arXiv subjects

Eliot Tron

Publications and source records attributed to Eliot Tron.

5 recordsLinked to original sources

arXiv2026

k-Volume: a Unified Active Learning Strategy

In the classification context where revealing the label of a sample is costly, we propose a new Active Learning (AL) strategy named k-Volume (kVol). It randomly selects a sample to be labeled according to the volume of the smallest ball containing its k nearest neighbors. kVol is highly versatile: it can be applied either in the input space or in the output space, and it enjoys the desirable properties of the three main families of AL strategies: density, diversity, and (when performed in the output space) uncertainty. From a theoretical perspective, we establish that kVol is asymptotically equivalent to uniform sampling over the support of the (input or output) data distribution. Based on experiments on various synthetic and real-life tabular datasets and images, we observe that kVol is surprisingly stable: whatever the classifier or the experimental setup, it still lies among the best AL strategies unlike other competitors that fail dramatically. In particular, kVol outperforms state-of-the-art competitors such as CoreSet, TypiClust or Uncertainty on several real-life datasets, at a low computational cost.

cs.LG↗

arXiv2026

The Geometry of Manifold Learning: Singular foliations via Deep Learning models

Deep learning algorithms have become widely used and successful across a wide range of tasks. A key challenge is determining whether these algorithms truly capture the intrinsic geometry of their training data in order to better understand what they learn from data. In this theoretical work, we explore a new geometric structure defined on the data space by a trained neural network through the Data Information Matrix (DIM), a variation of the Fisher Information Matrix. This new geometric structure is a foliation and the main focus of this work is to explore the non-smooth case of ReLU activation functions, where traditional geometric tools often fail. The foliation is defined through a distribution spanned by the columns of the DIM. We first prove that this distribution is non-smooth in the sense of distributions, rendering the existence of an associated foliation much more complex. To circumvent this issue, we introduce a smoothed version of this distribution that differs from the original only on a set of measure zero. Finally, we prove that this smoothed distribution induces a singular foliation on the data space, which is based on the data the network has been trained with. Our theoretical contributions provide a geometric framework to better understand how neural networks organize their input space in non-smooth settings, bridging differential geometry and deep learning theory.

math.DG↗

arXiv2026

Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix

Understanding how real data is distributed in high dimensional spaces is the key to many tasks in machine learning. We want to provide a natural geometric structure on the space of data employing a ReLU neural network trained as a classifier. Through the Data Information Matrix (DIM), a variation of the Fisher information matrix, the model will discern a singular foliation structure on the space of data. We show that the singular points of such foliation are contained in a measure zero set, and that a local regular foliation exists almost everywhere. Experiments show that the data is correlated with leaves of such foliation. Moreover we show the potential of our approach for knowledge transfer by analyzing the spectrum of the DIM to measure distances between datasets.

cs.LG↗

arXiv2024

Cartan moving frames and the data manifolds

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and through experiments, explanations on the response of a neural network are given by pointing out the output classes that are easily reachable from a given input. This emphasizes how the proposed mathematical relationship between the output of the network and the geometry of its inputs can be exploited as an explainable artificial intelligence tool.

stat.ML↗

arXiv2024

Adversarial attacks on neural networks through canonical Riemannian foliations

Deep learning models are known to be vulnerable to adversarial attacks. Adversarial learning is therefore becoming a crucial task. We propose a new vision on neural network robustness using Riemannian geometry and foliation theory. The idea is illustrated by creating a new adversarial attack that takes into account the curvature of the data space. This new adversarial attack, called the two-step spectral attack is a piece-wise linear approximation of a geodesic in the data space. The data space is treated as a (degenerate) Riemannian manifold equipped with the pullback of the Fisher Information Metric (FIM) of the neural network. In most cases, this metric is only semi-definite and its kernel becomes a central object to study. A canonical foliation is derived from this kernel. The curvature of transverse leaves gives the appropriate correction to get a two-step approximation of the geodesic and hence a new efficient adversarial attack. The method is first illustrated on a 2D toy example in order to visualize the neural network foliation and the corresponding attacks. Next, we report numerical results on the MNIST and CIFAR10 datasets with the proposed technique and state of the art attacks presented in Zhao et al. (2019) (OSSA) and Croce et al. (2020) (AutoAttack). The result show that the proposed attack is more efficient at all levels of available budget for the attack (norm of the attack), confirming that the curvature of the transverse neural network FIM foliation plays an important role in the robustness of neural networks. The main objective and interest of this study is to provide a mathematical understanding of the geometrical issues at play in the data space when constructing efficient attacks on neural networks.

stat.ML↗