially at the cost of a single matrixvector multiplication, with exact recovery guarantees. A first version of the corresponding algorithm was incorporated in the FA$\mu $ST software library (see Section 7) and is subject to software optimizations to further speed it up.
Finally, we investigated extensions of these results in several directions. To improve the flexibility of the algorithm of 20 for butterfly factorization, we adapted it to socalled deformable butterlies and studied its performance guarantees beyond the case of matrices admitting an exact factorization. To embrace deep ReLU neural networks with sparsity constraints, we showed that the identifiability results of 17, 2070can be leveraged to identify (up to natural scaling ambiguities) the parameters of such networks with a prescribed butterfly structure. Finally, we investigated the closedness properties of the set of realizations of networks with prescribed support. These result are the objects of articles in preparation.
8.3 Statistical learning, dimension reduction, and privacy preservation
8.3.1 Theoretical foundations of compressive learning: sketches, kernels, and optimal transport
Participants: Rémi Gribonval, Titouan Vayer, Ayoub Belhadji.
Collaboration with Gilles Blanchard (Univ. ParisSaclay)
The compressive learning framework proposes to deal with the large scale of datasets by compressing them into a single vector of generalized random moments, called a sketch, from which the learning task is then performed. In past works we established statistical guarantees on the generalization error of this procedure, first in a general abstract setting illustrated on PCA 5, then for the specific case of compressive $k$means and compressive Gaussian Mixture Modeling 56. A tutorial paper on the principle and the main guarantees of compressive learning was also finalized and published last year 6.
Theoretical guarantees in compressive learning fundamentally rely on comparing certain metrics between probability distributions. We established some conditions under which the Wasserstein distance can be controlled by Maximum Mean Discrepancy (MMD) norms, which are defined using reproducing kernel Hilbert spaces. Based on the relations between the MMD and the Wasserstein distance, we provide new guarantees for compressive statistical learning by introducing and studying the concept of Wasserstein learnability of the learning task. The preprint submitted last year 64 is under revision.
Dimension reduction in compressive learning also exploits the ability to approximate certain kernels by finite dimensional quadratures. We revisited existing proofs of the Restricted Isometry Property of sketching operators with respect to certain mixtures models. We proposed an alternative analysis that circumvents the need to assume importance sampling when drawing random Fourier features to build random sketching operators. Our analysis is based on new deterministic bounds on the restricted isometry constant that depend solely on the set of frequencies used to define the sketching operator. Our analysis opens the door to theoretical guarantees for structured sketching with frequencies associated to fast random linear operators 29. An other related approach that we investigated consists in exploiting Determinantal Point Processes (DPPs) to obtain quadrature rules for kernels in reproducing kernel Hilbert spaces 26.
8.3.2 Practical exploration of sketching and methods with limited resources
Participants: Rémi Gribonval, Titouan Vayer, Luc Giffon, Léon Zheng, Elisa Riccietti, Rémi Vaudaine.
Collaborations with Valeo AI; LightOn SAS; Hughes Van Assel (UMPA, ENS de Lyon); Marton Karsai (CEU, Vienne, Austria)
From a more empirical perspective, we pursued our efforts to make sketching for compressive learning and sketching more versatile and efficient. This notably involved exploring how to adapt the sketching pipeline to exploit optical processing units (OPUs) for energyefficient fast random projection 27, and investigating the ability to exploit sketching in largescale deep selfsupervised learning scenarios 35.
Sketching was explored for temporal network compression. In the context of temporal networks, which can model spreading processes such as epidemics, the outcomponent of a source node is the set of nodes reachable from this node, and the distribution of the size of outcomponents is an important characteristics which computation can be demanding for large networks. We proposed both an exact online matrix algorithm with controlled complexity footprint to compute this distriution, and a sketchingbased framework to estimate it from a highly compressed representation of the temporal network.
Moreover, making the connection between graph learning and sketching methods, we recently started to study the practical possibility and theoretical limitations of using a sketching technique to estimate the precision matrix involved in the Graphical Lasso algorithm. In particular, we showed that it was possible to estimate such matrices with limited memory from a sketch based on Gaussian quadratic measurements. We pursed the practical applications of such result with structured rankone measurements.
More generally, properties of kernels methods were also exploited in a more applicative context to reduce time and memory complexity: selfsupervised learning of image representations. We introduced a regularization loss based on kernel mean embeddings with rotationinvariant kernels on the hypersphere, promoting the embedding distribution to be close to the uniform distribution on the hypersphere, with respect to the maximum mean discrepancy pseudometric 35. Besides being fully competitive with the state of the art, our method significantly reduces the resources needed for training, making it implementable for very large embedding dimensions on existing devices and more easily adjustable than previous methods to settings with limited resources.
Finally, in collaboration with Hugues Van Assel (PhD student, UMPA), we proposed and investigated a novel dimension reduction method by leveraging the optimal transport framework and entropic affinities. Our work generalizes popular approaches such as tdistributed stochastic neighbor embedding (tSNE) and has empirical benefits.
8.3.3 Privacy preservation
Participants: Rémi Gribonval, Clément Lalanne.
Collaborations with Aurélien Garivier (UMPA, ENS de Lyon) and SARUS, Paris
Producing statistics that respect the privacy of the samples while still maintaining their accuracy is an important topic of research that we addressed under the framework of differential privacy with two complementary perspectives, on selected statistical problems : the design of concrete mechanims with controlled statistical utility and provable differential privacy guarantees; and the exhibition of lowerbounds on the achievable statistical performance of any mechanism with constrained differential privacy guarantees.
We addressed the problem of differentially private estimation of multiple quantiles (MQ) of a dataset 32, a key building block in modern data analysis. We showed how to implement the nonsmoothed Inverse Sensitivity (IS) mechanism for this specific problem and established that the resulting method is closely related to the recent JointExp algorithm, sharing in particular the same computational complexity and a similar efficiency. We also identified pitfalls of the two approaches on certain peaked distributions, and proposed a fix. Numerical experiments showed that the empirical efficiency of the resulting algorithms is similar to the nonsmoothed methods for nondegenerate datasets, but orders of magnitude better on real datasets with repeated values.
We studied minimax lower bounds when the class of estimators is restricted to the differentially private ones 31. In particular, we showed that characterizing the power of a distributional test under differential privacy can be done by solving a transport problem. With specific coupling constructions, this observation allowed us to derivate Le Camtype and Fanotype inequalities for both regular definitions of differential privacy and for divergencebased ones (based on Renyi divergence). We illustrated our results on three simple, fully worked out examples. For some problems, we showed that privacy leads to a provable degradation only when the rate of the privacy parameters is small enough whereas for other problems, the degradation systematically occurs under much looser hypotheses on the privacy parameters. Finally, we showed the near minimax optimality of the known guarantees for DPSGLD, a private convex solver for maximum likelihood estimation on logconcave models.
8.4 Largescale convex and nonconvex optimization
8.4.1 Multilevel schemes for image restoration
Participants: Elisa Riccietti, Paulo Gonçalves, Guillaume Lauga.
Collaboration with Nelly Pustelnik (CNRS, ENS de Lyon)
In the context of the Ph.D. work of Guillaume Lauga, we pursued the work started last year on the study of the combination of proximal methods and multiresolution analysis in largescale image denoising problems. In the spirit of multilevel gradient methods 3 we developed a family of multilevel inertial proximal methods, tailored for problems arising in imaging, which exploit waveletsbased transfer operators. Our methods are capable of handling also problems in which the proximal operators cannot be computed explicitly. Their ability to accelerate proximal algorithms was shown in several large dimensional problems 19, 33.8.4.2 Training of physics informed neural networks
Participants: Elisa Riccietti.
Collaboration with Serge Gratton, Valentin Mercier (IRIT, Toulouse), Stefania Bellavia (UNIFI, Italy), Mattéo Clémot (PLR internship with DANTE)
Physics informed neural networks (PINNs) are special network architectures designed for the solution of partial differential equations. We studied two aspects related to the training of these networks. On the one hand, in the context of the Ph.D. work of Valentin Mercier, we studied the integration of a multigrid approach in the training to improve the approximation of solutions with multiple frequency components. On the other hand, in the context of the internship of Mattéo Clémot, we investigated the ability of PINNs to solve illposed parameter identification inverse problems and the use of regularising training procedures to correctly fit noisy data in such a context.8.4.3 Reproducible benchmarking of optimization algorithms
Participants: Mathurin Massias, Badr Moufad.
Collaborations with Thomas Moreau (MIND, Inria Saclay), Alexandre Gramfort (MIND, Inria Saclay).
To improve numerical reproducibility of optimisation benchmarks, we proposed Benchopt 22, a collaborative framework to automate, reproduce and publish optimization benchmarks in machine learning across programming languages and hardware architectures. This alleviates the burden of having many methods to reimplement, nonpublished code, and diverging stances on best practices. Benchopt (see also Section 7.1.4) simplifies benchmarking for the community by providing an offtheshelf tool for running, sharing and extending experiments.
To demonstrate the benefits of using Benchopt, we showcased benchmarks on close to twenty standard machine learning tasks, such as ResNet18 training for image classification. We published open source implementations of stateoftheart solvers on those problems, and a detailed comparison of the regimes in which they succeed and fail respectively.
8.4.4 Algorithms for large scale sparse linear models
Participants: Mathurin Massias, Badr Moufad.
Collaboration with Quentin Bertrand (MILA, Montréal), Quentin Klopfenstein (UMB, Dijon), PierreAntoine Bannier and Gauthier Gidel (MILA, Montréal), Samuel Vaiter (UMB, Dijon), Alexandre Gramfort (MIND, Inria Saclay), Joseph Salmon (IMAG, Montpellier).In a series of works, we developed new fast algorithms that allow solving optimization problems with millions of variables in the context of sparse linear models. In 18 we proposed a new working set algorithm tailored to non convex sparse penalties such as the ${\ell}_{q}$ quasinorms ($0<q<1$). It exploits the support identification properties of coordinate descent to obtain faster convergence rates using Anderson acceleration 44 This algorithm was implemented in a high level python package, skglm (see Section 7.1.3), that was integrated into the ecosystem of the scikitlearn package. In 12, we proposed an efficient meta algorithm building on the previous work to automatically tune the regularization strength of sparse convex models such as the Lasso as sparse Logistic regression.
Finally, we provided dimension independent bound for stochastic gradient descent in a non convex setting, using Langevin dynamics in infinite dimension in 23.
9 Bilateral contracts and grants with industry
9.1 Bilateral grants with industry

CIFRE contract with Valeo AI, Paris on Frugal learning with applications to autonomous vehicles
Participants: Rémi Gribonval, Elisa Riccietti, Léon Zheng.
Duration: 3 years (20212024)Partners: Valeo AI, Paris; ENS de Lyon
Funding: Valeo AI, Paris; ANRT
Context: Chaire IA AllegroAssai 10.1.1
The overall objective of this thesis is to develop machine learning methods exploiting lowdimensional sketches and sparsity to address perceptionbased learning tasks in the context of autonomous vehicles.

Funding from Facebook Artificial Intelligence Research, Paris
Participants: Rémi Gribonval.
Duration: 4 years (20212024)Partners: Facebook Artificial Intelligence Research, Paris; ENS de Lyon
Funding: Facebook Artificial Intelligence Research, Paris
Context: Chaire IA AllegroAssai 10.1.1
This is supporting the research conducted in the framework of the Chaire IA AllegroAssai.
10 Partnerships and cooperations
10.1 National initiatives
10.1.1 ANR IA Chaire : AllegroAssai
Participants: Rémi Gribonval [correspondant], Paulo Gonçalves, Elisa Ricietti, Marion Foare, Mathurin Massias, Léon Zheng, QuocTung Le, Antoine Gonon, Titouan Vayer, Ayoub Belhadji, Luc Giffon, Clement Lalanne, Can Pouliquen.
Duration of the project: 2020  2025.
AllegroAssai focuses on the design of machine learning techniques endowed both with statistical guarantees (to ensure their performance, fairness, privacy, etc.) and provable resourceefficiency (e.g. in terms of bytes and flops, which impact energy consumption and hardware costs), robustness in adversarial conditions for secure performance, and ability to leverage domainspecific models and expert knowledge. The vision of AllegroAssai is that the versatile notion of sparsity, together with sketching techniques using random features, are key in harnessing these fundamental tradeoffs. The first pillar of the project is to investigate sparsely connected deep networks, to understand the tradeoffs between the approximation capacity of a network architecture (ResNet, Unet, etc.) and its “trainability” with provablygood algorithms. A major endeavor is to design efficient regularizers promoting sparsely connected networks with provable robustness in adversarial settings. The second pillar revolves around the design and analysis of provablygood endtoend sketching pipelines for versatile and resourceefficient largescale learning, with controlled complexity driven by the structure of the data and that of the task rather than the dataset size.
10.1.2 ANR DataRedux
Participants: Paulo Gonçalves [correspondant], Rémi Gribonval, Marion Foare, Rémi Vaudaine.
Duration of the project: February 2020  January 2024.
DataRedux puts forward an innovative framework to reduce networked data complexity while preserving its richness, by working at intermediate scales (“mesoscales”). Our objective is to reach a fundamental breakthrough in the theoretical understanding and representation of rich and complex networked datasets for use in predictive datadriven models. Our main novelty is to define network reduction techniques in relation with the dynamical processes occurring on the networks. To this aim, we will develop methods to go from data to information and knowledge at different scales in a humanaccessible way by extracting structures from highresolution, diverse and heterogeneous data. Our methodology will involve the identification of the most relevant subparts of timeresolved datasets while remapping the remaining parts of the system, the simultaneous structuraltemporal representations of timevarying networks, the development of parsimonious data representations extracting meaningful structures at mesoscales (“mesostructures”), and the building of models of interactions that include mesostructures of various types. Our aim is to identify data aggregation methods at intermediate scales and new types of data representations in relation with dynamical processes, that carry the richness of information of the original data, while keeping their most relevant patterns for their manageable integration in datadriven numerical models for decision making and actionable insights.
10.1.3 ANR Darling
Participants: Paulo Gonçalves [correspondant], Rémi Gribonval, Marion Foare.
Duration of the project: February 2020  January 2024.
This project meets the compelling demand of developing a unified framework for distributed knowledge extraction and learning from graph data streaming using innetwork adaptive processing, and adjoining powerful recent mathematical tools to analyze and improve performances. The project draws on three major parallel directions of research: network diffusion, signal processing on graphs, and random matrix theory which DARLING aims at unifying into a holistic dynamic network processing framework. Signal processing on graphs has recently provided a comprehensive set of basic instruments allowing for signal on graph filtering or sampling, but it is limited to static signal models. Network diffusion on the opposite inherently assumes models of time varying graphs and signals, and has pursued the path of proposing and understanding the performance of distributed dynamic inference on graphs. Both areas are however limited by their assuming either deterministic graph or signal models, thereby entailing often inflexible and difficulttograsp theoretical results. Random matrix theory for random graph inference has taken a parallel road in explicitly studying the performance, thereby drawing limitations and providing directions of improvement, of graphbased algorithms (e.g., spectral clustering methods). The ambition of DARLING lies in the development of network diffusiontype algorithms anchored in the graph signal processing lore, rather than heuristics, which shall systematically be analyzed and improved through random matrix analysis on elementary graph models. We believe that this original communion of as yet remote areas has the potential to path the pave to the emergence of the critically needed future field of dynamical network signal processing.
10.1.4 ANR JCJC MASSILIA
Participants: Titouan Vayer.
Duration of the project: December 2021  December 2025.
Collaboration with Arnaud Breloy (PI of the project, Univ. Paris Nanterre), Florent Bouchard (CentraleSupélec), Cédric Richard (Univ. Côte d'Azur), Rémi Flamary (Ecole Polytechnique) and Ammar Mian (Univ. Savoie Mont Blanc)
This project aims at tackling current problems related to graph learning and its applications in a unified way centered around the spectral decomposition of the graph Laplacian and/or adjacency matrices. The central objective of this project is to model graph structures (distributions on spectral parameters) and leverage this formalism in to two main directions 1) improve graph learning processes by directly learning structured spectral decompositions from the data 2) handle collections of graphs in order to compute structured graphs barycenters, compress graphs representations, and classify/cluster data using their graph as the main feature.
10.1.5 GDR ISIS project MOMIGS
Participants: Elisa Riccietti [correspondant], Marion Foare, Trieu Vy Le Hoang, Paulo Gonçalves.
Duration of the project: September 2021  September 2023.
This project focuses on large scale optimization problems in signal processing and imaging. A natural way to tackle them is to exploit their underlying structure, and to represent them at different resolution levels. The use of multiresolution schemes, such as wavelets transforms, is not new in imaging and is widely used to define regularization strategies. However, such techniques could be used to a wider extent, in order to accelerate the optimization algorithms used for their solution and to tackle large datasets. Techniques based on such ideas are usually called multilevel optimization methods and are wellknown and widely used in the field of smooth optimization and especially in the solution of partial differential equations. Optimization problems arising in image reconstruction are however usually nonsmooth and thus solved by proximal methods. Such approaches are efficient for smallscale problems but still computationally demanding for problems with very highdimensional data. The ambition of this project is thus to combine proximal methods and multiresolution analysis not only as a regularization, but as a solution to accelerate proximal algorithms.
10.2 Regional initiatives
10.2.1 Labex CominLabs LeanAI
Participants: Elisa Riccietti [correspondant], Rémi Gribonval.
Duration of the project: October 2021December 2024.
Collaboration with SilviuIoan Filip and Olivier Sentieys (IRISA, Rennes), Anastasia Volkova (LS2N Nantes)
The LeanAI project aims at developing a comprehensive and flexible framework for mixedprecision optimization. The project is motivated by the increasing demand for intelligent edge devices capable of onsite learning, driven by the recent developments in deep learning. The realization of such systems is a massive challenge due to the limited resources available in an embedded context and the massive training costs for stateoftheart deep neural networks. In this project we attack these problems at the arithmetic and algorithmic levels by exploring the design of new mixed numerical precision algorithms, energyefficient and capable of offering increased performance in a resourcerestricted environment. The ambition of the project is to develop more flexible and faster techniques than existing reducedprecision gradient algorithms, by determining the best numeric formats to be used in combination with this kind of methods, rules to dynamically adjust the precision and extension of such techniques to secondorder and multilevel strategies.
10.2.2 Labex Emerging Topics
Participants: Marion Foare [correspondant].
Duration of the project: April 2019December 2022.
Collaboration with Eric Van Reeth (Creatis, Lyon)
Magnetic Resonance Imaging (MRI) is an extremely important anatomical and functional imaging technique, widely used by physicists to establish medical diagnosis. Acquiring high resolution volumes is desirable in many clinical and preclinical applications to accurately adapt the treatment to the measurements, or simply obtain highly resolved images of small anatomical structures. However, directly acquiring highresolution volumes implies: i) long scanning times, which are often not tolerated by patients and children, and ii) images with low signaltonoise ratio. Therefore, it is of particular interest to quickly acquire lowresolution volumes, and enhance their resolution as a postprocessing step. This project aims at developing new techniques to build superresolution images for 3D MRI, that can take into account more physical constraints, such as prior medical knowledge, and to derive efficient machine learning algorithms suited for large scale data, with theoretical guarantees. In particular, we explore specialized piecewise smooth reconstruction variational methods, like the MumfordShah (MS) and the Total Variation (TV) variants, and to adapt their fitting terms as well as their optimization algorithms. The main originality of this project is to combine resolution enhancement and segmentation in MRI (usually performed as two distinct postprocessing steps), starting from the MS model, a seminal tool originally designed for image denoising and segmentation tasks. This approach will improve the quality of the reconstruction both in terms of sharpness and smoothness, and help the doctors with reaching a diagnosis.
11 Dissemination
Participants: Rémi Gribonval, Paulo Gonçalves, Marion Foare, Mathurin Massias, Elisa Riccietti, Titouan Vayer.
11.1 Promoting scientific activities
11.1.1 Scientific events: organisation
Member of the organizing committees
 Rémi Gribonval, Journées de Statistiques 2022, Lyon
 Mathurin Massias, Elisa Riccietti, Rémi Gribonval, Journées SMAIMODE 2024, Lyon
 Mathurin Massias, Learning and Optimization in Luminy 2024, CIRM, Marseille
 Titouan Vayer, Graph Learning & Learning with Graphs, special session at GRETSI conference 2023.
11.1.2 Scientific events: selection
Member of the conference program committees
 Rémi Gribonval, GRETSI 2022, GRETSI 2023
 Rémi Gribonval, 10th SMAISIGMA conference on Curves and Surfaces
 Rémi Gribonval, 2022 Spring School on Machine Learning (EPIT22), CIRM, Spring 2022
 Rémi Gribonval, MiLYON Spring School on Machine Learning, SaintEtienne, Spring 2021 (postponed to 2022 then cancelled due to Covid19)
11.1.3 Journal
Member of the editorial boards
 Rémi Gribonval: Associate Editor for Constructive Approximation (Springer), Senior Area Editor for the IEEE Signal Processing Magazine
 Mathurin Massias: Associate editor for Computo (French Statistical Society)
11.1.4 Invited talks
 Titouan Vayer was an invited speaker at the 2nd InriaDFKI European Summer School on AI (IDESSAI 2022) and at the conference Machine Learning and Signal Processing on Graphs (CIRM 2022).
 Mathurin Massias was an invited speaker at the Learning and Optimization in Luminy 2022 conference (CIRM) and an invited lecturer at the Computation and Modelling (Wrocław University of Science and Technology, Poland)
 Elisa Riccietti was an invited speaker at the workshop LowRank Models and Applications (LRMA 2022) in Mons, Belgium.
11.1.5 Leadership within the scientific community
 Rémi Gribonval is a member of the Scientific Committee of RT MIA (formerly GDR MIA)
 Rémi Gribonval is a member of the Comité de Liaison SIGMASMAI
 Rémi Gribonval is a member of the Cellule ERC of INS2I, mentoring for ERC candidates in the STIC domain
11.1.6 Scientific expertise
 Rémi Gribonval is a member of the Scientific Advisory Board (vicepresident) of the Acoustics Research Institute of the Austrian Academy of Sciences, and a member of the Commission Prospective of Institut de Mathématiques de Marseille
 Elisa Riccietti is a member of the "commission formation" of the labex MILyon
11.1.7 Research administration
 Paulo Gonçalves is Deputy Scientific Director of the new research center of Inria in Lyon and member of the Inria Evaluation Committee since sept. 2022.
11.2 Teaching  Supervision  Juries
11.2.1 Teaching
 Master :
 Rémi Gribonval: Inverse problems and high dimension; Mathematical foundations of deep neural networks; Concentration of measure in probability and highdimensional statistical learning; M2, ENS Lyon
 Mathurin Massias: Large scale optimization for Machine Learning; M2, ENS Lyon
 Mathurin Massias: Python for Datascience; M1, Ecole Polytechnique/HEC
 Engineer cycle (Bac+3 to Bac+5):
 Paulo Gonçalves: Traitement du Signal (déterministe, aléatoire, numérique), Estimation statistique. 80 heures Eq. TD. CPE Lyon, France
 Marion Foare: Traitement du Signal (déterministe, numérique, aléatoire), Traitement et analyse d'images, Optimisation, Compression, Projets. 280 heures Eq. TD. CPE Lyon, France
 Elisa Riccietti: M1 courses on Optimization and Approximation and Fundamentals of Machine Learning. 19h of tutor responsibility at ENS Lyon
 Other formations: ‘‘Fondements et pratique du machine learning et du deep learning”, CNRS formation, 5 days (20h) with Rémi Gribonval, Mathurin Massias and Titouan Vayer.
11.2.2 Supervision
All PhD students of the team are cosupervised by at least one team member. In addition, some team members are involved in the cosupervision of students hosted in other labs.
 Marion Foare is involved in the cosupervision of the Ph.D. of Hoang Trieu Vy Le since 2021 (Laboratoire de Physique, Lyon).
 Elisa Riccietti is involved in the cosupervision of the Ph.D. of Valentin Mercier since 2021 (IRIT, Toulouse).
 Elisa Riccietti is involved in the cosupervision of the Ph.D. of Paul Estano since 2022 (IRISA, Rennes).
 Rémi Gribonval is involved in the cosupervision of the Ph.D. of Sibylle Marcotte since 2022 (Center for Data Science, ENS Paris).
No PhD was defended in DANTE in 2022
11.2.3 Juries
Members of the DANTE team participated to the following juries:
 PhD juries: Antoine Mazarguil (Université Paris Cité, member); ElMehdi Achour (Université de Toulouse, member); Nina Göttschling (University of Cambridge, external reviewer); Paul Viallard (Université de Lyon, chair); PierreHugo Vial (Université de Toulouse, chair); Ruben Ohana (Sorbonne Université, reviewer); Thomas Debarre (EPFL, external examiner); Gilles Bareilles (Université GrenobleAlpes, member); Florent Bascou (Université de Montpellier, member); Florian Mouret (Université de Toulouse, member); Mikhail Kamalov (Université Côte d'Azur, member)
 Habilitation juries: Antoine Liutkus (Université de Montpellier, reviewer); Vincent Duval (Université ParisDauphine PSL, member)
12 Scientific production
12.1 Major publications

1
article
${L}^{}$ PageRank for SemiSupervised Learning.Applied Network Science4572019, 120  2 articleImplicit differentiation for fast hyperparameter selection in nonsmooth convex learning.Journal of Machine Learning Research23149April 2022, 143
 3 articleOn a multilevel Levenberg–Marquardt method for the training of artificial neural networks and its application to the solution of partial differential equations.Optimization Methods and Software2020, 126
 4 articleSemiLinearized Proximal Alternating Minimization for a Discrete MumfordShah Model.IEEE Transactions on Image ProcessingOctober 2019, 113
 5 articleCompressive Statistical Learning with Random Feature Moments.Mathematical Statistics and LearningMain novelties between version 1 and version 2: improved concentration bounds, improved sketch sizes for compressive kmeans and compressive GMM that now scale linearly with the ambient dimensionMain novelties of version 3: all content on compressive clustering and compressive GMM is now developed in the companion paper hal02536818; improved statistical guarantees in a generic framework with illustration of the improvements on compressive PCA2021
 6 articleSketching Data Sets for LargeScale Learning: Keeping only what you need.IEEE Signal Processing Magazine385September 2021, 1236
 7 articleApproximation spaces of deep neural networks.Constructive Approximation2020
 8 articleDual Extrapolation for Sparse Generalized Linear Models.Journal of Machine Learning Research21234October 2020, 133
 9 articleFourier could be a Data Scientist: from Graph Fourier Transform to Signal Processing on Graphs.Comptes Rendus. PhysiqueSeptember 2019, 474488
 10 inproceedingsSemirelaxed Gromov Wasserstein divergence with applications on graphs.ICLR 2022  10th International Conference on Learning RepresentationsVirtual, FranceApril 2022, 128
 11 inproceedingsTemplate based Graph Neural Network with Optimal Transport Distances.NeurIPS 2022 – 36th Conference on Neural Information Processing SystemsNew Orleans, United States2022
12.2 Publications of the year
International journals
 12 articleImplicit differentiation for fast hyperparameter selection in nonsmooth convex learning.Journal of Machine Learning Research23149April 2022, 143
 13 articleSpurious Valleys, NPhardness, and Tractability of Sparse Matrix Factorization With Fixed Support.SIAM Journal on Matrix Analysis and Applications2022
 14 articleNonsmooth convex optimization to estimate the Covid19 reproduction number spacetime evolution with robustness against low quality data.IEEE Transactions on Signal Processing70June 2022, 28592868
 15 articleAn Embedding of ReLU Networks and an Analysis of their Identifiability.Constructive Approximation2022
 16 articleTime Series Alignment with Global Invariances.Transactions on Machine Learning ResearchOctober 2022
 17 articleEfficient Identification of Butterfly Sparse Matrix Factorizations.SIAM Journal on Mathematics of Data Science2022
International peerreviewed conferences
 18 inproceedingsBeyond L1: Faster and Better Sparse Models with skglm.NeurIPSNew Orleans, United StatesNovember 2022
 19 inproceedingsMéthodes proximales multiniveaux pour la restauration d'images.GRETSI'22  28ème Colloque Francophone de Traitement du Signal et des ImagesNancy, FranceSeptember 2022
 20 inproceedingsFast learning of fast transforms, with guarantees.ICASSP 2022  IEEE International Conference on Acoustics, Speech and Signal ProcessingSingapore, SingaporeMay 2022
 21 inproceedingsFast Multiscale Diffusion on Graphs.ICASSP 2022  IEEE International Conference on Acoustics, Speech and Signal ProcessingSingapore, SingaporeMay 2022
 22 inproceedingsBenchopt: Reproducible, efficient and collaborative optimization benchmarks.NeurIPS 2022  36th Conference on Neural Information Processing SystemsNew Orleans, United StatesNovember 2022
 23 inproceedingsDimensionfree convergence rates for gradient Langevin dynamics in RKHS.COLT 2022  35th Annual Conference on Learning TheoryLondon, United KingdomJuly 2022
 24 inproceedingsSemirelaxed Gromov Wasserstein divergence with applications on graphs.ICLR 2022  10th International Conference on Learning RepresentationsVirtual, FranceApril 2022, 128
 25 inproceedingsTemplate based Graph Neural Network with Optimal Transport Distances.NeurIPS 2022 – 36th Conference on Neural Information Processing SystemsNew Orleans, United States2022
National peerreviewed Conferences
 26 inproceedingsDes règles de quadrature dans les RKHSs à base de DPPs.GRETSI 2022  XXVIIIème Colloque Francophone de Traitement du Signal et des ImagesNancy, FranceSeptember 2022, 14
 27 inproceedingsCompressive Clustering with an Optical Processing Unit.GRETSI 2022  XXVIIIème Colloque Francophone de Traitement du Signal et des ImagesNancy, FranceSeptember 2022
 28 inproceedingsSemirelaxed GromovWasserstein divergence for graphs classification.Colloque GRETSI 2022  XXVIIIème Colloque Francophone de Traitement du Signal et des ImagesNancy, FranceSeptember 2022
Reports & preprints
 29 miscRevisiting RIP guarantees for sketching operators on mixture models.November 2022
 30 miscApproximation speed of quantized vs. unquantized ReLU neural networks and beyond.May 2022
 31 miscOn the Statistical Complexity of Estimation and Testing under Privacy Constraints.October 2022
 32 miscPrivate Quantiles Estimation in the Presence of Atoms.February 2022
 33 miscMultilevel FISTA for image restoration.October 2022
 34 miscA theory of optimal convex regularization for lowdimensional recovery.December 2022
 35 miscSelfsupervised learning with rotationinvariant kernels.October 2022
12.3 Other
Softwares
 36 softwareCode for the paper "Structured Support Exploration For Multilayer Sparse Matrix Factorization".February 2022BSD3 Clause License
 37 softwareCode for reproducible research  "Spurious Valleys, NPhardness, and Tractability of Sparse Matrix Factorization With Fixed Support".May 2022BSD 2Clause FreeBSD License
 38 softwareCode for reproducible research  Fast Multiscale Diffusion on Graphs.February 2022BSD 3Clause License
 39 softwareCode for reproducible research  Fast learning of fast transforms, with guarantees.February 2022BSD 3Clause License
 40 softwareCode for reproducible research  Selfsupervised learning with rotationinvariant kernels.July 2022Apache License 2.0
 41 softwareCode for reproducible research  Efficient Identification of Butterfly Sparse Matrix Factorizations.March 2022BSD 3Clause License
12.4 Cited publications
 42 articleSpatial and temporal regularization to estimate COVID19 reproduction number R(t): Promoting piecewise smoothness via convex optimization.PLoS ONE158August 2020, e0237901
 43 bookConvex analysis and monotone operator theory in Hilbert spaces.408Springer2011
 44 inproceedingsAnderson acceleration of coordinate descent.International Conference on Artificial Intelligence and StatisticsPMLR2021, 12881296
 45 bookH.Holger BocheR.Robert CalderbankG.Gitta KutyniokJ.Jan VybiralCompressed Sensing and its Applications.Series: Applied and Numerical Harmonic AnalysisMATHEON Workshop 2013ISSN: 22965009Please note that you have the right to download and disseminate single chapters from the book that are authored by you and that are created and provided by Springer only for your private and professional noncommercial research and classroom use (e.g. sharing the chapter by mail or in hardcopy form with research colleagues for their professional noncommercial research and classroom use, or to use it for presentations or handouts for students). You are also entitled to use single chapters for the further development of your scientific career (e.g. by copying and attaching chapters to an electronic or hardcopy job or grant application). If you are an editor, book author or chapter author, please ask the (co)author(s) of the respective individual chapter for approval before you share it with other scientists since sharing chapters requires the prior consent of any coauthor(s) of the chapter. Posting of the book or a chapter on your homepage or deposit on repositories of third parties is not allowed.ChamBirkhäuser, Cham2015, URL: http://books.google.cz/books?id=6KoYCgAAQBAJ&pg=PA340&dq=intitle:Compressed+Sensing+and+its+Applications&hl=&cd=1&source=gbs_api
 46 articleExact Reconstruction using Beurling Minimal Extrapolation.arXiv.orgarXiv: 1103.4951v2March 2011, URL: http://arxiv.org/abs/1103.4951v2
 47 articleCompressive Learning with Privacy Guarantees.Information and Inference2021
 48 incollectionProximal splitting methods in signal processing.Fixedpoint algorithms for inverse problems in science and engineeringSpringer2011, 185212
 49 articleDistributed Adaptive Learning of Graph Signals .IEEE Transaction on Signal Processing65162017
 50 bookCooperative and Graph Signal Processing: Principle and Applications.Academic Press2018
 51 bookSparse and Redundant Representations.From Theory to Applications in Signal and Image ProcessingSpringer2010, URL: http://books.google.fr/books?id=d5b6lJI9BvAC&printsec=frontcover&dq=sparse+and+redundant+representations&hl=&cd=1&source=gbs_api
 52 bookA Mathematical Introduction to Compressive Sensing.New York, NYSpringer2013, URL: http://link.springer.com/10.1007/9780817649487
 53 articleSparse inverse covariance estimation with the graphical lasso .Biostatistics932008, 432441
 54 articleTranslation on Graphs: An Isometric Shift Operator.IEEE Signal Processing Letters2212December 2015, 2416  2420
 55 articleCompressive Statistical Learning with Random Feature Moments.Mathematical Statistics and Learning2021, URL: https://hal.inria.fr/hal01544609
 56 articleStatistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling.Mathematical Statistics and Learning32This preprint results from a split and profound restructuring and improvements of of https://hal.inria.fr/hal01544609v2It is a companion paper to https://hal.inria.fr/hal01544609v3August 2021, 165257
 57 articleStructured Variable Selection with SparsityInducing Norms.Journal of Machine Learning Research12Publisher: Massachusetts Institute of Technology Press2011, 27772824URL: http://hal.inria.fr/inria00377732
 58 articleA unified Framework for Structured Graph Learning via Spectral Constraints .Journal of Machine Learning Research212020, 160
 59 inproceedingsRandom features for largescale kernel machines.NIPSReplace implicit mapping of kernel trick by explicit nonlinear mapping from R⌃2007
 60 articleSubsampled Newton methods.Math. Program.1742019, 293326
 61 articleThe Emerging Field of Signal Processing on Graphs .IEEE Signal Processing MagazineMay 2013, 8398
 62 articleHilbert Space Embeddings and Metrics on Probability Measures..JMLR11Theorem 21 relates Wasserstein metric to Kernel metric2010, 15171561URL: http://dblp.org/rec/journals/jmlr/SriperumbudurGFSL10
 63 articleDictionary Learning.IEEE Signal Processing Magazine2822738URL: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5714407
 64 unpublishedControlling Wasserstein distances by Kernel norms with application to Compressive Statistical Learning.December 2021, working paper or preprint
 65 inproceedingsOnline Graph Dictionary Learning.ICML 2021  38th International Conference on Machine LearningVirtual Conference, United StatesJuly 2021
 66 inproceedingsE2train: Training stateoftheart cnns with over 80% energy savings.Advances in Neural Information Processing Systems2019, 51385150
 67 inproceedingsSWALP: Stochastic weight averaging in low precision training.International Conference on Machine Learning2019, 70157024
 68 articleADAHESSIAN: An adaptive second order optimizer for machine learning.arXiv preprint arXiv:2006.007192020
 69 inproceedingsNonconvex Sparse Graph Learning under Laplacian Constrained Graphical Model.34th Conference on Neural Information Processing Systems2020
 70 unpublishedIdentifiability in TwoLayer Sparse Matrix Factorization.November 2021, working paper or preprint