Guillaume Tochon

Anthony Frion, Lucas Drumetz, Mauro Dalla Mura et al.

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

Anthony Frion, Lucas Drumetz, Mauro Dalla Mura et al.

Over the last few years, several works have proposed deep learning architectures to learn dynamical systems from observation data with no or little knowledge of the underlying physics. A line of work relies on learning representations where the dynamics of the underlying phenomenon can be described by a linear operator, based on the Koopman operator theory. However, despite being able to provide reliable long-term predictions for some dynamical systems in ideal situations, the methods proposed so far have limitations, such as requiring to discretize intrinsically continuous dynamical systems, leading to data loss, especially when handling incomplete or sparsely sampled data. Here, we propose a new deep Koopman framework that represents dynamics in an intrinsically continuous way, leading to better performance on limited training data, as exemplified on several datasets arising from dynamical systems.

Raghav Mehta, Angelos Filos, Ujjwal Baid et al.

Deep learning (DL) models have provided state-of-the-art performance in various medical imaging benchmarking challenges, including the Brain Tumor Segmentation (BraTS) challenges. However, the task of focal pathology multi-compartment segmentation (e.g., tumor and lesion sub-regions) is particularly challenging, and potential errors hinder translating DL models into clinical workflows. Quantifying the reliability of DL model predictions in the form of uncertainties could enable clinical review of the most uncertain regions, thereby building trust and paving the way toward clinical translation. Several uncertainty estimation methods have recently been introduced for DL medical image segmentation tasks. Developing scores to evaluate and compare the performance of uncertainty measures will assist the end-user in making more informed decisions. In this study, we explore and evaluate a score developed during the BraTS 2019 and BraTS 2020 task on uncertainty quantification (QU-BraTS) and designed to assess and rank uncertainty estimates for brain tumor multi-compartment segmentation. This score (1) rewards uncertainty estimates that produce high confidence in correct assertions and those that assign low confidence levels at incorrect assertions, and (2) penalizes uncertainty measures that lead to a higher percentage of under-confident correct assertions. We further benchmark the segmentation uncertainties generated by 14 independent participating teams of QU-BraTS 2020, all of which also participated in the main BraTS segmentation task. Overall, our findings confirm the importance and complementary value that uncertainty estimates provide to segmentation algorithms, highlighting the need for uncertainty quantification in medical image analyses. Finally, in favor of transparency and reproducibility, our evaluation code is made publicly available at: https://github.com/RagMeh11/QU-BraTS.

Anthony Frion, Lucas Drumetz, Guillaume Tochon et al.

In the context of an increasing popularity of data-driven models to represent dynamical systems, many machine learning-based implementations of the Koopman operator have recently been proposed. However, the vast majority of those works are limited to deterministic predictions, while the knowledge of uncertainty is critical in fields like meteorology and climatology. In this work, we investigate the training of ensembles of models to produce stochastic outputs. We show through experiments on real remote sensing image time series that ensembles of independently trained models are highly overconfident and that using a training criterion that explicitly encourages the members to produce predictions with high inter-model variances greatly improves the uncertainty quantification of the ensembles.

Anthony Frion, Lucas Drumetz, Mauro Dalla Mura et al.

Following the introduction of Dynamic Mode Decomposition and its numerous extensions, many neural autoencoder-based implementations of the Koopman operator have recently been proposed. This class of methods appears to be of interest for modeling dynamical systems, either through direct long-term prediction of the evolution of the state or as a powerful embedding for downstream methods. In particular, a recent line of work has developed invertible Koopman autoencoders (IKAEs), which provide an exact reconstruction of the input state thanks to their analytically invertible encoder, based on coupling layer normalizing flow models. We identify that the conservation of the dimension imposed by the normalizing flows is a limitation for the IKAE models, and thus we propose to augment the latent state with a second, non-invertible encoder network. This results in our new model: the Augmented Invertible Koopman AutoEncoder (AIKAE). We demonstrate the relevance of the AIKAE through a series of long-term time series forecasting experiments, on satellite image time series as well as on a benchmark involving predictions based on a large lookback window of observations.

Anthony Frion, Lucas Drumetz, Guillaume Tochon et al.

Over the last few years, massive amounts of satellite multispectral and hyperspectral images covering the Earth's surface have been made publicly available for scientific purpose, for example through the European Copernicus project. Simultaneously, the development of self-supervised learning (SSL) methods has sparked great interest in the remote sensing community, enabling to learn latent representations from unlabeled data to help treating downstream tasks for which there is few annotated examples, such as interpolation, forecasting or unmixing. Following this line, we train a deep learning model inspired from the Koopman operator theory to model long-term reflectance dynamics in an unsupervised way. We show that this trained model, being differentiable, can be used as a prior for data assimilation in a straightforward way. Our datasets, which are composed of Sentinel-2 multispectral image time series, are publicly released with several levels of treatment.

Alexandre Kirszenberg, Guillaume Tochon, Elodie Puybareau et al.

Integrating mathematical morphology operations within deep neural networks has been subject to increasing attention lately. However, replacing standard convolution layers with erosions or dilations is particularly challenging because the min and max operations are not differentiable. Relying on the asymptotic behavior of the counter-harmonic mean, p-convolutional layers were proposed as a possible workaround to this issue since they can perform pseudo-dilation or pseudo-erosion operations (depending on the value of their inner parameter p), and very promising results were reported. In this work, we present two new morphological layers based on the same principle as the p-convolutional layer while circumventing its principal drawbacks, and demonstrate their potential interest in further implementations within deep convolutional neural network architectures.

Simon Barthelmé, Pierre-Olivier Amblard, Nicolas Tremblay

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size "$k$-DPPs" the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work we show that as the size of the ground set grows, $k$-DPPs and DPPs become equivalent, meaning that their inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in $k$-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that $k$-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.