Searcharxiv⌕ Search

arXiv subjects

Johanna Marie Gegenfurtner

Publications and source records attributed to Johanna Marie Gegenfurtner.

7 recordsLinked to original sources
arXiv2026

The Symmetries of Three-Layer ReLU Networks

We develop a framework for analyzing parameter symmetries in deep ReLU networks and obtain a complete characterization of the generic parameter fibers for three-layer bottleneck architectures. Our approach provides explicit semi-algebraic descriptions of these fibers and yields a polynomial time algorithm for deciding functional equivalence of two parameters. The symmetries include discrete and continuous transformations arising from layer composition, and depend on whether deeper layers hide or preserve geometric structure from preceding layers. Finally, we show that some of these symmetries induce local conservation laws along gradient flow, while others do not.

cs.LG↗
arXiv2026

Don't Stop Me Yet: Sampling Loss Minima via Dissipative Riemannian Mechanics

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.

cs.LG↗
arXiv2026

Reducing Memorisation in Generative Models via Riemannian Bayesian Inference

Modern generative models can produce realistic samples, however, balancing memorisation and generalisation remains an open problem. We approach this challenge from a Bayesian perspective by focusing on the parameter space of flow matching and diffusion models and constructing a predictive posterior that better captures the variability of the data distribution. In particular, we capture the geometry of the loss using a Riemannian metric and leverage a flexible approximate posterior that adapts to the local structure of the loss landscape. This approach allows us to sample generative models that resemble the original model, but exhibit reduced memorisation. Empirically, we demonstrate that the proposed approach reduces memorisation while preserving generalisation. Further, we provide a theoretical analysis of our method, which explains our findings. Overall, our work illustrates how considering the geometry of the loss enables effective use of the parameter space, even for complex high-dimensional generative models.

cs.LG↗
arXiv2025

Staying on the Manifold: Geometry-Aware Noise Injection

It has been shown that perturbing the input during training implicitly regularises the gradient of the learnt function, leading to smoother models and enhancing generalisation. However, previous research mostly considered the addition of ambient noise in the input space, without considering the underlying structure of the data. In this work, we propose several strategies of adding geometry-aware input noise that accounts for the lower dimensional manifold the input space inhabits. We start by projecting ambient Gaussian noise onto the tangent space of the manifold. In a second step, the noise sample is mapped on the manifold via the associated geodesic curve. We also consider Brownian motion noise, which moves in random steps along the manifold. We show that geometry-aware noise leads to improved generalisation and robustness to hyperparameter selection on highly curved manifolds, while performing at least as well as training without noise on simpler manifolds. Our proposed framework extends to data manifolds approximated by generative models and we observe similar trends on the MNIST digits dataset.

cs.LG↗
arXiv2024

Minimal Submanifolds of the Classical Compact Riemannian Symmetric Spaces

Minimal submanifolds constitute a central area within the realm of differential geometry, due to their many applications in various branches of physics. In this thesis we will employ a recent result of S. Gudmundsson and T.J. Munn to construct minimal submanifolds of the classical compact Riemannian symmetric spaces using eigenfunctions.

math.DG↗
arXiv2022

CMC Tori in the Generalised Berger Spheres and their Duals

The study of minimal surfaces has a long history, due to the important applications. Given a fixed boundary, one wants to minimise the surface area: this can be used, for example, to minimise the area of the roof of a building. Similarly, looking for constant mean curvature (CMC) provides us with many interesting applications in physics: one of the easiest examples are soap bubbles. In this work however we occupy ourselves with minimal and constant mean curvature surfaces in the three-dimensional sphere $S^3$ and its dual space $Σ^3$. In Chapter 1 we give a brief overview of the tools of Riemannian and Lorentzian geometry that we will use. We then take a closer look at $S^3$, computing its Levi-Civita connection and sectional curvatures: in Chapter 2 with respect to the Riemannian metric g and in Chapter 4 with respect to the Lorentzian metric h. Further, we determine some minimal and CMC tori inside ($S^3$, g) in Chapter 3 and in ($S^3$, h) in Chapter 5. We then proceed with the dual space $Σ^3$ of $S^3$. In Chapter 6, we calculate the Levi-Civita connection and sectional curvatures with respect to g, and with respect to h in Chapter 8. Again we look for minimal and CMC tori of a certain family in ($Σ^3$, g) in Chapter 7 and in ($Σ^3$, h) in Chapter 9. In the appendix, the reader will find a Maple program. It was written to check the computations of the $S^3$ cases, but it can easily be adapted to$Σ^3$.

math.DG↗