arXiv subjects

Nelly Pustelnik

Publications and source records attributed to Nelly Pustelnik.

At least 19 recordsLinked to original sources

arXiv2026

Correlation flow governs learning at criticality

The initialization of deep neural networks determines whether information and gradients can propagate across depth, yet a unified theory connecting these properties to learning dynamics remains elusive. Combining mean-field theory and random matrix theory, we establish a direct link between correlation propagation and the Neural Tangent Kernel (NTK) that governs learning in the sequential limit of infinitely wide, infinitely deep networks. Correlation propagation to infinite depth is possible only at a single, critical point in the weight-bias variance plane. At this point, we leverage the algebraic decay of the end-to-end Jacobian with depth to prove that the NTK becomes exactly proportional to the output correlation at infinite depth, tying together information propagation and learning dynamics. We further show that orthogonal initialization suppresses the leading finite-size corrections present under Gaussian initialization, clarifying the respective roles of the two initialization ensembles in this limit. These theoretical predictions are validated quantitatively on finite-width, finite-depth networks. Together, these results demonstrate that orthogonal initialization and criticality are required to control the asymptotic dynamics of deep learning.

cs.LG↗

arXiv2026

Multiresolution Adaptive Block-Coordinate Forward-Backward for Image Reconstruction

Classical first-order optimization methods for imaging inverse problems scale poorly with image resolution. Wavelet based multilevel strategies can accelerate convergence under strong blur, but their fixed coarse-to-fine schedules lose effectiveness in moderate-blur or noise-dominated regimes. In this work, we propose an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration. Multiresolution block selection is driven by the local magnitude of the proximal update via a stochastic non-smooth Gauss-Southwell rule applied to the wavelet decomposition of the image. This adaptive selection strategy dynamically balances updates across scales, emphasizing coarse or fine blocks according to the degradation regime. As a result, the proposed method automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

eess.SP↗

arXiv2026

Plug-and-play forward backward algorithm to restore Landsat images: A preliminary step to uncover the history of surface waters

The temporal and spatial analysis of river dynamics is a key factor for studying and understanding human impacts on floodplains. To assess the changes taking place, it is necessary to have high-resolution images with a large spatial coverage and a high temporal revisit frequency over the long term. Satellite imagery meets several of these criteria. For instance, Sentinel data provide high-resolution images but only after 2015. Therefore, to study water surface evolution prior to this date, it is necessary to rely on other satellite images such as Landsat, which offers longer historical coverage, albeit with lower spatial resolution. In this study, we aim to increase the spatial resolution of Landsat data from 30 to 10 meters (resolution of Sentinel images). To achieve this goal, we develop an innovative single image super-resolution method based on a plug-and-play approach.

eess.SP↗

arXiv2025

A new initialisation to Control Gradients in Sinusoidal Neural network

Proper initialisation strategy is of primary importance to mitigate gradient explosion or vanishing when training neural networks. Yet, the impact of initialisation parameters still lacks a precise theoretical understanding for several well-established architectures. Here, we propose a new initialisation for networks with sinusoidal activation functions such as \texttt{SIREN}, focusing on gradients control, their scaling with network depth, their impact on training and on generalization. To achieve this, we identify a closed-form expression for the initialisation of the parameters, differing from the original \texttt{SIREN} scheme. This expression is derived from fixed points obtained through the convergence of pre-activation distribution and the variance of Jacobian sequences. Controlling both gradients and targeting vanishing pre-activation helps preventing the emergence of inappropriate frequencies during estimation, thereby improving generalization. We further show that this initialisation strongly influences training dynamics through the Neural Tangent Kernel framework (NTK). Finally, we benchmark \texttt{SIREN} with the proposed initialisation against the original scheme and other baselines on function fitting and image reconstruction. The new initialisation consistently outperforms state-of-the-art methods across a wide range of reconstruction tasks, including those involving physics-informed neural networks.

cs.LG↗

arXiv2025

A flexible block-coordinate forward-backward algorithm for non-smooth and non-convex optimization

Block coordinate descent (BCD) methods are prevalent in large scale optimization problems due to the low memory and computational costs per iteration, the predisposition to parallelization, and the ability to exploit the structure of the problem. The theoretical and practical performance of BCD relies heavily on the rules defining the choice of the blocks to be updated at each iteration. We propose a new deterministic BCD framework that allows for very flexible updates, while guaranteeing state-of-the-art convergence guarantees on non-smooth nonconvex optimization problems. While encompassing several update rules from the literature, this framework allows for priority on updates of particular blocks and correlations in the block selection between iterations, which is not permitted under the classical convergent stochastic framework. This flexibility is leveraged in the context of multilevel optimization algorithms and, in particular, in multilevel image restoration problems, where the efficiency of the approach is illustrated.

math.OC↗

arXiv2025

Diffusion models for polarimetric reconstruction of circumstellar environments

In this paper, we propose an approach combining diffusion models and inverse problems for the reconstruction of circumstellar disk images. Our method builds upon the Rhapsodie framework for polarimetric imaging, substituting its classical prior with a diffusion model trained on synthetic data. Our formulation explicitly incorporates stellar leakage while efficiently handling missing data and high level noise inherent to high-contrast polarimetric imaging. Experiments show significant improvement over conventional methods within our framework of assumptions, opening new perspectives for studying circumstellar environments.

astro-ph.IM↗

arXiv2025

Deep learning in the abyss: a stratified Physics Informed Neural Network for data assimilation

The reconstruction of deep ocean currents is a major challenge in data assimilation due to the scarcity of interior data. In this work, we present a proof of concept for deep ocean flow reconstruction using a Physics-Informed Neural Network (PINN), a machine learning approach that offers an alternative to traditional data assimilation methods. We introduce an efficient algorithm called StrAssPINN (for Stratified Assimilation PINNs), which assigns a separate network to each layer of the ocean model while allowing them to interact during training. The neural network takes spatiotemporal coordinates as input and predicts the velocity field at those points. Using a SIREN architecture (a multilayer perceptron with sine activation functions), which has proven effective in various contexts, the network is trained using both available observational data and dynamical priors enforced at several collocation points. We apply this method to pseudo-observed ocean data generated from a 3-layer quasi-geostrophic model, where the pseudo-observations include surface-level data akin to SWOT observations of sea surface height, interior data similar to ARGO floats, and a limited number of deep ARGO-like measurements in the lower layers. Our approach successfully reconstructs ocean flows in both the interior and surface layers, demonstrating a strong ability to resolve key ocean mesoscale features, including vortex rings, eastward jets associated with potential vorticity fronts, and smoother Rossby waves. This work serves as a prelude to applying StrAssPINN to real-world observational data.

physics.ao-ph↗

arXiv2024

Identifying a piecewise affine signal from its nonlinear observation -- application to DNA replication analysis

DNA replication stands as one of the fundamental biological processes crucial for cellular functioning. Recent experimental developments enable the study of replication dynamics at the single-molecule level for complete genomes, facilitating a deeper understanding of its main parameters. In these new data, replication dynamics is reported by the incorporation of an exogenous chemical, whose intra-cellular concentration follows a nonlinear function. The analysis of replication traces thus gives rise to a nonlinear inverse problem, presenting a nonconvex optimization challenge. We demonstrate that under noiseless conditions, the replication dynamics can be uniquely identified by the proposed model. Computing a global solution to this optimization problem is specially challenging because of its multiple local minima. We present the DNA-inverse optimization method that is capable of finding this global solution even in the presence of noise. Comparative analysis against state-of-the-art optimization methods highlights the superior computational efficiency of our approach. DNA-inverse enables the automatic recovery of all configurations of the replication dynamics, which was not possible with previous methods.

math.OC↗

arXiv2024

A multilevel framework for accelerating uSARA in radio-interferometric imaging

This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.

math.OC↗

arXiv2024

Comparison of Proximal First-Order Primal and Primal-Dual algorithms via Performance Estimation

Selecting the fastest algorithm for a specific signal/image processing task is a challenging question. We propose an approach based on the Performance Estimation Problem framework that numerically and automatically computes the worst-case performance of a given optimization method on a class of functions. We first propose a computer-assisted analysis and comparison of several first-order primal optimization methods, namely, the gradient method, the forward-backward, Peaceman-Rachford, and Douglas-Rachford splittings. We tighten the existing convergence results of these algorithms and extend them to new classes of functions. Our analysis is then extended and evaluated in the context of the primal-dual Chambolle-Pock and Condat-V\~u methods.

math.OC↗

arXiv2023

Equivariant plug-and-play image reconstruction

Plug-and-play algorithms constitute a popular framework for solving inverse imaging problems that rely on the implicit definition of an image prior via a denoiser. These algorithms can leverage powerful pre-trained denoisers to solve a wide range of imaging tasks, circumventing the necessity to train models on a per-task basis. Unfortunately, plug-and-play methods often show unstable behaviors, hampering their promise of versatility and leading to suboptimal quality of reconstructed images. In this work, we show that enforcing equivariance to certain groups of transformations (rotations, reflections, and/or translations) on the denoiser strongly improves the stability of the algorithm as well as its reconstruction quality. We provide a theoretical analysis that illustrates the role of equivariance on better performance and stability. We present a simple algorithm that enforces equivariance on any existing denoiser by simply applying a random transformation to the input of the denoiser and the inverse transformation to the output at each iteration of the algorithm. Experiments on multiple imaging modalities and denoising networks show that the equivariant plug-and-play algorithm improves both the reconstruction performance and the stability compared to their non-equivariant counterparts.

eess.IV↗

arXiv2023

Unfolded proximal neural networks for robust image Gaussian denoising

A common approach to solve inverse imaging problems relies on finding a maximum a posteriori (MAP) estimate of the original unknown image, by solving a minimization problem. In thiscontext, iterative proximal algorithms are widely used, enabling to handle non-smooth functions and linear operators. Recently, these algorithms have been paired with deep learning strategies, to further improve the estimate quality. In particular, proximal neural networks (PNNs) have been introduced, obtained by unrolling a proximal algorithm as for finding a MAP estimate, but over a fixed number of iterations, with learned linear operators and parameters. As PNNs are based on optimization theory, they are very flexible, and can be adapted to any image restoration task, as soon as a proximal algorithm can solve it. They further have much lighter architectures than traditional networks. In this article we propose a unified framework to build PNNs for the Gaussian denoising task, based on both the dual-FB and the primal-dual Chambolle-Pock algorithms. We further show that accelerated inertial versions of these algorithms enable skip connections in the associated NN layers. We propose different learning strategies for our PNN framework, and investigate their robustness (Lipschitz property) and denoising efficiency. Finally, we assess the robustness of our PNNs when plugged in a forward-backward algorithm for an image deblurring problem.

cs.CV↗

arXiv2023

IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration

This paper presents a multilevel framework for inertial and inexact proximal algorithms, that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems and we show that it considerably accelerates the convergence of the corresponding one-level (i.e. standard) version of the methods, for large-scale images.

math.OC↗

arXiv2022

Multilevel fista for image restoration

This paper presents a multilevel FISTA algorithm, based on the use of the Moreau envelope to build the correction brought by the coarse models, which is easy to compute when the explicit form of the proximal operator of the considered functions is known. This approach is supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We evaluate our approach on image restoration problems and we show that it outperforms classical FISTA for large-scale images.

math.OC↗

arXiv2022

Covid19 Reproduction Number: Credibility Intervals by Blockwise Proximal Monte Carlo Samplers

Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. The intensity of the pandemic on a given territory is efficiently measured by the reproduction number, quantifying the rate of growth of daily new infections. Recently, estimates for the time evolution of the reproduction number were produced using an inverse problem formulation with a nonsmooth functional minimization. While it was designed to be robust to the limited quality of the Covid19 data (outliers, missing counts), the procedure lacks the ability to output credibility interval based estimates. This remains a severe limitation for practical use in actual pandemic monitoring by epidemiologists that the present work aims to overcome by use of Monte Carlo sampling. After interpretation of the nonsmooth functional into a Bayesian framework, several sampling schemes are tailored to adjust the nonsmooth nature of the resulting posterior distribution. The originality of the devised algorithms stems from combining a Langevin Monte Carlo sampling scheme with Proximal operators. Performance of the new algorithms in producing relevant credibility intervals for the reproduction number estimates and denoised counts are compared. Assessment is conducted on real daily new infection counts made available by the Johns Hopkins University. The interest of the devised monitoring tools are illustrated on Covid19 data from several different countries.

cs.LG↗

arXiv2022

Alternative design of DeepPDNet in the context of image restoration

This work designs an image restoration deep network relying on unfolded Chambolle-Pock primal-dual iterations. Each layer of our network is built from Chambolle-Pock iterations when specified for minimizing a sum of a $\ell_2$-norm data-term and an analysis sparse prior. The parameters of our network are the step-sizes of the Chambolle-Pock scheme and the linear operator involved in sparsity-based penalization, including implicitly the regularization parameter. A backpropagation procedure is fully described. Preliminary experiments illustrate the good behavior of such a deep primal-dual network in the context of image restoration on BSD68 database.

eess.IV↗

arXiv2022

Temporal evolution of the Covid19 pandemic reproduction number: Estimations from proximal optimization to Monte Carlo sampling

Monitoring the evolution of the Covid19 pandemic constitutes a critical step in sanitary policy design. Yet, the assessment of the pandemic intensity within the pandemic period remains a challenging task because of the limited quality of data made available by public health authorities (missing data, outliers and pseudoseasonalities, notably), that calls for cumbersome and ad-hoc preprocessing (denoising) prior to estimation. Recently, the estimation of the reproduction number, a measure of the pandemic intensity, was formulated as an inverse problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that formulation lacks robustness against the limited quality of the Covid19 data and confidence assessment. The present work aims to address both limitations: First, it discusses solutions to produce a robust assessment of the pandemic intensity by accounting for the low quality of the data directly within the inverse problem formulation. Second, exploiting a Bayesian interpretation of the inverse problem formulation, it devises a Monte Carlo sampling strategy, tailored to a nonsmooth log-concave a posteriori distribution, to produce relevant credibility intervalbased estimates for the Covid19 reproduction number. Clinical relevance Applied to daily counts of new infections made publicly available by the Health Authorities for around 200 countries, the proposed procedures permit robust assessments of the time evolution of the Covid19 pandemic intensity, updated automatically and on a daily basis.

q-bio.PE↗

arXiv2021

Hyperparameter selection for Discrete Mumford-Shah

This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford-Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images.

eess.IV↗