riemannian-geometry

This paper develops intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking, showing that curvature—via Ricci curvature thresholds and geodesic capacity scalings—determines what such a memory retains, creates, and forgets, with demonstrations on WordNet noun hierarchy.

A new paper proposes TMLN (Trajectory-Modulatory Landscape Navigation), which treats continual learning as an optimal control problem over a curved loss landscape, using a diagonal empirical Fisher Information Matrix and trajectory-based gradient preconditioning to mitigate catastrophic forgetting without additive regularization penalties.

Heat Kernel Textures (HKTex) is a UV-free texture representation using anisotropic heat kernels on manifold surfaces, winning the Best Paper Award at ECCV 2026. It overcomes UV mapping issues and is optimized from multi-view images or existing textures.

This paper unifies Euclidean residual connections and geodesic normalization on the hypersphere, introducing a one-parameter family of angular retractions called p-SpheretNorm. The proposed norm-preserving algebraic methods outperform existing deep connection schemes on nanoGPT, showing the exponential map is just one end of a spectrum.

Proposes LieBN, a framework for batch normalization over Lie groups, applicable to SPD, rotation, and correlation manifolds, with theoretical guarantees and extensive experiments.

This paper reveals that PPO-Clipping's use of Euclidean metric causes exploration collapse in LLM RL, and proposes Riemannian Isometric Policy Optimization (RIPO) to ensure geometrically consistent policy updates, achieving up to 60% improvement over GRPO on AIME24.

The paper introduces Riemannian Mean Pooling (RMP), a method that aggregates token embeddings from pre-trained language models using Riemannian geometry via pullback metrics, showing improved performance over Euclidean pooling on sentence classification tasks.

I²RiMA is a novel intra-inter Riemannian manifold attention network for EEG-based mental stress detection. It constructs frequency-specific spatial covariance and uses temporal attention to improve cross-subject stress classification, achieving up to 82.78% balanced accuracy.

SPDM introduces a geometry-aware state-space model that uses manifold constraints on the symmetric positive definite manifold for time series forecasting, achieving state-of-the-art performance on 11 benchmarks.

This paper introduces a Riemannian version of archetypal analysis using data-driven pullback geometry to combine interpretability with non-linear expressiveness, proposing the Riemannian Archetypal Mapping (RAM) and demonstrating its effectiveness on synthetic data and MNIST.