Mireille El Gheche

Matthias Minder, Zahra Farsijani, Dhruti Shah et al.

In many applications, a dataset can be considered as a set of observed signals that live on an unknown underlying graph structure. Some of these signals may be seen as white noise that has been filtered on the graph topology by a graph filter. Hence, the knowledge of the filter and the graph provides valuable information about the underlying data generation process and the complex interactions that arise in the dataset. We hence introduce a novel graph signal processing framework for jointly learning the graph and its generating filter from signal observations. We cast a new optimisation problem that minimises the Wasserstein distance between the distribution of the signal observations and the filtered signal distribution model. Our proposed method outperforms state-of-the-art graph learning frameworks on synthetic data. We then apply our method to a temperature anomaly dataset, and further show how this framework can be used to infer missing values if only very little information is available.

Mattia Rossi, Mireille El Gheche, Andreas Kuhn et al.

Depth estimation is an essential component in understanding the 3D geometry of a scene, with numerous applications in urban and indoor settings. These scenes are characterized by a prevalence of human made structures, which in most of the cases, are either inherently piece-wise planar, or can be approximated as such. In these settings, we devise a novel depth refinement framework that aims at recovering the underlying piece-wise planarity of the inverse depth map. We formulate this task as an optimization problem involving a data fidelity term that minimizes the distance to the input inverse depth map, as well as a regularization that enforces a piece-wise planar solution. As for the regularization term, we model the inverse depth map as a weighted graph between pixels. The proposed regularization is designed to estimate a plane automatically at each pixel, without any need for an a priori estimation of the scene planes, and at the same time it encourages similar pixels to be assigned to the same plane. The resulting optimization problem is efficiently solved with ADAM algorithm. Experiments show that our method leads to a significant improvement in depth refinement, both visually and numerically, with respect to state-of-the-art algorithms on Middlebury, KITTI and ETH3D multi-view stereo datasets.

Guillermo Ortiz-Jimenez, Mireille El Gheche, Effrosyni Simou et al.

Optimal transport aims to estimate a transportation plan that minimizes a displacement cost. This is realized by optimizing the scalar product between the sought plan and the given cost, over the space of doubly stochastic matrices. When the entropy regularization is added to the problem, the transportation plan can be efficiently computed with the Sinkhorn algorithm. Thanks to this breakthrough, optimal transport has been progressively extended to machine learning and statistical inference by introducing additional application-specific terms in the problem formulation. It is however challenging to design efficient optimization algorithms for optimal transport based extensions. To overcome this limitation, we devise a general forward-backward splitting algorithm based on Bregman distances for solving a wide range of optimization problems involving a differentiable function with Lipschitz-continuous gradient and a doubly stochastic constraint. We illustrate the efficiency of our approach in the context of continuous domain adaptation. Experiments show that the proposed method leads to a significant improvement in terms of speed and performance with respect to the state of the art for domain adaptation on a continually rotating distribution coming from the standard two moon dataset.

Hermina Petric Maretic, Mireille El Gheche, Pascal Frossard

Graph inference methods have recently attracted a great interest from the scientific community, due to the large value they bring in data interpretation and analysis. However, most of the available state-of-the-art methods focus on scenarios where all available data can be explained through the same graph, or groups corresponding to each graph are known a priori. In this paper, we argue that this is not always realistic and we introduce a generative model for mixed signals following a heat diffusion process on multiple graphs. We propose an expectation-maximisation algorithm that can successfully separate signals into corresponding groups, and infer multiple graphs that govern their behaviour. We demonstrate the benefits of our method on both synthetic and real data.