Hubert Leterme

Selected Publications

1. Journal article

   Distribution-Free Uncertainty Quantification for Inverse Problems: Application to Weak Lensing Mass Mapping

   Leterme, Hubert, Fadili, Jalal,  and Starck, Jean-Luc

   Astronomy & Astrophysics (in press) Jan 2025

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   In inverse problems, distribution-free uncertainty quantification (UQ) aims to obtain error bars with coverage guarantees that are independent of any prior assumptions about the data distribution. In the context of mass mapping, uncertainties could lead to errors that affects our understanding of the underlying mass distribution, or could propagate to cosmological parameter estimation, thereby impacting the precision and reliability of cosmological models. Current surveys, such as Euclid or Rubin, will provide new weak lensing datasets of very high quality. Accurately quantifying uncertainties in mass maps is therefore critical to perform reliable cosmological parameter inference. In this paper, we extend the conformalized quantile regression (CQR) algorithm, initially proposed for scalar regression, to inverse problems. We compare our approach with another distribution-free approach based on risk-controlling prediction sets (RCPS). Both methods are based on a calibration dataset, and offer finite-sample coverage guarantees that are independent of the data distribution. Furthermore, they are applicable to any mass mapping method, including blackbox predictors. In our experiments, we apply UQ on three mass-mapping method: the Kaiser-Squires inversion, iterative Wiener filtering, and the MCALens algorithm. Our experiments reveal that RCPS tends to produce overconservative confidence bounds with small calibration sets, whereas CQR is designed to avoid this issue. Although the expected miscoverage rate is guaranteed to stay below a user-prescribed threshold regardless of the mass mapping method, selecting an appropriate reconstruction algorithm remains crucial for obtaining accurate estimates, especially around peak-like structures, which are particularly important for inferring cosmological parameters. Additionally, the choice of mass mapping method influences the size of the error bars.

2. Conference paper

   From CNNs to Shift-Invariant Twin Models Based on Complex Wavelets

   Leterme, Hubert, Polisano, Kévin, Perrier, Valérie,  and Alahari, Karteek

   In 2024 32nd European Signal Processing Conference (EUSIPCO) Aug 2024

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   We propose a novel antialiasing method to increase shift invariance in convolutional neural networks (CNNs). More precisely, we replace the conventional combination "real-valued convolutions + max pooling" ({}mathbb R\Max) by "complex-valued convolutions + modulus" ({}mathbb C\Mod), which produce stable feature representations for band-pass filters with well-defined orientations. In a recent work, we proved that, for such filters, the two operators yield similar outputs. Therefore, {}mathbb C\Mod can be viewed as a stable alternative to {}mathbb R\Max. To separate band-pass filters from other freely-trained kernels, in this paper, we designed a "twin" architecture based on the dual-tree complex wavelet packet transform, which generates similar outputs as standard CNNs with fewer trainable parameters. In addition to improving stability to small shifts, our experiments on AlexNet and ResNet showed increased prediction accuracy on natural image datasets such as ImageNet and CIFAR10. Furthermore, our approach outperformed recent antialiasing methods based on low-pass filtering by preserving high-frequency information, while reducing memory usage.

3. Preprint

   On the Shift Invariance of Max Pooling Feature Maps in Convolutional Neural Networks

   Leterme, Hubert, Polisano, Kévin, Perrier, Valérie,  and Alahari, Karteek

   arXiv:2209.11740 Sep 2022

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   In this paper, we aim to improve the mathematical interpretability of convolutional neural networks for image classification. When trained on natural image datasets, such networks tend to learn parameters in the first layer that closely resemble oriented Gabor filters. By leveraging the properties of discrete Gabor-like convolutions, we prove that, under specific conditions, feature maps computed by the subsequent max pooling operator tend to approximate the modulus of complex Gabor-like coefficients, and as such, are stable with respect to certain input shifts. We then compute a probabilistic measure of shift invariance for these layers. More precisely, we show that some filters, depending on their frequency and orientation, are more likely than others to produce stable image representations. We experimentally validate our theory by considering a deterministic feature extractor based on the dual-tree wavelet packet transform, a particular case of discrete Gabor-like decomposition. We demonstrate a strong correlation between shift invariance on the one hand and similarity with complex modulus on the other hand.