Offres de stages

La compilation de connaissances est une méthode de d’exploration de problèmes d’optimisation (problèmes de planification, de configuration de produits, de diagnostic) exprimes sous une forme logique ou sous la forme d’un problème de satisfaction de contraintes consistant en la transformation de ces problèmes vers un autre langage (appelé langage cible). Ces langages de représentation ont pour intérêt que la résolution du problème y est garantie en temps polynomial. Le processus de compilation lui même n’étant pas polynomial, il n’est pas garanti d’aboutir (risque d’explosion en espace et en temps), mais en cas de succès, permet une exploitation grandement accélérée du problème. Cette méthode peut s’avérer particulièrement utile lorsque le nombre de requêtes à effectuer sur un même problème est grand (c’est typiquement le cas dans les domaines du diagnostic, de la configuration ou de la planification cités plus haut).

Il existe aujourd’hui une certaine variété de compilateurs, permettant la compilation vers différents langages cibles, et chacun avec des méthodes de compilation différentes. De ce fait, les performances peuvent varier grandement d’un compilateur à l’autre pour chaque instance à compiler. De même, beaucoup de ces compilateurs génèrent une forme compilée mais n’implémentent pas les algorithmes qui permettraient à un utilisateur de réaliser l’ensemble des requêtes et transformations qui intéressent son application.

Ce stage se compose de deux parties plus ou moins indépendantes.

– Une première partie, plus orientée logiciel, consistera à l’élaboration d’une API et l’implémentation d’une bibliothèque regroupant l’ensemble des compilateurs, et permettant à l’utilisateur de simplement utiliser le compilateur de son choix, voir l’ensemble des compilateurs afin d’obtenir le meilleur résultat.

– Une deuxième partie, plus orientée algorithmie, consistera en l’élaboration d’un outil permettant l’exploitation de ces formes compilées. L’outil devra pouvoir interpréter la forme compilée fournie par les différents compilateurs, et pouvoir y appliquer différents algos correspondant aux différentes requêtes ou transformations dont peut avoir besoin l’utilisateur.

Contact

Helene Fargier, IRIT/CNRS, 3IA ANITI Senior Chair, fargier@irit.fr – Nicolas Schmidt IRIT & 3IA ANITI, nicolas.schmidt@irit.fr

Profil du candidat

Stage de Master 2 ou de dernière année d’ingénieur. Le candidat idéal serait intéressé par la programmation par contraintes (e.g. PLNE/CSP solvers), la manipulation de graphes, l’optimisation combanoire et la recherche opérationnelle.

La théorie des jeux se propose d’étudier des situations (appelées « jeux ») où des agents (les « joueurs ») prennent des décisions (on parle de « strategies ») , chacun étant conscient que le résultat de son propre choix (son « gain ») dépend de celui des autres. Si les joueurs peuvent passer des accords le jeu est dit jeu coopératif. Si c’est non (par exemple parce que les agents ne peuvent pas communiquer, ou parce qu’il n’est pas possible de garantir un accord ), le jeu est dit non coopératif. La distinction est fondée sur le contexte, pas du tout sur la manière de jouer. Il ne faut pas confondre jeu de coordination et jeu de coopéation. Dans un jeu de coopération, les joueurs ont la possibilité de se concerter afin d’obtenir le meilleur paiement. Ce n’est pas le cas dans un jeu de coordination, où les joueurs ont interêt à jouer la même action (c’est à dire à se coordonner), car ils ont la même fonction de paiement, mais sans pouvoir se concerter. L’utilisation d’un réseau social est un exemple typique de jeu de coopération dans une grand graphe de joueurs, certains ne connaissant même pas l’existence d’autres : on a intérêt à utiliser le même outil que ses proches, lesquels ont des accointances avec l’autres etc.

Dans ce stage on considérera des jeux à issue incertaine, où les gains dependent non seulement des stratégies/décisions des joueurs mais aussi circonstances extérieures sur lesquelles ils ont une connaissance limitée (par exemple le niveau sécurité des données dans l’outil utilisé, son exposition aux attaques, etc), commune ou pas – typiquement, le résultat des actions jointes est incertain. .

Dans le cadre de ce stage, on mènera une étude des jeux de cooperation sous incertitude probabiliste. Il s’agira dans une premier temps de developper des algorithmes pour les jeux de cooperation Bayesiens où les interactions entre joueurs sont pair à pair – dans ce cas, il est en effet théoriquement possible de diminuer la complexité du problème de recherche d’équilibres de Nash en donnant une formulation plus compacte du problème.

Dans ce second temps, on étendra cette étude aux jeux de cooperation non Bayesiens – il existe en effet de nombreuses situations où l’utilité espérée n’est pas (ou ne peut pas être) utilisée, car la connaissance ne peut être exprimée par une unique distribution, comme mis en evidence par le paradoxe de Ellsberg . On se placera alors dans le cadre de la théorie des fonctions de croyance pour developper une nouvelle approche des jeux de coordination sous incertitude.

Nous recherchons donc, pour un stage de 4 à 6 mois, un/e candidat/e en cours de Master II ou de dernière année d’ecole d’ingénieur, intéressé par la décision sous incertitude ; des connaissances en optimisation (e.g.PLNE, Solveurs SAT/CSP) seraient appréciees

Ces recherches prennent place dans le cadre de la chaire « Compilation de Connaissances »

Contact (Hélène Fargier (helene.fargier@irit.fr)) – IRIT/ ADRIA –

Decision aid systems, like on-line configurators or recommender systems, need to adapt themselves to each user in order to offer a better interaction and guide the user quickly to the best decision for her: the system should be able to gather a model of the preferences of the user, and be able to show her, almost instantly, during the interaction, what seems to be her best, most preferred possible alternatives. Several models of preferences have been developed in the literature on Artificial Intelligence and Operations Research, offering the possibility to represent complex preferences over multi-attribute domains in some rather compact form. The richness of the models comes at a cost: finding the optimal alternatives is, in general, a computationnally hard problem – at least NP-nard – except for some quite restrictive models.

Computationally hard queries are also typical of many logic-based systems. In the last two decades, a new approach, called knowledge compilation has been studied to cope with such hard queries in the context of real-time systems: logical bases can be compiled, off-line, into some less compact representation, which enables fast (polynomial time) query answering. Although, in theory, the size of these less compact representations can grow exponentially fast with the number of variables, this explosion does not often occur in practice.

The topic of this internship is to extend compilation approaches for decision aiding with preferences. There have been some works on properties of real-valued languages for representing preferences, and on some logic-based languages, like penalty logic; and, more recently, for languages based on conditional preference statements. The aim of this internship is to complete the picture. One classical direction of research in knowledge compilation i the comparison of the difficulty of the requests (here, decision aiding requests) which depends one the families of langages considered (generally, the more the langage is expressive, le higher the complexity of the resuqests). But we shall now consider another important properties of these langages : the elicitation of preferences, or how they can be learnt from data, like customer ratings or sales history for on-line decision-aid systems.

We are thus proposing an internship of 4 to 6 months, for candidates in their last year Master II engineering school, interested in hybrid IA. Konwledge in computational complexity theory, machine learning and/or logics would be appreciated

This research takes place within the framework of the « Knowledge Compilation » chair

Contact Hélène Fargier (helene.fargier@irit.fr) & Jérôme Mengin ( jerome.mengin @ irit.fr)- IRIT/ ADRIA –

Dans la population, le syndrome d’apnées du sommeil non traité est associé à un risque de morbi-mortalité cardiovasculaire. Chez les personnes âgées de plus de 65 ans, le lien entre la pathologie et l’augmentation du risque de mortalité n’a pas été clairement établi et les indices de sévérité utilisés en clinique (AHI, ODI) restent controversés. La prise en compte d’informations subjectives supplémentaires telles que celles représentant la qualité de vie ressentie par les sujets, pourrait permettre de mieux discriminer les personnes à risque d’évènements cardiovasculaires et d’optimiser le traitement. Dans le cadre du formalisme des fonctions de croyance, l’objectif de ce stage est d’élaborer un modèle prédictif du risque de morbi-mortalité chez le patient âgé à partir de la fusion d’informations incertaines de sommeil et de qualité de vie.

Le premier objectif de ce stage de master est d’appliquer ce nouveau formalisme aux données qualitatives issues d’un questionnaire de qualité de vie réalisé chez les sujets de l’étude PROOF par l’équipe SNA-EPIS de Saint Etienne [6]. Chaque réponse à une question est considérée comme une source d’information qualitative incertaine. Le degré d’incertitude provient de la subjectivité liée à l’interprétation des questions par le patient. Il s’agit dès lors d’étudier différentes stratégies de fusion d’informations qualitatives dans le cadre du formalisme des fonctions de croyance.

La seconde étape consiste à fusionner les données quantitatives et qualitatives : d’une part des indices de sévérité indices de sévérité (AHI, ODI) issus du suivi de la morbi-mortalité d’une cohorte de sujets âgés non traités pour l’apnée du sommeil, pendant 16 ans, après un examen polygraphique et d’autre part les informations qualitatives issues des questionnaires .

Enfin, on pourra prendre en compte les résultats de la fusion afin d’élaborer un modèle prédictif du risque de morbi-mortalité chez le patient âgé. La prise en compte de critères supplémentaires pourrait permettre de mieux discriminer les personnes à risque d’évènements cardiovasculaires, et donc, à terme, de cibler ceux pour lesquels le traitement serait le plus justifié et efficace.

Nous recherchons donc, pour un stage de 4 à 6 mois, un/e candidat/e en cours de Master II ou de derniere année d’ecole d’ingénieur, intéressé par la décision sous incertitude ; des connaissances en optimisation (e.g. PLNE, Solveurs SAT/CSP) et/ou statistiques seraient appréciees. Ces recherches prennent place dans le cadre de la chaire « Compilation de Connaissances » d’ANITI

Les recherches s’effectueront à l’IRIT au sein de l’équipe ADRIA. Pour candidater merci d’envoyer un CV à helene.fargier@irit.fr et francis.faux@irit.fr

The automation of robotic tasks in the aviation industry requires the development of techniques combining the automatic generation of movement elements to execute tasks with the optimal scheduling of these tasks. The Gepetto team at LAAS has a long experience in motion planning for systems with a large number of degrees of freedom in environments occupied by obstacles. The ROC team has a long experience in combinatorial optimization problems such as task scheduling, which is also the subject of the KC@ANITI chair.

The objective of this internship is to combine these techniques in order to produce

movements performing robotic tasks in optimal order. Two difficulties specific to this combination are:

1. the a priori ignorance of the cost of transition from one task to another,

2. the fact that a given task can be performed by an infinite number of configurations of the robot due to its kinematic redundancy.

The work will be illustrated by an application of drilled holes deburring in an Airbus A 320 reactor mast (picture above).

REQUIRED SKILLS

The candidate must have programming skills in python. Knowledge of C++ programming and combinatorial optimization will be appreciated.

Pour les besoins des activités de recherche en cartographie de l’occupation du sol et de suivi des cultures réalisées au CESBIO (www.cesbio.cnrs.fr), il serait utile de disposer d’une méthodologie automatique permettant, à partir d’une série temporelle d’images Sentinel2, de produire une base de données géographique des parcelles agricoles.

> Consultez l’annonce en ligne sur le site du Cnes

We are looking for Master students interested in the development of technologies for civic participation. Candidates should be passionate about the design and development of online systems, and should think critically about the social impact of technologies. Candidates should be interested in working in a multidisciplinary environment combining computer scientists, lawyers, UI/UX designers, natural and social scientists. We are looking for candidates with web development skills (e.g. Javascript, React, Node, etc.), design sensibilities, and/or the ability to critically analyze data using statistical and graphical methods. The working language of the group is English. 

Formal applications should include a CV, letter of motivation, a list of three possible recommenders, and links to previous work. They should be directed to professor Cesar Hidalgo (cesar.hidalgo@univ-toulouse.fr). The work will be performed at ANITI’s Collective Learning group and is supported by ANITI’s Augmented Society chair. 

Summary: 

A large number of problems from diverse fields such as optimization, probability and statistics, dynamical systems and control or quantum physics can be tackled within the unied and powerful framework of the Lasserre hierarchy [1, 2], which allows one to solve challenging nonconvex and nonlinear problems by a sequence of convex optimization problems, in a uni_ed and very systematic fashion. Additional research [5] investigated the ability of Christo_el-Darboux kernels to capture information about the support of an unknown probability measure. A distinguishing feature of this approach is to allow one to infer support characteristics, based on the knowledge of infinitely many moments of the underlying measure. A major open question remains whether this approach can be used in a data-driven setting, where the underlying model is unknown and only observed data are available. This project will investigate this direction, building on the very recent work [3]. Progress in this direction would be an enabling factor in bringing the elegant and powerful tools of the Lasserre hierarchy to the realm of the present-day big-data applications, which are currently typically tackled using ad-hoc heuristic techniques with limited mathematical foundation.

 Goals:

The goal of this M2 internship is the extension of the Lasserre hierarchy and Christo_el-Darboux kernels to a data-driven setting, developing new methods furnished with a theoretical analysis (convergence rate, non-asymptotic out-of-sample error etc). One first possible investigation track will consist of studying Christo_el-Darboux kernels to extend the approach from [4] for measures supported on specic classes of mathematical varieties. We intend to apply this framework to deep learning network models, for which latent representation correspond to such low-dimensional varieties. Numerical experiments will be performed on several benchmark suites, including MNIST, CIFAR10 or fashion MNIST. Other main steps may include, but are not limited to, the investigation of adaptive sampling techniques and basis choice for the approach developed in [3] as well as extension of the proposed methodology beyond the problem class considered, e.g., to data-driven optimal control. Other directions include complexity reduction based on sparsity, symmetries and other more advanced structural properties of the problem at hand. A significant degree of freedom will be given to the student to create and pursue his/her ideas broadly within this scope.

 Requirements:

A successful M2 candidate will have a strong background in applied mathematics or physics, having a very good knowledge of probability and statistics as well as a working knowledge of convex optimization, real analysis and basic measure theory. The candidate should be highly motivated and creative.

Funding: This M2 internship will be funded by the arti_cial intelligence center of Toulouse (ANITI) and co-supervised with LAAS CNRS and University Toulouse 3, Paul Sabatier. A PhD can be foreseen. The M2 candidate will be hosted by LAAS and the DEEL team (DEpendable and Explainable Learning), a joint project team between Toulouse and Montr_eal, in building B612, IRT Saint Exupery.

References

[1] J.B. Lasserre (2001). Global optimization with polynomials and the problem of moments. SIAM

Journal on optimization, 11(3), 796-817.

[2] J.B. Lasserre (2010). Moments, positive polynomials and their applications. World Scientific, 2010.

[3] M. Korda (2019). Data-driven computation of the maximum positively invariant set for nonlinear dynamical systems. FEANICSES workshop, https://cavale.enseeiht.fr/feanicses/files/

W2019/4-korda.pdf.

[4] E. Pauwels, M. Putinar and J.-B. Lasserre (2019). Data analysis from empirical moments and the Christo_el function. hal-01845137

[5] E. Pauwels and J.-B. Lasserre (2019). The empirical Christo_el function with appication in data analysis. To appear in Advances in Computational Mathematics.

SUPERVISORS
The internship will be supervised by Prof. H. Goulart (henrique.goulart@irit.fr) at IRIT/ENSEEIHT, in remote collaboration with two members of the LargeDATA (DataScience) chair at 3IA MIAI/Univ. Grenoble-Alpes : Prof. R. Couillet (head, romain.couillet@gipsa-lab.grenobleinp.fr) and Dr. P. Comon (pierre.comon@gipsa-lab.grenoble-inp.fr).
FUNDING
This internship will be funded by the ArtiFicial and Natural Intelligence Toulouse Institute (3IA ANITI), as part of the AI Research Chair lead by N. Dobigeon.
CONTEXT
Tensor models are powerful tools for addressing many problems in signal processing, machine learning and beyond [1]–[3]. Yet, their use in these applications typically requires estimating a low-rank tensor from a set of observations corrupted by noise, which is often a difficult task. Moreover, in most cases there is currently no theory for predicting the actual estimation performance that can be attained.
To overcome this gap, in recent years several researchers have studied the asymptotic statistical performance of ideal and practical estimators in the large-dimensional regime,
where the size of the tensor grows large [4]–[6]. In particular, these works have uncovered the abrupt phase transition that the performance of an ideal estimator may undergo as the
signal-to-noise ratio grows (see Figure 1 below for an illustration). While some important advancements have been achieved, many scenarios of practical interest remain unexplored, as well as the practical implications of the existing results in applications.
OBJECTIVES
The overall goal of this internship is to study extensions and applications of the existing results, as a first step for pushing the existing theory beyond its current limits. We will in particular
consider extensions to more general tensor models that apply to larger classes of realworld problems, including e.g. asymmetric models. Application to practical machine learning problems—such as community detection in hypergraphs [7], latent variable model estimation [2] and high-order co-clustering [8]—will also be considered.

The intern will initially perform computer simulations aimed at understanding the behavior of ideal and practical estimators in the target scenarios/applications. Some theoretical results may then be derived on the basis of these experimental findings. Scientific dissemination of these findings will also be encouraged, via publication of papers and/or participation
in scientific events.
A PhD thesis may be proposed to the intern at the end.

CANDIDATE PROFILE
We look for strongly motivated candidates with a solid background on mathematics and statistics, having good programming skills in scientific computing languages (Python, Matlab, Julia). Basic knowledge or interest in random matrix theory is a strong plus.
PRACTICAL INFORMATION
• The intern will be hosted at the ENSEEIHT site of IRIT, located at 2 rue Charles Camichel,Toulouse.
• The monthly internship gratification is of 600€.
• Application procedure: please send your CV, your report card and a motivation letter to Henrique Goulart

References
[1] N. D. Sidiropoulos, L. De Lathauwer, X. Fu, et al., “Tensor decomposition for signal processing and machine
learning,” IEEE Transactions on Signal Processing, vol. 65, no. 13, pp. 3551–3582, 2017.
[2] A. Anandkumar, D. Hsu, S. M. Kakade, et al., “Tensor decompositions for learning latent variable models,”
Journal of Machine Learning Research, vol. 15, pp. 2773–2832, 2014.
[3] S. Rabanser, O. Shchur, and S. Günnemann, “Introduction to tensor decompositions and their applications
in machine learning,” arXiv preprint arXiv:1711.10781, 2017.
[4] A. Jagannath, P. Lopatto, and L. Miolane, “Statistical thresholds for Tensor PCA,” The Annals of Applied Probability,
vol. 30, no. 4, pp. 1910–1933, 2020.
[5] A. Perry, A. S. Wein, and A. S. Bandeira, “Statistical limits of spiked tensor models,” Annales de l’Institut Henri
Poincaré, Probabilités et Statistiques, vol. 56, no. 1, pp. 230–264, 2020.
[6] A. Montanari and E. Richard, “A statistical model for tensor PCA,” Advances in Neural Information Processing
Systems, vol. 4, pp. 2897–2905, 2014. eprint: 1411.1076.
[7] C. Kim, A. S. Bandeira, and M. X. Goemans, “Community detection in hypergraphs, spiked tensor models, and
sum-of-squares,” in 2017 International Conference on Sampling Theory and Applications (SampTA), Tallinn,
Estonia, Jul. 2017, pp. 124–128.
[8] E. E. Papalexakis, N. D. Sidiropoulos, and R. Bro, “From k-means to higher-way co-clustering: Multilinear
decomposition with sparse latent factors,” IEEE Transactions on Signal Processing, vol. 61, no. 2, pp. 493–
506, 2013.

DESCRIPTION
This internship aims at designing, developing and testing a distributed software architecture for a Multi-Robot System (MRS), in order to achieve a patrolling mission.
This architecture will be based on two elements:
• a Market-Based Auction (MBA) mechanism for multi-robot task allocation; the tasks to allocate consist in patrol trajectories, observation of points of interest, identification of objects, communication relays, … MBA approaches are based on the following principle: tasks are proposed by an auctioneer, which can be a human operator or a robot of the MRS, and other robots can bid on this task, depending on their capability to achieve it; the auctioneer then decides who wins the auction for each proposed task;
• a Machine Learning (ML) based approach to plan the trajectories of robots; the objective is to learn trajectories from either actual field data or a robot simulator, and to use this function with two objectives: first to compute more precise bids in the MBA approach, and second, to better execute trajectories.

During this internship, the student will then have to develop and test both the MBA and the ML approaches, and to design a global MRS architecture that integrates both. The developments will be evaluated at least on a robotic simulator, and possibly on actual robotic platforms on a small arena available at ONERA.

The student will be integrated within the robotics team at ONERA, among researchers and students working on several projects, robots, and applications.

APPLICATION PROCEDURE
Formal applications should include detailed cv, a motivation letter, and transcripts of Bachelor’s degree.
Applications should be sent by email to: advisor emails

CONTEXT
In the last years, the advent of Earth observation satellite missions with short revisit time and increased spatial resolution has led to an unprecedented amount of remote sensing images of heterogeneous physical nature (e.g., optical & radar Sentinel time series . . . ) at various scales (e.g., submetric, decametric . . . ). Furthermore, satellite image archives, such as Spot Heritage, are made available by many space agencies. Such massive data extend existing satellite and in-situ acquisition system used to understand, explain and predict the states and trends of our environment.
In order to address challenges raised by such applicative domains, the interdisciplinary institute in artificial intelligence of Toulouse, named the Artificial and Natural Intelligence Toulouse Institute (ANITI) of whom CNES is partner, has proposed to develop innovative
solutions using theoretical advances in core AI scientific areas. The CESBIO lab, with J. Inglada and M. Fauvel, is part of the ANITI Chair entitled “Data-driven approximate Bayesian computation for fusion-based inference from heterogeneous (remote sensing) data” hold by Prof. N.Dobigeon. This chair’s objective is to develop learning algorithms able to extract meaningful information from multi-source, multi-scale and multi-temporal data.
For 4 years now, CESBIO has used machine learning tools to provide automatically the fist land cover map of France from satellite image time series (http://osr-cesbio.ups-tlse. fr/~oso/ and https://www.theia-land.fr/en/ceslist/land-cover-sec/). The master internship is related to the integration of massive multi-source/scale satellite image time series
in learning algorithms for the production of such land use and land cover maps.
OBJECTIVES
The goal of the internship is to investigate the definition of an auto-encoder architectures adapted to satellite image time series. Such data are structured into three dimensions:
1 spatial, spectral and temporal. Hence, each pixel of a given data cube can be represented as a collection of spectral measurements over the time and spatial domains (see figure 1 for instance).

While several neural architectures have been proposed in the remote sensing literature, most of them have been applied to hyperspectral images of reduced size. Thus, they hardly take into account the temporal dimension and do not scale well w.r.t. the number of pixels.
Therefore, there is a need for specific work in satellite image time-series analysis. The recruited candidate will do a review of auto-encoder architectures used in satellite image times series and will implement the appropriate ones. Tests on large scale data sets will be performed for the classification of land cover types. Depending on the advancement, modification of existing architectures will be considered.
The candidate will be located at the “Maison de la recherche et de la valorisation” close to the CESBIO lab. He/She will be with other master students within the ANITI project. He/She will benefit of a full access to CESBIO and CNES high performance computers.

CANDIDATE PROFIL
The candidate must have a solid background in at least one of the following subjects:

• Statistical signal and image processing,
•  Machine learning,
•  Remote sensing data processing.

A good knowledge of English and scientific programming (Python, C/C++) is required.

PRACTICAL DETAILS

• The grant is approximately 620 euros per month.
•  Beginning of the internship: February or March 2021, for 6 months.
•  To apply, send a CV and a motivation letter to Mathieu Fauvel, mathieu.fauvel@inrae.fr.

TOPIC
Most of the significant advances in imaging in recent years rely on the use of numerical computation. Examples include super-resolution in microscopy, compressed sensing in magnetic resonance imaging or radar interferometry for earth observation. These techniques
share the prior need to design a mathematical model of the observation system. The acquired data is often described by an equation of the form y = Ax where A is a linear operator that describes the physics of the acquisition. Recovering x then requires inverting the operator. Although these imaging modalities have allowed solving problems beyond reach in the past, their expansion is often severely limited by an important problem: it is impossible to accurately control imaging conditions, resulting in errors on the operator A that make the numerical methods unreliable.
The main objective of this project is to develop theories and algorithms to overcome these difficulties and apply them to microscopy and astronomical imaging.
This problem is one of the most important current challenges in the field of inverse problems. Depending on the candidate interest, it will be possible to concentrate the efforts on rather theoretical issues or practical issues.
THEORY
The supervisors have begun investigating various open questions in this field. Two of them will be explored.
Recently [4], Kech and Krahmer provided necessary and sufficient conditions guaranteeing the ability to recover the operator A and the signal x from the measurements y. This implies studying the global injectivity of a bilinear mapping. The setting is however too
restricted for practical applications and we plan to study the local injectivity of the mapping.
A second trail is about the design of optimization algorithms to recover A and x. A standard approach [1] consists in lifting the problem on the space of rank-1 matrices. A traditional approach to solve the resulting problem consists in relaxing the rank 1 constraint by penalizing the nuclear norm. We recently observed that simpler projected gradient descents [3] tend to produce better results. Following [5], we plan to explore the use of continuous relaxations, which discard spurious minimizer and simplify the minimization.

APPLICATION
The supervisors of this project have begun a collaboration with the Centre de Biologie Intégrative in Toulouse (CBI), to improve the resolution of microscopes of the imaging facility. In particular, they are interested in the field of Single Molecule Localization Microscopy (SMLM) [2] and Structured Illumination microscopy [6].
The dominant approach to solve the associated inverse problems consists in assuming that the observation operator has been perfectly calibrated before the acquisition. This hypothesis is seldom satisfied and we expect to significantly improve the reconstructions by
using a finer modelling of the system and an identification of the operator. In addition, real improvements of the microscopes might result in new collaborations with the biologists to answer practical problems.

PRACTICAL ASPECTS
We are looking for a highly motivated student, willing to continue with a PhD thesis, with a background in electrical engineering (signal/image processing, harmonic analysis) or mathematics (optimization, probability an statistics, geometry). Strong abilities in mathematics or computer sciences will be appreciated. A taste for optics or biological
problems is a plus since applied projects might emerge.
If the candidate is successful, this internship will be pursued by a PhD. Various funding sources will be considered (ADI, FRM, Labex CIMI, doctoral school, ANR, ANITI).
This internship will take place either at IRIT or at INSA (in Toulouse, France), depending on the candidate’s interest. It will be co-supervised by Emmanuel Soubies (CR Toulouse) and Pierre Weiss (CR Toulouse). In addition, the candidate will have access to the CBI (Centre de Biologie Intégrative) with Thomas Mangeat for an access to the microscopy resources and biological problems. Do not hesitate to contact us for more information.
BIBLIOGRAPHY
[1] Ali Ahmed, Benjamin Recht, and Justin Romberg. Blind deconvolution using convex programming. IEEE Transactions on Information Theory, 60(3):1711–1732, 2013.
[2] Valentin Debarnot, Paul Escande, Thomas Mangeat, and Pierre Weiss. Learning low- dimensional models of microscopes. IEEE Transactions on Computational Imaging, 2020.
[3] Valentin Debarnot and Weiss Pierre. Blind inverse problems with isolated spikes. ArXiv, 2020.
[4] Michael Kech and Felix Krahmer. Optimal injectivity conditions for bilinear inverse problems with applications to identifiability of deconvolution problems. SIAM Journal on Applied Algebra and Geometry, 1(1):20–37, 2017.
[5] Emmanuel Soubies, Laure Blanc-Féraud, and Gilles Aubert. A continuous exact \ell_0 penalty (cel0) for least squares regularized problem. SIAM Journal on Imaging Sciences, 8(3):1607–1639, 2015.
[6] Emmanuel Soubies and Michael Unser. Computational super-sectioning for single-slice structuredillumination microscopy. IEEE Transactions on Computational Imaging, 5(2):240–250, 2018.

APPLICATION PROCEDURE
Formal applications should include detailed cv, a motivation letter, and transcripts of Bachelor’s degree.
Applications should be sent by email to: advisor emails