Program

Abstract: "Physics-informed neural networks (PINNs) combine the expressiveness of neural networks with the interpretability of physical modeling. Their good practical performance has been demonstrated both in the context of solving partial differential equations and more generally in the context of hybrid modeling, which consists of combining an imperfect physical model with noisy observations. However, most of their theoretical properties remain to be established. We offer some food for thought and statistical insight into the proper use of PINNs." 

Abstract: "A Multivariate Multiscale model for Locally Stationary Processes is introduced. The model allows correlation between increments across different time series and scales, a novel achievement in the field. Identification and estimation theories are obtained thanks to the new concept of Cross-Correlation Wavelet Functions (CCWF), serving as a generalization of the concept of Autocorrelation Wavelet Functions (ACWF) to measure redundancy levels among sets of non-decimated discrete wavelet functions. Owing to the linear independence of the CCWF, a unique asymptotic representation of the Evolutionary Wavelet Spectrum (EWS) is obtained in the CCWF domain for the proposed model. A consistent estimator for the EWS, called the corrected and smoothed wavelet periodogram, is studied. Simulations and an econometric application demonstrate the practical utility of the method."

Abstract: "We propose a fully automated system to detect, aggregate, and classify sunspot groups according to the McIntosh scheme using ground-based white light (WL) observations from the USET facility  located at the Royal Observatory of Belgium. The sunspot detection uses a Convolutional Neural Network (CNN), trained from segmentation maps obtained with an unsupervised method based on mathematical morphology and image thresholding.
Given the sunspot mask, a mean-shift algorithm is used to aggregate individual sunspots into sunspot groups. This algorithm accounts for the area of each sunspot as well as for prior knowledge regarding the shape of sunspot group. A sunspot group, defined by its bounding box and  location on the Sun, is finally fed into a CNN multitask classifier. The latter predicts the three characters Z, p, and c in the McIntosh system. The tasks are organized hierarchically to mimic the dependency of the second (p) and third (c) characters on the first (Z).
The resulting CNN-based segmentation is more accurate than classical unsupervised methods, with an enhancement up to 16% of F1 score in detection of the smallest sunspots, and it is robust to the presence of clouds. The automated clustering method was able to separate  groups with an accuracy of 80%, when compared to hand-made USET sunspot group catalog. The CNN-based sunspot classifier shows comparable performances to methods using continuum as well as magnetogram images recorded by instruments on space mission. We also show that ensemble of classifiers allows differentiating reliable and potentially incorrect predictions."

Abstract: "Solving the classical image restoration problem aims at estimating an unknown pristine image from few corrupted data samples. The data formation model typically includes optical blur, detector noise, downscaling and quantization of measured intensities. From scarce degraded data, an upscaled deblurred and denoised higher resolution image is estimated using a Bayesian maximum-likelihood estimation framework and a novel adaptive image representation model based on freely positioned, oriented and scaled image elements. Regularization is introduced by limiting adaptively the number of such image elements depending on local image details. First experiments demonstrate encouraging results for joint image deblurring denoising and upscaling by outperforming exemplar-based methods such as non-local mean BM3D denoising and recent deep-learning image upscaling."

Abstract: "Sparsity promotion is a popular regularization technique for inverse problems, reflecting the prior knowledge that the exact solution is expected to have few non-vanishing components, e.g. in a suitable wavelet basis. In this talk, I will present a convolutional neural network designed for sparsity-promoting regularization for linear inverse problems. The key idea motivating the proposed architecture is to unroll the Iterative Soft Thresholding Algorithm (ISTA) for sparsity promotion, introducing a learnable correction on the forward operator. By employing a multiresolution wavelet representation of the signals, we can represent the learned correction as a (suitably designed) convolutional layer, and by microlocal analysis, we can interpret it as a pseudodifferential operator, motivating the name of our novel architecture: PsiDONet. I will discuss the main theoretical results associated with the resulting algorithm, as well as some numerical examples in the main considered case study, namely, limited-angle computed tomography. 

Finally, I will describe some recent extensions of the project, both in terms of a more efficient parametrization of the architecture and of a more general class of considered regularization functionals. 

This is a joint project with T. A. Bubba (University of Bath), M. Lassas, S. Siltanen (University of Helsinki), and M. Galinier, M. Prato (Università di Modena)."

Abstract: "The refinable B-spline basis is extended to the arbitrary assembling of segments from a set of smooth building functions. In the next stage, factoring the refinement matrix associated to this scaling basis into lifting steps enables the construction of a wavelet transform on a nonequidistant grid and a finite interval, adapted to the irregularity and the presence of boundaries. In contrast to wavelets defined by subdivision, the smoothness of the wavelet basis functions is fully controlled. We discuss the near-necessity of father functions and possible applications in adaptive signal analysis and processing and in interpretable deep neural networks."

Abstract: "We address global high-dimensional optimization problems in which the objective is mostly varying along a low-dimensional subspace of the search space. These problems appear, e.g., in complex engineering and physical simulation/inverse problems. We propose a random-subspace algorithmic framework (referred to as X-REGO) that randomly projects, in a sequential or simultaneous manner, the high-dimensional original problem into low-dimensional subproblems that can then be solved with any global, or even local, optimization solver. For Lipschitz-continuous objectives, we analyse its convergence using novel tools from probability theory as well as conic integral geometry; our analysis relies on an estimation of the probability that the randomly-embedded subproblem shares (approximately) the same global optimum as the original problem. This success probability is then used to show almost sure convergence of X-REGO to an approximate global solution of the original problem, under weak assumptions on the problem (having a strictly feasible global solution) and on the solver (guaranteed to find an approximate global solution of the reduced problem with sufficiently high probability).

This is joint work with C. Cartis (University of Oxford) and A. Otemissov (Nazarbayev University)."

Abstract: "Low-rank matrix approximations are widely used in many fields such as data analysis and machine learning. When dealing with a large data, stored in a matrix say X, it is conventional to perform dimensionality reduction by approximating X by a low-rank matrix. In many applications, the data matrix X happens to be nonnegative and sparse. This gives rise to a widely recurring nonlinear matrix decomposition (NMD) problem: Given a sparse nonnegative matrix X, find a low-rank matrix Θ such that X ≈ f (Θ), where f is an element-wise nonlinear function. We focus on the case where f (·) = max(0, ·), the rectified unit (ReLU) non-linear activation. We refer to the corresponding problem as ReLU-NMD. 

The talk will provide a brief overview of the existing approaches that were developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1) Aggressive Accelerated NMD (A-NMD) which uses an adaptive Nesterov extrapolation to accelerate an existing algorithm, and (2) Three-Block NMD (3B-NMD) which parametrizes Θ = W H and leads to a significant reduction in the computational cost. We also propose an effective initialization strategy

based on the nuclear norm as a proxy for the rank function. We illustrate the effectiveness of the proposed algorithms on synthetic and real-world data sets."