Séminaire de mathématiques appliquées (archives)

This talk aims at introducing a new multi-frequency method to reconstruct width defects in waveguides. Different inverse methods already exist. However, those methods are not using some frequencies, called resonant frequencies, where propagation equations are known to be ill-conditioned. Since waves seem very sensible to defects at these particular frequencies, we exploit them instead. After studying the forward problem at these resonant frequencies, we approximate the wavefield and focus on the inverse problem. Given partial wavefield measurements, we reconstruct slowly varying width defects in a stable and precise way and provide numerical validations and comparisons with existing methods.

Abstract

A method for constructing signatures of random reparametrizations of periodic functions is presented.

The proposed signatures are functions, which contain information about the height and order of local extrema of the observation. In contrast to other statistical methods for reparametrized curves, the observations can be of different lengths and the construction does not involve aligning them.

The signature is shown to be stable with respect to changes in the distribution of reparametrizations and to enjoy standard CLT properties, including in the case of dependent observations.

The positioning of a vehicle based on magnetic signals is the industrial application which motivated this work.

Ongoing work with Frédéric Chazal and Bertrand Michel.

References

1. Berry, E., Chen, Y.-C., Cisewski-Kehe, J. & Fasy, B. T. Functional Summaries of Persistence Diagrams. <arXiv:1804.01618> [stat] (2018).
2. Bristeau, P.-J. Techniques d’estimation du déplacement d’un véhicule sans GPS et autres exemples de conception de systèmes de navigation MEMS. (2012).
3. Chazal, F., Fasy, B. T., Lecci, F., Rinaldo, A. & Wasserman, L. Stochastic Convergence of Persistence Landscapes and Silhouettes. in Annual Symposium on Computational Geometry - SOCG’14 474–483 (ACM Press, 2014). doi:10.1145/2582112.2582128.
4. Kosorok, M. R. Introduction to Empirical Processes and Semiparametric Inference. (Springer New York, 2008). doi:10.1007/978-0-387-74978-5.
5. Perez, D. On C0-persistent homology and trees. 41 (2022).
6. Chazal F, Michel B. An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists. Front Artif Intell. 2021 Sep 29;4:667963. doi: <10.3389/frai.2021.667963>. PMID: 34661095; PMCID: PMC8511823.

Heterogeneous problems that take place at multiple scales are ubiquitous in science and engineering. Examples are wind turbines made from composites or groundwater flow relevant e.g., for the design of flood prevention measures. However, finite element or finite volume methods require an often prohibitively large amount of computational time for such tasks. Multiscale methods that are based on ansatz functions which incorporate the local behavior of the (numerical) solution of the partial differential equations (PDEs) have been developed to tackle these heterogeneous problems. Localizable multiscale methods that allow controlling the error due to localization and the (global) approximation error at a (quasi-optimal) rate and do not rely on structural assumptions such as scale separation or periodicity have only been developed within the last decade. Here, localizable multiscale methods allow the efficient construction of the basis functions by solving the PDE (in parallel) on several small subdomains at low cost.

While there has been a significant progress in recent years for these types of multiscale methods for linear PDEs, very few results have been obtained so far for nonlinear PDEs. In this talk, we will show how randomized methods and their probabilistic numerical analysis can be exploited for the construction and numerical analysis of such types of multiscale methods for nonlinear PDEs.

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level $\alpha \in (0,1)$. From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikol'skii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.

References:

-> This presentation will be based on the paper :

Adaptive test of independence based on HSIC measures (https://arxiv.org/abs/1902.06441)

-> To be prepared :

• First part of the talk deals with optimality of non parametric independence statistical tests, see Sections 0.1 and 0.2 of the PhD thesis of Melisande Albert

https://perso.math.univ-toulouse.fr/albert/files/2016/11/These-malbert.pdf

• Second part of the talk deals with kernel based measures of independence, see

Measuring Statistical Dependence with Hilbert-Schmidt Norms (https://is.mpg.de/publications/3774)

In this talk, I will consider a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system, which arises in population dynamics. This system has entropy dissipation properties on which one can rely to design a robust and convergent numerical scheme for its numerical simulation. In terms of numerical analysis, I will present discrete compactness techniques, entropy-dissipation estimates and a new adaptation of the so-called duality estimates for parabolic equations in Laplacian form. I will also present numerical experiments illustrating the influence of the nonlocality in the system: on convergence properties, as an approximation of the local system and on the development of diffusive instabilities. This is a joint work with Antoine Zurek (UTC).

Single-cell RNA sequencing (scRNAseq) is a high-throughput technology quantifying gene expression at the single-cell level, for thousands of cells and tens of thousands of genes. A major statistical challenge in scRNAseq data analysis is to distinguish biological information from technical noise in order to compare conditions or tissues. Differential Expression Analysis (DEA) is usually performed with univariate two-sample tests and thus does not account for the multivariate aspect of scRNAseq data that carries information about gene dependencies and underlying regulatory networks and pathways. Applying multivariate two-sample tests would allow to perform Differential Transcriptome Analysis (DTA), to assess for the global similarity of the compared datasets.

We propose a kernel based two-sample test that can be used for DEA as well as for DTA. The Maximum Mean Discrepency (MMD) test is the most famous kernel two-sample test [1], it is based on the distance between the mean embeddings of the empirical distributions in an high-dimensional feature space, obtained through a non-linear embedding called the feature map. Our package implements a normalized version of the MMD test derived from the non-linear classification method KFDA [2], then regularized by a kernel PCA-like dimension reduction [3]. Besides reaching state of the art performances in DEA with competitive computational cost, the non-linear discriminant transformation obtained from the KFDA approach offers visualization tools highlighting the main differences between the two conditions in terms of cells, allowing to identify condition-specific sub-populations.

[1] Arthur Gretton, Karsten M Borgwardt, Malte Rasch, Bernhard Schölkopf, and Alex J Smola. A Kernel Method for the Two-Sample-Problem. page 8, 2007.

[2] Zaid Harchaoui, Francis Bach, and Eric Moulines. Testing for Homogeneity with Kernel Fisher Discriminant Analysis. arXiv:0804.1026 [stat], April 2008. arXiv: 0804.1026.

[3] Zaid Harchaoui, Felicien Vallet, Alexandre Lung-Yut-Fong, and Olivier Cappe. A regularized kernel-based approach to unsupervised audio segmentation. In 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 1665–1668, Taipei, Taiwan, April 2009. IEEE.

For kernel methods:

[4] Le cours de Jean-Philippe Vert et Julien Mairal https://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/course/2021mva/index.html

[5] Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, and Bernhard Sch ̈olkopf. Kernel Mean Embedding of Distributions: A Review and Beyond. Foundations and Trends in Machine Learning, 10(1-2):1–141, 2017. arXiv: 1605.09522.

We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces.

In case you want to read more about this topic or get prepared for the talk, you can have a look at the following two articles:

• Poisat, J. and Simenhaus, F., 2020. A ℓimit theorem for the survival probability of a simple random walk among power-law renewal obstacles. The Annals of Applied Probability, 30(5), pp.2030–68.
• Poisat, J. and Simenhaus, F., 2022. Localization of a one-dimensional simple random walk among power-law renewal obstacles. arXiv preprint <arXiv:2201.05377>.

L’optimisation stochastique englobe des méthodes permettant de minimiser une fonction de coût avec un caractère aléatoire, problème qui intervient souvent en machine learning et en particulier dans l’entraînement des réseaux de neurones. L'exemple le plus connu et le plus étudié d'une telle méthode est l'algorithme de la descente du gradient introduit par Robbins et Monro en 1951. Les algorithmes dits adaptatifs sont des extensions de cette descente de gradient stochastique classique qui visent à améliorer ses propriétés de convergence en déterminant automatiquement à chaque étape le taux d’apprentissage. Dans cette présentation on s’intéressera au comportement asymptotique des algorithmes de type RmsProp et Adagrad quand la fonction de coût est non-convexe. On montrera en particulier qu'ils convergent presque sûrement vers l'ensemble des points critiques de la fonction cible et (sous quelques hypothèses supplémentaires) vers un minimum local.

Quelques références : - pour la descente du gradient stochastique classique : Robbins et Monro(1951) A stochastic approximation method, The Annals of Mathematical Statistics - un article qui contient des résultats utiles pour l'étude des algorithmes : M. Benaim, Dynamics of stochastic approximation algorithms, publié dans Séminaire de probabilités XXXIII - un livre : M. Duflo (1996) Algorithmes stochastiques, volume 23 de Mathématiques & Applications (Berlin) - l'article correspondant à la présentation : S. Gadat et I. Gavra, Asymptotic study of stochastic adaptive algorithms in non-convex landscape, Journal of Machine Learning Research

TBA

Abstract

Recent works in time-frequency analysis proposed to switch the focus from the maxima of the spectrogram toward its zeros, which form a random point pattern with a very stable structure. Several signal processing tasks, such as component disentanglement and signal detection procedures, have already been renewed by using modern spatial statistics on the pattern of zeros. Tough, they require cautious choice of both the discretization strategy and the observation window in the time-frequency plane. To overcome these limitations, we propose a generalized time-frequency representation: the Kravchuk transform, especially designed for discrete signals analysis, whose phase space is the unit sphere, particularly amenable to spatial statistics. We show that it has all desired properties for signal processing, among which covariance, invertibility and symmetry, and that the point process of the zeros of the Kravchuk transform of complex white Gaussian noise coincides with the zeros of the spherical Gaussian Analytic Function. Elaborating on this theorem, we finally develop a Monte Carlo envelope test procedure for signal detection based on the spatial statistics of the zeros of the Kravchuk spectrogram.

References

The presentation will be based upon a journal paper and a conference paper:

• Pascal, B., & Bardenet, R. (2022). A covariant, discrete time-frequency representation tailored for zero-based signal detection. IEEE Transactions on Signal Processing. https://arxiv.org/pdf/2202.03835.pdf
• Pascal, B., & Bardenet, R. Une famille de représentations covariantes de signaux discrets et son application à la détection de signaux à partir de leurs zéros. Colloque GRETSI, Nancy, 6-9 Sept. 2022. http://gretsi.fr/data/colloque/pdf/2022_pascal810.pdf

For those who want to get prepared about point processes and signal processing:

• Flandrin, P. (2015). Time–frequency filtering based on spectrogram zeros. IEEE Signal Processing Letters, 22(11), 2137-2141.
• Bardenet, R., Flamant, J., & Chainais, P. (2020). On the zeros of the spectrogram of white noise. Applied and Computational Harmonic Analysis, 48(2), 6- 82-705. Bardenet, R., & Hardy, A. (2021). Time-frequency transforms of white noises and Gaussian analytic functions. Applied and computational harmonic analysis, 50, 73-104.