Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Our approach involves deriving the local latent basis within $\mathcal{X}$ by leveraging the pullback metric associated with their encoding feature maps. Remarkably, our discovered local latent basis enables image editing capabilities by moving $\mathbf{x}_t$, the latent space of DMs, along the basis vector at specific timesteps. We further analyze how the geometric structure of DMs evolves over diffusion timesteps and differs across different text conditions. This confirms the known phenomenon of coarse-to-fine generation, as well as reveals novel insights such as the discrepancy between $\mathbf{x}_t$ across timesteps, the effect of dataset complexity, and the time-varying influence of text prompts.

Robust decoding and classification of brain patterns measured with electroencephalography (EEG) remains a major challenge for real-world (i.e. outside scientific lab and medical facilities) brain-computer interface (BCI) applications due to well documented inter- and intra-participant variability. Here, we present a large-scale benchmark evaluating over 340,000+ unique combinations of spatial and nonlinear EEG classification. Our methodological pipeline consists in combinations of Common Spatial Patterns (CSP), Riemannian geometry, functional connectivity, and fractal- or entropy-based features across three open-access EEG datasets. Unlike prior studies, our analysis operates at the per-participant level and across multiple frequency bands (8-15 Hz and 8-30 Hz), enabling direct assessment of both group-level performance and individual variability. Covariance tangent space projection (cov-tgsp) and CSP consistently achieved the highest average classification accuracies. However, their effectiveness was strongly dataset-dependent, and marked participant-level differences persisted, particularly in the most heterogeneous of the datasets. Importantly, nonlinear methods outperformed spatial approaches for specific individuals, underscoring the need for personalized pipeline selection. Our findings highlight that no universal 'one-size-fits-all' method can optimally decode EEG motor imagery patterns across all users or datasets. Future work will require adaptive, multimodal, and possibly novel approaches to fully address neurophysiological variability in practical BCI applications where the system can automatically adapt to what makes each user unique.

Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here, we show that signatures of such computations emerge in the representational geometry of neural networks as they learn. By analysing the Riemannian pullback metric across layers of a neural network, we find that network computation can be decomposed into two functions: discretising continuous input features and performing logical operations on these discretised variables. Furthermore, we demonstrate how different learning regimes (rich vs. lazy) have contrasting metric and curvature structures, affecting the ability of the networks to generalise to unseen inputs. Overall, our work provides a geometric framework for understanding how neural networks learn to perform discrete computations on continuous manifolds.

This study deals with neural networks in the sense of geometric transformations acting on the coordinate representation of the underlying data manifold which the data is sampled from. It forms part of an attempt to construct a formalized general theory of neural networks in the setting of Riemannian geometry. From this perspective, the following theoretical results are developed and proven for feedforward networks. First it is shown that residual neural networks are finite difference approximations to dynamical systems of first order differential equations, as opposed to ordinary networks that are static. This implies that the network is learning systems of differential equations governing the coordinate transformations that represent the data. Second it is shown that a closed form solution of the metric tensor on the underlying data manifold can be found by backpropagating the coordinate representations learned by the neural network itself. This is formulated in a formal abstract sense as a sequence of Lie group actions on the metric fibre space in the principal and associated bundles on the data manifold. Toy experiments were run to confirm parts of the proposed theory, as well as to provide intuitions as to how neural networks operate on data.

Neural Information Processing Systems http://nips.cc/

HP A consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space.

High-dimensional data that exhibit an intrinsic low-dimensional structure are ubiquitous in machine learning and data science. While various approaches allow for learning the corresponding data manifold from finite samples, performing downstream tasks such as optimization directly on these learned manifolds presents a significant challenge. This work introduces a principled framework for optimization on learned data manifolds using iso-Riemannian geometry. Our approach addresses key limitations of classical Riemannian optimization in this setting, specifically, that the Levi-Civita connection fails to yield constant-speed geodesics, and that geodesic convexity assumptions break down under the learned pullback constructions commonly used in practice. To overcome these challenges, we propose new notions of monotonicity and Lipschitz continuity tailored to the iso-Riemannian setting and propose iso-Riemannian descent algorithms for which we provide a detailed convergence analysis. We demonstrate the practical effectiveness of those algorithms on both synthetic and real datasets, including MNIST under a learned pullback structure. Our approach yields interpretable barycentres, improved clustering, and provably efficient solutions to inverse problems, even in high-dimensional settings. These results establish that optimization under iso-Riemannian geometry can overcome distortions inherent to learned manifold mappings.

Message Passing Neural Networks (MPNNs) is the building block of graph foundation models, but fundamentally suffer from oversmoothing and oversquashing. There has recently been a surge of interest in fixing both issues. Existing efforts primarily adopt global approaches, which may be beneficial in some regions but detrimental in others, ultimately leading to the suboptimal expressiveness. In this paper, we begin by revisiting oversquashing through a global measure -- spectral gap $λ$ -- and prove that the increase of $λ$ leads to gradient vanishing with respect to the input features, thereby undermining the effectiveness of message passing. Motivated by such theoretical insights, we propose a \textbf{local} approach that adaptively adjusts message passing based on local structures. To achieve this, we connect local Riemannian geometry with MPNNs, and establish a novel nonhomogeneous boundary condition to address both oversquashing and oversmoothing. Building on the Robin condition, we design a GBN network with local bottleneck adjustment, coupled with theoretical guarantees. Extensive experiments on homophilic and heterophilic graphs show the expressiveness of GBN. Furthermore, GBN does not exhibit performance degradation even when the network depth exceeds $256$ layers.