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113 results on '"Soubies, Emmanuel"'

1. Convexity in ReLU Neural Networks: beyond ICNNs?

Author

Gagneux, Anne, Massias, Mathurin, Soubies, Emmanuel, and Gribonval, Rémi

Subjects
Computer Science - Machine Learning
Abstract

Convex functions and their gradients play a critical role in mathematical imaging, from proximal optimization to Optimal Transport. The successes of deep learning has led many to use learning-based methods, where fixed functions or operators are replaced by learned neural networks. Regardless of their empirical superiority, establishing rigorous guarantees for these methods often requires to impose structural constraints on neural architectures, in particular convexity. The most popular way to do so is to use so-called Input Convex Neural Networks (ICNNs). In order to explore the expressivity of ICNNs, we provide necessary and sufficient conditions for a ReLU neural network to be convex. Such characterizations are based on product of weights and activations, and write nicely for any architecture in the path-lifting framework. As particular applications, we study our characterizations in depth for 1 and 2-hidden-layer neural networks: we show that every convex function implemented by a 1-hidden-layer ReLU network can be also expressed by an ICNN with the same architecture; however this property no longer holds with more layers. Finally, we provide a numerical procedure that allows an exact check of convexity for ReLU neural networks with a large number of affine regions.

Published
2025

2. Exact continuous relaxations of l0-regularized criteria with non-quadratic data terms

Author

Essafri, M'hamed, Calatroni, Luca, and Soubies, Emmanuel

Subjects
Mathematics - Optimization and Control
Abstract

We propose a new class of exact continuous relaxations of l0-regularized criteria involving non-quadratic data terms such as the Kullback-Leibler divergence and the logistic regression, possibly combined with an l2 regularization. We first prove the existence of global minimizers for such problems and characterize their local minimizers.Then, we propose the l0 Bregman Relaxation (B-rex), a continuous approximation of the l0 pseudo-norm defined in terms of suitable Bregman distances, which leads to an exact continuous relaxations of the original l0-regularized problem in the sense that it does not alter its set of global minimizers and reduces the non-convexity by eliminating certain local minimizers. Both features make the relaxed problem more amenable to be solved by standard non-convex optimization algorithms. In this spirit, we consider the proximal gradient algorithm and provide explicit computation of proximal points for the B-rex penalty in several cases. Finally, we report a set of numerical results illustrating the geometrical behavior of the proposed B-rex penalty for different choices of the underlying Bregman distance, its relation with convex envelopes, as well as its exact relaxation properties in 1D/2D and higher dimensions.

Published
2024

3. Optical Diffraction Tomography Meets Fluorescence Localization Microscopy

Author

Pham, Thanh-An, Soubies, Emmanuel, Soulez, Ferréol, and Unser, Michael

Subjects
Electrical Engineering and Systems Science - Signal Processing, Physics - Optics
Abstract

We show that structural information can be extracted from single molecule localization microscopy (SMLM) data. More precisely, we reinterpret SMLM data as the measures of a phaseless optical diffraction tomography system for which the illumination sources are fluorophores within the sample. Building upon this model, we propose a joint optimization framework to estimate both the refractive index map and the position of fluorescent molecules from the sole SMLM frames., Comment: Presented in ISCS23

Published
2023

4. Deep Unrolling for Nonconvex Robust Principal Component Analysis

Author

Tan, Elizabeth Z. C., Chaux, Caroline, Soubies, Emmanuel, and Tan, Vincent Y. F.

Subjects
Electrical Engineering and Systems Science - Signal Processing, Computer Science - Machine Learning
Abstract

We design algorithms for Robust Principal Component Analysis (RPCA) which consists in decomposing a matrix into the sum of a low rank matrix and a sparse matrix. We propose a deep unrolled algorithm based on an accelerated alternating projection algorithm which aims to solve RPCA in its nonconvex form. The proposed procedure combines benefits of deep neural networks and the interpretability of the original algorithm and it automatically learns hyperparameters. We demonstrate the unrolled algorithm's effectiveness on synthetic datasets and also on a face modeling problem, where it leads to both better numerical and visual performances., Comment: 7 pages, 3 figures; Accepted to the 2023 IEEE International Workshop on Machine Learning for Signal Processing

Published
2023

5. Accelerating Non-Negative and Bounded-Variable Linear Regression Algorithms with Safe Screening

Author

Dantas, Cassio F., Soubies, Emmanuel, and Févotte, Cédric

Subjects
Computer Science - Machine Learning, Statistics - Machine Learning
Abstract

Non-negative and bounded-variable linear regression problems arise in a variety of applications in machine learning and signal processing. In this paper, we propose a technique to accelerate existing solvers for these problems by identifying saturated coordinates in the course of iterations. This is akin to safe screening techniques previously proposed for sparsity-regularized regression problems. The proposed strategy is provably safe as it provides theoretical guarantees that the identified coordinates are indeed saturated in the optimal solution. Experimental results on synthetic and real data show compelling accelerations for both non-negative and bounded-variable problems.

Published
2022

6. Leveraging Joint-Diagonalization in Transform-Learning NMF

Author

Zhang, Sixin, Soubies, Emmanuel, and Févotte, Cédric

Subjects
Computer Science - Machine Learning, Electrical Engineering and Systems Science - Signal Processing
Abstract

Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approach -- termed as JD+NMF -- that estimates the transform through JD, prior to NMF computation. In contrast, we found that when the number of data realizations is limited, not only is JD+NMF no longer equivalent to TL-NMF, but the inherent low-rank constraint of TL-NMF turns out to be an essential ingredient to learn meaningful transforms for NMF.

Published
2021
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7. Expanding boundaries of Gap Safe screening

Author

Dantas, Cassio F., Soubies, Emmanuel, and Févotte, Cédric

Subjects
Computer Science - Machine Learning, Electrical Engineering and Systems Science - Signal Processing, Mathematics - Optimization and Control
Abstract

Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).

Published
2021

8. Joint Angular Refinement and Reconstruction for Single-Particle Cryo-EM

Author

Zehni, Mona, Donati, Laurène, Soubies, Emmanuel, Zhao, Zhizhen J., and Unser, Michael

Subjects
Electrical Engineering and Systems Science - Image and Video Processing, Electrical Engineering and Systems Science - Signal Processing
Abstract

Single-particle cryo-electron microscopy (cryo-EM) reconstructs the three-dimensional (3D) structure of bio-molecules from a large set of 2D projection images with random and unknown orientations. A crucial step in the single-particle cryo-EM pipeline is 3D refinement, which resolves a high-resolution 3D structure from an initial approximate volume by refining the estimation of the orientation of each projection. In this work, we propose a new approach that refines the projection angles on the continuum. We formulate the optimization problem over the density map and the orientations jointly. The density map is updated using the efficient alternating-direction method of multipliers, while the orientations are updated through a semi-coordinate-wise gradient descent for which we provide an explicit derivation of the gradient. Our method eliminates the requirement for a fine discretization of the orientation space and does away with the classical but computationally expensive template-matching step. Numerical results demonstrate the feasibility and performance of our approach compared to several baselines.

Published
2020
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9. Three-Dimensional Optical Diffraction Tomography with Lippmann-Schwinger Model

Author

Pham, Thanh-an, Soubies, Emmanuel, Ayoub, Ahmed, Lim, Joowon, Psaltis, Demetri, and Unser, Michael

Subjects
Electrical Engineering and Systems Science - Image and Video Processing, Physics - Optics
Abstract

A broad class of imaging modalities involve the resolution of an inverse-scattering problem. Among them, three-dimensional optical diffraction tomography (ODT) comes with its own challenges. These include a limited range of views, a large size of the sample with respect to the illumination wavelength, and optical aberrations that are inherent to the system itself. In this work, we present an accurate and efficient implementation of the forward model. It relies on the exact (nonlinear) Lippmann-Schwinger equation. We address several crucial issues such as the discretization of the Green function, the computation of the far field, and the estimation of the incident field. We then deploy this model in a regularized variational-reconstruction framework and show on both simulated and real data that it leads to substantially better reconstructions than the approximate models that are traditionally used in ODT., Comment: 17 pages

Published
2019

10. Pocket Guide to Solve Inverse Problems with GlobalBioIm

Author

Soubies, Emmanuel, Soulez, Ferréol, McCann, Michael T., Pham, Thanh-an, Donati, Laurène, Debarre, Thomas, Sage, Daniel, and Unser, Michael

Subjects
Computer Science - Mathematical Software
Abstract

GlobalBioIm is an open-source MATLAB library for solving inverse problems. The library capitalizes on the strong commonalities between forward models to standardize the resolution of a wide range of imaging inverse problems. Endowed with an operator-algebra mechanism, GlobalBioIm allows one to easily solve inverse problems by combining elementary modules in a lego-like fashion. This user-friendly toolbox gives access to cutting-edge reconstruction algorithms, while its high modularity makes it easily extensible to new modalities and novel reconstruction methods. We expect GlobalBioIm to respond to the needs of imaging scientists looking for reliable and easy-to-use computational tools for solving their inverse problems. In this paper, we present in detail the structure and main features of the library. We also illustrate its flexibility with examples from multichannel deconvolution microscopy.

Published
2018
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11. The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

Author

Denoyelle, Quentin, Duval, Vincent, Peyré, Gabriel, and Soubies, Emmanuel

Subjects
Mathematics - Numerical Analysis, Mathematics - Optimization and Control
Abstract

This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

Published
2018

12. Efficient Inversion of Multiple-Scattering Model for Optical Diffraction Tomography

Author

Soubies, Emmanuel, Pham, Thanh-An, and Unser, Michael

Subjects
Computer Science - Computational Engineering, Finance, and Science, Computer Science - Numerical Analysis, Physics - Data Analysis, Statistics and Probability, Physics - Optics
Abstract

Optical diffraction tomography relies on solving an inverse scattering problem governed by the wave equation. Classical reconstruction algorithms are based on linear approximations of the forward model (Born or Rytov), which limits their applicability to thin samples with low refractive-index contrasts. More recent works have shown the benefit of adopting nonlinear models. They account for multiple scattering and reflections, improving the quality of reconstruction. To reduce the complexity and memory requirements of these methods, we derive an explicit formula for the Jacobian matrix of the nonlinear Lippmann-Schwinger model which lends itself to an efficient evaluation of the gradient of the data- fidelity term. This allows us to deploy efficient methods to solve the corresponding inverse problem subject to sparsity constraints.

Published
2017
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14. Surpassing light inhomogeneities in structured‐illumination microscopy with FlexSIM.

Author

Soubies, Emmanuel, Nogueron, Alejandro, Pelletier, Florence, Mangeat, Thomas, Leterrier, Christophe, Unser, Michael, and Sage, Daniel

Subjects
*IMAGE reconstruction, *MICROSCOPY, *FLUORESCENCE, *NOISE
Abstract

Super‐resolution structured‐illumination microscopy (SIM) is a powerful technique that allows one to surpass the diffraction limit by up to a factor two. Yet, its practical use is hampered by its sensitivity to imaging conditions which makes it prone to reconstruction artefacts. In this work, we present FlexSIM, a flexible SIM reconstruction method capable to handle highly challenging data. Specifically, we demonstrate the ability of FlexSIM to deal with the distortion of patterns, the high level of noise encountered in live imaging, as well as out‐of‐focus fluorescence. Moreover, we show that FlexSIM achieves state‐of‐the‐art performance over a variety of open SIM datasets. [ABSTRACT FROM AUTHOR]

Published
2024
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22. Non-identifiability visualisation algorithms

Author

Munier, Nathanaël, Soubies, Emmanuel, Weiss, Pierre, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Centre de Biologie Intégrative (CBI), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Groupe de Recherche et d'Etudes de Traitement du Signal et des Images, ANR-21-CE48-0008,MICROBLIND,Problèmes inverses aveugles et microscopie optique(2021), and ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011)

Subjects
Problème inverse, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Identifiabilité, [INFO]Computer Science [cs], [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG], [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing
Abstract

International audience; The identification of a set of parameters x from observed data y is often reduced to the resolution of a problem of the type f(x) = y, where f is a continuous application. This equation appears in domains such as (blind) inverse problems, data assimilation, or optimal control. When the solution is not unique, it may be important to inform the user about the set of plausible solutions. We propose here an algorithm to locally describe a set of solutions of this equation. It is essentially based on automatic differentiation and on a local parameterization of the solution manifold. We illustrate the interest of this approach on two toy examples.; L'identification d'un ensemble de paramètres x à partir de données observées y est souvent réduite à la résolution d'un problème du type f(x) = y, où f est une application continue. Cette équation apparaît dans des domaines tels que les problèmes inverses (aveugles), l'assimilation de données ou encore le contrôle optimal. Lorsque la solution n'est pas unique, il peut être important d'en informer l'utilisateur et de le renseigner sur l'ensemble des solutions plausibles. Nous proposons ici un algorithme permettant de décrire localement un ensemble de solutions de cette équation. Il repose essentiellement sur la différentiation automatique et sur une paramétrisation locale de la variété des solutions. Nous illustrons l'intérêt de cette approche sur deux exemples jouets.

Published
2023

26. The MLE is a reliable source: sharp performance guarantees for localization problems

Author

Munier, Nathanaël, Soubies, Emmanuel, Weiss, Pierre, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Centre de Biologie Intégrative (CBI), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), ANR-21-CE48-0008,MICROBLIND,Problèmes inverses aveugles et microscopie optique(2021), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Munier, Nathanaël, Problèmes inverses aveugles et microscopie optique - - MICROBLIND2021 - ANR-21-CE48-0008 - AAPG2021 - VALID, and Centre International de Mathématiques et d'Informatique (de Toulouse) - - CIMI2011 - ANR-11-LABX-0040 - LABX - VALID

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [STAT]Statistics [stat], [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, [STAT] Statistics [stat], [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing
Abstract

28 pages; Single source localization from low-pass filtered measurements is ubiquitous in optics, wireless communications and sound processing. We analyse the performance of the maximum likelihood estimator (MLE) in this context with additive white Gaussian noise. We derive necessary conditions and sufficient conditions on the maximum admissible noise level to reach a given precision with high probability. The two conditions match closely, with a discrepancy related to the conditioning of a noiseless cost function. They tightly surround the Cramér-Rao lower bound for low noise levels. However, they are significantly more precise for larger levels. An outcome is new optimization criteria for the design of point spread functions in single molecule microscopy.

Published
2022

27. On the Relationships between Transform-Learning NMF and Joint-Diagonalization

Author

Zhang, Sixin, Soubies, Emmanuel, Févotte, Cédric, Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, and European Project: CoG-6681839,ERC FACTORY

Subjects
[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO]Computer Science [cs]
Abstract

Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approach -- termed as JD+NMF -- that estimates the transform through JD, prior to NMF computation. In contrast, we found that when the number of data realizations is limited, not only is JD+NMF no longer equivalent to TL-NMF, but the inherent low-rank constraint of TL-NMF turns out to be an essential ingredient to learn meaningful transforms for NMF.

Published
2021

30. Expanding boundaries of Gap Safe screening

Author

Cássio Dantas, Soubies, Emmanuel, Fevotte, Cedric, Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), European Project: 681839,H2020,ERC-2015-CoG,FACTORY(2016), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)

Subjects
Signal Processing (eess.SP), FOS: Computer and information sciences, Computer Science - Machine Learning, non-negativity, sparse regression, Machine Learning (cs.LG), Convex optimization, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, Optimization and Control (math.OC), FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, safe screening rules, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Electrical Engineering and Systems Science - Signal Processing, Mathematics - Optimization and Control, beta-divergence
Abstract

International audience; Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).

Published
2021

31. User-friendly Building of Reconstruction Algorithms with GlobalBioIm

Author

Donati, Laurène, Soubies, Emmanuel, Unser, Michael, Soubies, Emmanuel, Biomedical Imaging Group [Lausanne], Ecole Polytechnique Fédérale de Lausanne (EPFL), Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), and DAS-SAB

Subjects
[INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]
Abstract

International audience; The current shift towards computational imaging has made reconstruction procedures an integral part of many advanced imaging systems. A consequence is that imaging scientists now commonly require efficient and reliable computational tools for solving their inverse problems. To this end, we recently developed GlobalBioIm, an open-source Matlab library that standardizes the resolution of a wide range of imaging problems [1]. This toolbox gives access to cutting-edge reconstruction algorithms, and can be extended to new modalities and methods by combining elementary modules. The versatility and efficiency of GlobalBioIm have been highlighted in a series of recent high-impact works [2-4]. Driven by these encouraging applications, we have devoted our efforts towards improving the usability of GlobalBioIm by those with limited expertise in inverse problems and optimization theory. The outcome is a new user-friendly Matlab interface (Figure 1) that allows non-experts to intuitively build tailored reconstruction algorithms with minimal effort. Figure 1: The new GlobalBioIm GUI (left) can be used to solve various inverse problems (right).

Published
2020

32. Inner-Loop Free ADMM for Cryo-EM

Author

Donati, Lauréne, Soubies, Emmanuel, Unser, Michael, Biomedical Imaging Group [Lausanne], Ecole Polytechnique Fédérale de Lausanne (EPFL), and Soubies, Emmanuel

Subjects
regularization, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], Cryo-EM reconstruction, splitting, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], inverse problem, inner-loop-free, convergence speed, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, alternating-direction of multipliers method (ADMM), [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing
Abstract

International audience; Thanks to recent advances in signal processing, the interest for fast l1-regularized reconstruction algorithms in cryo-electron mi-croscopy (cryo-EM) has intensified. The approaches based on the alternating-direction of multipliers method (ADMM) are particularly well-suited due to the prime convergence speed and flexibility of use of this algorithm. Yet, the standard ADMM scheme still relies on a nested conjugate gradient (CG) to solve the linear step in its alternating-minimization procedure, which can be costly when handling large-scale problems. In this work, we present an inner-loop-free ADMM algorithm for 3D reconstruction in cryo-EM. By using an appropriate splitting scheme, we are able to avoid the use of CG for solving the linear step. This leads to a substantial increase in algorithmic speed, as demonstrated by our experiments.

Published
2019

38. The Sliding Frank-Wolfe Algorithm for the BLASSO

Author

Denoyelle, Quentin, Duval, Vincent, Peyré, Gabriel, Soubies, Emmanuel, Biomedical Imaging Group [Lausanne], Ecole Polytechnique Fédérale de Lausanne (EPFL), Laboratoire Traitement et Communication de l'Information (LTCI), Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centre National de la Recherche Scientifique (CNRS), CNRS, Grélaud, Françoise, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École normale supérieure - Paris (ENS-PSL), Université Toulouse Capitole (UT Capitole), Université Fédérale Toulouse Midi-Pyrénées-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris, and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris)

Subjects
[INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, Traitement du signal et de l'image, [INFO.INFO-LG] Computer Science [cs]/Machine Learning [cs.LG]
Abstract

International audience; This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

Published
2019

39. On the Local Minimizers of the CEL0 Relaxation

Author

Soubies, Emmanuel, Blanc-Féraud, Laure, Aubert, Gilles, Signal et Communications (IRIT-SC), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Toulouse Mind & Brain Institut (TMBI), Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Centre National de la Recherche Scientifique (CNRS), Morphologie et Images (MORPHEME), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Biologie Valrose (IBV), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Signal, Images et Systèmes (Laboratoire I3S - SIS), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), ANR-19-P3IA-0002,3IA@cote d'azur,3IA Côte d'Azur(2019), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Signal, Images et Systèmes (Laboratoire I3S - SIS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract

International audience; We study the strict local minimizers of the CEL0 func-tional, an exact continuous relaxation of the`0-regularized least-squarescriterion. More precisely, we derive a necessary and sufficient conditionfor strict local optimality, recalling that global minimizers are strict.Moreover, we quantify the number of strict local (not global) minimizersof the initial functional that are eliminated by the relax.

Published
2019

40. Quelles relaxations continues pour le critère $l2-l0$ ?

Author

Soubies, Emmanuel, Blanc-Féraud, Laure, Aubert, Gilles, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Morphologie et Images (MORPHEME), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Biologie Valrose (IBV), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Signal, Images et Systèmes (Laboratoire I3S - SIS), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS)-Signal, Images et Systèmes (Laboratoire I3S - SIS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Soubies, Emmanuel, and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
[INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, [SPI.SIGNAL] Engineering Sciences [physics]/Signal and Image processing
Abstract

For more than two decades, several continuous (and generally separable) penalties approximating (relaxing) the l0-pseudo norm have been proposed. Although some "good" properties for such penalties have been highlighted, the choice of one relaxation rather than another one remains unclear. One approach to compare them is to investigate their fidelity to the initial problem. In other words, do they preserve global minimizers of the initial criteria without adding new local ones? Within the context of the l0 penalized least squares, we have recently studied this question resulting in a class of penalties said exact. In this communication, we present these results and complete them with a study concerning the local minimizers eliminated by such relaxations. In particular, we show that the CEL0 penalty is the one removing the largest number of local minimizers., Depuis plus de deux décennies, de nombreuses pénalités continues (et généralement séparables) ont été proposées afin d'approcher (relaxer) la pseudo-norme l0. Bien que certaines «bonnes» propriétés pour ces pénalités aient été mises en évidence, le choix d'une relaxation par rapport à une autre reste peu clair. Une approche pour les comparer est de s'intéresser à la fidélité de la relaxation au problème initial. En d'autres termes, est-ce que cette dernière préserve les minimiseurs globaux du critère initial sans en ajouter de locaux ? Dans le contexte des moindres carrés pénalisés en norme-l0, nous avons récemment étudié cette question et défini une classe de pénalités qualifiés d'exactes. Dans cette communication, nous présentons ces résultats et les complétons par une étude des minimiseurs locaux qui sont éliminés par de telles relaxations. En particulier, nous montrons que la pénalité CEL0 est celle éliminant le plus de minimiseurs locaux.

Published
2017

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