arXiv subjects

Nicolas Papadakis

Publications and source records attributed to Nicolas Papadakis.

At least 55 records · Page 3Linked to original sources

arXiv2021

GraphXCOVID: Explainable Deep Graph Diffusion Pseudo-Labelling for Identifying COVID-19 on Chest X-rays

Can one learn to diagnose COVID-19 under extreme minimal supervision? Since the outbreak of the novel COVID-19 there has been a rush for developing Artificial Intelligence techniques for expert-level disease identification on Chest X-ray data. In particular, the use of deep supervised learning has become the go-to paradigm. However, the performance of such models is heavily dependent on the availability of a large and representative labelled dataset. The creation of which is a heavily expensive and time consuming task, and especially imposes a great challenge for a novel disease. Semi-supervised learning has shown the ability to match the incredible performance of supervised models whilst requiring a small fraction of the labelled examples. This makes the semi-supervised paradigm an attractive option for identifying COVID-19. In this work, we introduce a graph based deep semi-supervised framework for classifying COVID-19 from chest X-rays. Our framework introduces an optimisation model for graph diffusion that reinforces the natural relation among the tiny labelled set and the vast unlabelled data. We then connect the diffusion prediction output as pseudo-labels that are used in an iterative scheme in a deep net. We demonstrate, through our experiments, that our model is able to outperform the current leading supervised model with a tiny fraction of the labelled examples. Finally, we provide attention maps to accommodate the radiologist's mental model, better fitting their perceptual and cognitive abilities. These visualisation aims to assist the radiologist in judging whether the diagnostic is correct or not, and in consequence to accelerate the decision.

cs.LG↗

arXiv2021

Nonlinear Power Method for Computing Eigenvectors of Proximal Operators and Neural Networks

Neural networks have revolutionized the field of data science, yielding remarkable solutions in a data-driven manner. For instance, in the field of mathematical imaging, they have surpassed traditional methods based on convex regularization. However, a fundamental theory supporting the practical applications is still in the early stages of development. We take a fresh look at neural networks and examine them via nonlinear eigenvalue analysis. The field of nonlinear spectral theory is still emerging, providing insights about nonlinear operators and systems. In this paper we view a neural network as a complex nonlinear operator and attempt to find its nonlinear eigenvectors. We first discuss the existence of such eigenvectors and analyze the kernel of ReLU networks. Then we study a nonlinear power method for generic nonlinear operators. For proximal operators associated to absolutely one-homogeneous convex regularization functionals, we can prove convergence of the method to an eigenvector of the proximal operator. This motivates us to apply a nonlinear method to networks which are trained to act similarly as a proximal operator. In order to take the non-homogeneity of neural networks into account we define a modified version of the power method. We perform extensive experiments for different proximal operators and on various shallow and deep neural networks designed for image denoising. Proximal eigenvectors will be used for geometric analysis of graphs, as clustering or the computation of distance functions. For simple neural nets, we observe the influence of training data on the eigenvectors. For state-of-the-art denoising networks, we show that eigenvectors can be interpreted as (un)stable modes of the network, when contaminated with noise or other degradations.

math.SP↗

arXiv2021

Overrelaxed Sinkhorn-Knopp Algorithm for Regularized Optimal Transport

This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely-used iterative Bregman projections algorithm (or Sinkhorn--Knopp algorithm). We first propose to rely on regularized nonlinear acceleration schemes. In practice, such approaches lead to fast algorithms, but their global convergence is not ensured. Hence, we next propose a new algorithm with convergence guarantees. The idea is to overrelax the Bregman projection operators, allowing for faster convergence. We propose a simple method for establishing global convergence by ensuring the decrease of a Lyapunov function at each step. An adaptive choice of overrelaxation parameter based on the Lyapunov function is constructed. We also suggest a heuristic to choose a suitable asymptotic overrelaxation parameter, based on a local convergence analysis. Our numerical experiments show a gain in convergence speed by an order of magnitude in certain regimes.

math.NA↗

arXiv2021

Learning local regularization for variational image restoration

In this work, we propose a framework to learn a local regularization model for solving general image restoration problems. This regularizer is defined with a fully convolutional neural network that sees the image through a receptive field corresponding to small image patches. The regularizer is then learned as a critic between unpaired distributions of clean and degraded patches using a Wasserstein generative adversarial networks based energy. This yields a regularization function that can be incorporated in any image restoration problem. The efficiency of the framework is finally shown on denoising and deblurring applications.

eess.IV↗

arXiv2021

On the Existence of Optimal Transport Gradient for Learning Generative Models

The use of optimal transport cost for learning generative models has become popular with Wasserstein Generative Adversarial Networks (WGAN). Training of WGAN relies on a theoretical background: the calculation of the gradient of the optimal transport cost with respect to the generative model parameters. We first demonstrate that such gradient may not be defined, which can result in numerical instabilities during gradient-based optimization. We address this issue by stating a valid differentiation theorem in the case of entropic regularized transport and specify conditions under which existence is ensured. By exploiting the discrete nature of empirical data, we formulate the gradient in a semi-discrete setting and propose an algorithm for the optimization of the generative model parameters. Finally, we illustrate numerically the advantage of the proposed framework.

stat.ML↗

arXiv2020

Block based refitting in $\ell_{12}$ sparse regularisation

In many linear regression problems, including ill-posed inverse problems in image restoration, the data exhibit some sparse structures that can be used to regularize the inversion. To this end, a classical path is to use $\ell_{12}$ block based regularization. While efficient at retrieving the inherent sparsity patterns of the data -- the support -- the estimated solutions are known to suffer from a systematical bias. We propose a general framework for removing this artifact by refitting the solution towards the data while preserving key features of its structure such as the support. This is done through the use of refitting block penalties that only act on the support of the estimated solution. Based on an analysis of related works in the literature, we introduce a new penalty that is well suited for refitting purposes. We also present a new algorithm to obtain the refitted solution along with the original (biased) solution for any convex refitting block penalty. Experiments illustrate the good behavior of the proposed block penalty for refitting solutions of Total Variation and Total GeneralizedVariation models.

eess.SP↗

arXiv2020

Approximation of Wasserstein distance with Transshipment

An algorithm for approximating the p-Wasserstein distance between histograms defined on unstructured discrete grids is presented. It is based on the computation of a barycenter constrained to be supported on a low dimensional subspace, which corresponds to a transshipment problem. A multi-scale strategy is also considered. The method provides sparse transport matrices and can be applied to large scale and non structured data.

math.NA↗

arXiv2020

Learning to segment microscopy images with lazy labels

The need for labour intensive pixel-wise annotation is a major limitation of many fully supervised learning methods for segmenting bioimages that can contain numerous object instances with thin separations. In this paper, we introduce a deep convolutional neural network for microscopy image segmentation. Annotation issues are circumvented by letting the network being trainable on coarse labels combined with only a very small number of images with pixel-wise annotations. We call this new labelling strategy `lazy' labels. Image segmentation is stratified into three connected tasks: rough inner region detection, object separation and pixel-wise segmentation. These tasks are learned in an end-to-end multi-task learning framework. The method is demonstrated on two microscopy datasets, where we show that the model gives accurate segmentation results even if exact boundary labels are missing for a majority of annotated data. It brings more flexibility and efficiency for training deep neural networks that are data hungry and is applicable to biomedical images with poor contrast at the object boundaries or with diverse textures and repeated patterns.

cs.CV↗

arXiv2020

GraphX$^{NET}-$ Chest X-Ray Classification Under Extreme Minimal Supervision

The task of classifying X-ray data is a problem of both theoretical and clinical interest. Whilst supervised deep learning methods rely upon huge amounts of labelled data, the critical problem of achieving a good classification accuracy when an extremely small amount of labelled data is available has yet to be tackled. In this work, we introduce a novel semi-supervised framework for X-ray classification which is based on a graph-based optimisation model. To the best of our knowledge, this is the first method that exploits graph-based semi-supervised learning for X-ray data classification. Furthermore, we introduce a new multi-class classification functional with carefully selected class priors which allows for a smooth solution that strengthens the synergy between the limited number of labels and the huge amount of unlabelled data. We demonstrate, through a set of numerical and visual experiments, that our method produces highly competitive results on the ChestX-ray14 data set whilst drastically reducing the need for annotated data.

cs.LG↗

arXiv2020

Multi-task deep learning for image segmentation using recursive approximation tasks

Fully supervised deep neural networks for segmentation usually require a massive amount of pixel-level labels which are manually expensive to create. In this work, we develop a multi-task learning method to relax this constraint. We regard the segmentation problem as a sequence of approximation subproblems that are recursively defined and in increasing levels of approximation accuracy. The subproblems are handled by a framework that consists of 1) a segmentation task that learns from pixel-level ground truth segmentation masks of a small fraction of the images, 2) a recursive approximation task that conducts partial object regions learning and data-driven mask evolution starting from partial masks of each object instance, and 3) other problem oriented auxiliary tasks that are trained with sparse annotations and promote the learning of dedicated features. Most training images are only labeled by (rough) partial masks, which do not contain exact object boundaries, rather than by their full segmentation masks. During the training phase, the approximation task learns the statistics of these partial masks, and the partial regions are recursively increased towards object boundaries aided by the learned information from the segmentation task in a fully data-driven fashion. The network is trained on an extremely small amount of precisely segmented images and a large set of coarse labels. Annotations can thus be obtained in a cheap way. We demonstrate the efficiency of our approach in three applications with microscopy images and ultrasound images.

cs.CV↗

arXiv2020

Variational Osmosis for Non-linear Image Fusion

We propose a new variational model for non-linear image fusion. Our approach is based on the use of an osmosis energy term related to the one studied in Vogel et al. (2013) and Weickert et al. (2013) The minimization of the proposed non-convex energy realizes visually plausible image data fusion, invariant to multiplicative brightness changes. On the practical side, it requires minimal supervision and parameter tuning and can encode prior information on the structure of the images to be fused. For the numerical solution of the proposed model, we develop a primal-dual algorithm and we apply the resulting minimization scheme to solve multi-modal face fusion, color transfer and cultural heritage conservation problems. Visual and quantitative comparisons to state-of-the-art approaches prove the out-performance and the flexibility of our method.

cs.CV↗

arXiv2019

Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications

The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data analysis. The Sinkhorn divergence allows the fast computation of an entropically regularized Wasserstein distance between two probability distributions supported on a finite metric space of (possibly) high-dimension. For data sampled from one or two unknown probability distributions, we derive the distributional limits of the empirical Sinkhorn divergence and its centered version (Sinkhorn loss). We also propose a bootstrap procedure which allows to obtain new test statistics for measuring the discrepancies between multivariate probability distributions. Our work is inspired by the results of Sommerfeld and Munk (2016) on the asymptotic distribution of empirical Wasserstein distance on finite space using unregularized transportation costs. Incidentally we also analyze the asymptotic distribution of entropy-regularized Wasserstein distances when the regularization parameter tends to zero. Simulated and real datasets are used to illustrate our approach.

math.ST↗

arXiv2019

Data-driven regularization of Wasserstein barycenters with an application to multivariate density registration

We present a framework to simultaneously align and smooth data in the form of multiple point clouds sampled from unknown densities with support in a d-dimensional Euclidean space. This work is motivated by applications in bioinformatics where researchers aim to automatically homogenize large datasets to compare and analyze characteristics within a same cell population. Inconveniently, the information acquired is most certainly noisy due to mis-alignment caused by technical variations of the environment. To overcome this problem, we propose to register multiple point clouds by using the notion of regularized barycenters (or Fréchet mean) of a set of probability measures with respect to the Wasserstein metric. A first approach consists in penalizing a Wasserstein barycenter with a convex functional as recently proposed in Bigot and al. (2018). A second strategy is to transform the Wasserstein metric itself into an entropy regularized transportation cost between probability measures as introduced in Cuturi (2013). The main contribution of this work is to propose data-driven choices for the regularization parameters involved in each approach using the Goldenshluger-Lepski's principle. Simulated data sampled from Gaussian mixtures are used to illustrate each method, and an application to the analysis of flow cytometry data is finally proposed. This way of choosing of the regularization parameter for the Sinkhorn barycenter is also analyzed through the prism of an oracle inequality that relates the error made by such data-driven estimators to the one of an ideal estimator.

stat.ME↗

arXiv2019

Semi-supervised Learning with Graphs: Covariance Based Superpixels For Hyperspectral Image Classification

In this paper, we present a graph-based semi-supervised framework for hyperspectral image classification. We first introduce a novel superpixel algorithm based on the spectral covariance matrix representation of pixels to provide a better representation of our data. We then construct a superpixel graph, based on carefully considered feature vectors, before performing classification. We demonstrate, through a set of experimental results using two benchmarking datasets, that our approach outperforms three state-of-the-art classification frameworks, especially when an extremely small amount of labelled data is used.

cs.CV↗

arXiv2019

Penalized Barycenters in the Wasserstein Space

In this paper, a regularization of Wasserstein barycenters for random measures supported on $\mathbb{R}^{d}$ is introduced via convex penalization. The existence and uniqueness of such barycenters is first proved for a large class of penalization functions. The Bregman divergence associated to the penalization term is then considered to obtain a stability result on penalized barycenters. This allows the comparison of data made of $n$ absolutely continuous probability measures, within the more realistic setting where one only has access to a dataset of random variables sampled from unknown distributions. The convergence of the penalized empirical barycenter of a set of $n$ iid random probability measures towards its population counterpart is finally analyzed. This approach is shown to be appropriate for the statistical analysis of either discrete or absolutely continuous random measures. It also allows to construct, from a set of discrete measures, consistent estimators of population Wasserstein barycenters that are absolutely continuous.

math.ST↗

arXiv2019

Refitting solutions promoted by $\ell_{12}$ sparse analysis regularization with block penalties

In inverse problems, the use of an $\ell_{12}$ analysis regularizer induces a bias in the estimated solution. We propose a general refitting framework for removing this artifact while keeping information of interest contained in the biased solution. This is done through the use of refitting block penalties that only act on the co-support of the estimation. Based on an analysis of related works in the literature, we propose a new penalty that is well suited for refitting purposes. We also present an efficient algorithmic method to obtain the refitted solution along with the original (biased) solution for any convex refitting block penalty. Experiments illustrate the good behavior of the proposed block penalty for refitting.

math.OC↗

arXiv2018

Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification.

stat.ML↗

arXiv2017

Log-PCA versus Geodesic PCA of histograms in the Wasserstein space

This paper is concerned by the statistical analysis of data sets whose elements are random histograms. For the purpose of learning principal modes of variation from such data, we consider the issue of computing the PCA of histograms with respect to the 2-Wasserstein distance between probability measures. To this end, we propose to compare the methods of log-PCA and geodesic PCA in the Wasserstein space as introduced by Bigot et al. (2015) and Seguy and Cuturi (2015). Geodesic PCA involves solving a non-convex optimization problem. To solve it approximately, we propose a novel forward-backward algorithm. This allows a detailed comparison between log-PCA and geodesic PCA of one-dimensional histograms, which we carry out using various data sets, and stress the benefits and drawbacks of each method. We extend these results for two-dimensional data and compare both methods in that setting.

stat.ME↗