Cartan moving frames and the data manifolds

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and substantiate the geometric and topological view of the learning process of neural networks. Our attention is focused on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of the data manif...

Data sets tend to live in low-dimensional non-linear subspaces. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The symmetric Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of non-linear geometries that has been shown to be able to capture the data geometry through empirical evidence from classical non-linear embedding. Seco...

Artificial neural networks (ANNs) are powerful tools capable of approximating any arbitrary mathematical function, but their interpretability remains limited, rendering them as black box models. To address this issue, numerous methods have been proposed to enhance the explainability and interpretability of ANNs. In this study, we introduce the application of information geometric framework to investigate phase transition-like behavior during the training of ANNs and relate th...

Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric constraints can be expressed in the language of Riemannian geometry, where conventional vector space machine learning does not apply directly. The central question this paper deals with is: How does one train deep neural nets whose final outp...

We discover that deep ReLU neural network classifiers can see a low-dimensional Riemannian manifold structure on data. Such structure comes via the local data matrix, a variation of the Fisher information matrix, where the role of the model parameters is taken by the data variables. We obtain a foliation of the data domain and we show that the dataset on which the model is trained lies on a leaf, the data leaf, whose dimension is bounded by the number of classification labels...

This paper introduces a novel approach to statistics and data analysis, departing from the conventional assumption of data residing in Euclidean space to consider a Riemannian Manifold. The challenge lies in the absence of vector space operations on such manifolds. Pennec X. et al. in their book Riemannian Geometric Statistics in Medical Image Analysis proposed analyzing data on Riemannian manifolds through geometry, this approach is effective with structured data like medica...

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 deep 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...

While the last five years have seen considerable progress in understanding the internal representations of deep learning models, many questions remain. This is especially true when trying to understand the impact of model design choices, such as model architecture or training algorithm, on hidden representation geometry and dynamics. In this work we present a new approach to studying such representations inspired by the idea of a frame on the tangent bundle of a manifold. Our...

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), ...

We introduce Riemannian Integrated Gradients (RIG); an extension of Integrated Gradients (IG) to Riemannian manifolds. We demonstrate that RIG restricts to IG when the Riemannian manifold is Euclidean space. We show that feature attribution can be phrased as an eigenvalue problem where attributions correspond to eigenvalues of a symmetric endomorphism.