HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts

Large language models (LLMs) have shown great success in text modeling tasks across domains. However, natural language exhibits inherent semantic hierarchies and nuanced geometric structure, which current LLMs do not capture completely owing to their reliance on Euclidean operations. Recent studies have also shown that not respecting the geometry of token embeddings leads to training instabilities and degradation of generative capabilities. These findings suggest that shifting to non-Euclidean geometries can better align language models with the underlying geometry of text. We thus propose to operate fully in Hyperbolic space, known for its expansive, scale-free, and low-distortion properties. We thus introduce HELM, a family of HypErbolic Large Language Models, offering a geometric rethinking of the Transformer-based LLM that addresses the representational inflexibility, missing set of necessary operations, and poor scalability of existing hyperbolic LMs. We additionally introduce a Mixture-of-Curvature Experts model, HELM-MICE, where each expert operates in a distinct curvature space to encode more fine-grained geometric structure from text, as well as a dense model, HELM-D. For HELM-MICE, we further develop hyperbolic Multi-Head Latent Attention (HMLA) for efficient, reduced-KV-cache training and inference. For both models, we develop essential hyperbolic equivalents of rotary positional encodings and RMS normalization. We are the first to train fully hyperbolic LLMs at billion-parameter scale, and evaluate them on well-known benchmarks such as MMLU and ARC, spanning STEM problem-solving, general knowledge, and commonsense reasoning. Our results show consistent gains from our HELM architectures -- up to 4% -- over popular Euclidean architectures used in LLaMA and DeepSeek, highlighting the efficacy and enhanced reasoning afforded by hyperbolic geometry in large-scale LM pretraining.

InfoGain Wavelets: Furthering the Design of Diffusion Wavelets for Graph-Structured Data

Diffusion wavelets extract information from graph signals at different scales of resolution by utilizing graph diffusion operators raised to various powers, known as diffusion scales. Traditionally, the diffusion scales are chosen to be dyadic integers, $\mathbf{2^j}$. Here, we propose a novel, unsupervised method for selecting the diffusion scales based on ideas from information theory. We then show that our method can be incorporated into wavelet-based GNNs via graph classification experiments.

HiPoNet: A Topology-Preserving Multi-View Neural Network For High Dimensional Point Cloud and Single-Cell Data

In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Single-cell data can have high dimensionality exceeding the capabilities of existing methods point cloud tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e. one on every patient), necessitating models that can process large, high-dimensional point clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric information. In contrast, HiPoNet forms higher-order simplicial complexes through learnable feature reweighting, generating multiple data views that disentangle distinct biological processes. It then employs simplicial wavelet transforms to extract multi-scale features - capturing both local and global topology. We empirically show that these components preserve topological information in the learned representations, and that HiPoNet significantly outperforms state-of-the-art point-cloud and graph-based models on single cell. We also show an application of HiPoNet on spatial transcriptomics datasets using spatial co-ordinates as one of the views. Overall, HiPoNet offers a robust and scalable solution for high-dimensional data analysis.

Principal Curvatures Estimation with Applications to Single Cell Data

The rapidly growing field of single-cell transcriptomic sequencing (scRNAseq) presents challenges for data analysis due to its massive datasets. A common method in manifold learning consists in hypothesizing that datasets lie on a lower dimensional manifold. This allows to study the geometry of point clouds by extracting meaningful descriptors like curvature. In this work, we will present Adaptive Local PCA (AdaL-PCA), a data-driven method for accurately estimating various notions of intrinsic curvature on data manifolds, in particular principal curvatures for surfaces. The model relies on local PCA to estimate the tangent spaces. The evaluation of AdaL-PCA on sampled surfaces shows state-of-the-art results. Combined with a PHATE embedding, the model applied to single-cell RNA sequencing data allows us to identify key variations in the cellular differentiation.

Exploring the Manifold of Neural Networks Using Diffusion Geometry

Drawing motivation from the manifold hypothesis, which posits that most high-dimensional data lies on or near low-dimensional manifolds, we apply manifold learning to the space of neural networks. We learn manifolds where datapoints are neural networks by introducing a distance between the hidden layer representations of the neural networks. These distances are then fed to the non-linear dimensionality reduction algorithm PHATE to create a manifold of neural networks. We characterize this manifold using features of the representation, including class separation, hierarchical cluster structure, spectral entropy, and topological structure. Our analysis reveals that high-performing networks cluster together in the manifold, displaying consistent embedding patterns across all these features. Finally, we demonstrate the utility of this approach for guiding hyperparameter optimization and neural architecture search by sampling from the manifold.

ProtSCAPE: Mapping the landscape of protein conformations in molecular dynamics

Understanding the dynamic nature of protein structures is essential for comprehending their biological functions. While significant progress has been made in predicting static folded structures, modeling protein motions on microsecond to millisecond scales remains challenging. To address these challenges, we introduce a novel deep learning architecture, Protein Transformer with Scattering, Attention, and Positional Embedding (ProtSCAPE), which leverages the geometric scattering transform alongside transformer-based attention mechanisms to capture protein dynamics from molecular dynamics (MD) simulations. ProtSCAPE utilizes the multi-scale nature of the geometric scattering transform to extract features from protein structures conceptualized as graphs and integrates these features with dual attention structures that focus on residues and amino acid signals, generating latent representations of protein trajectories. Furthermore, ProtSCAPE incorporates a regression head to enforce temporally coherent latent representations.

Convergence of Manifold Filter-Combine Networks

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity.

Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds

Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold. GAGA shows competitive performance in simulated and real world datasets, including a 30% improvement over the state-of-the-art methods in single-cell population-level trajectory inference.

DiffKillR: Killing and Recreating Diffeomorphisms for Cell Annotation in Dense Microscopy Images

The proliferation of digital microscopy images, driven by advances in automated whole slide scanning, presents significant opportunities for biomedical research and clinical diagnostics. However, accurately annotating densely packed information in these images remains a major challenge. To address this, we introduce DiffKillR, a novel framework that reframes cell annotation as the combination of archetype matching and image registration tasks. DiffKillR employs two complementary neural networks: one that learns a diffeomorphism-invariant feature space for robust cell matching and another that computes the precise warping field between cells for annotation mapping. Using a small set of annotated archetypes, DiffKillR efficiently propagates annotations across large microscopy images, reducing the need for extensive manual labeling. More importantly, it is suitable for any type of pixel-level annotation. We will discuss the theoretical properties of DiffKillR and validate it on three microscopy tasks, demonstrating its advantages over existing supervised, semi-supervised, and unsupervised methods.

Latent Representation Learning for Multimodal Brain Activity Translation

Neuroscience employs diverse neuroimaging techniques, each offering distinct insights into brain activity, from electrophysiological recordings such as EEG, which have high temporal resolution, to hemodynamic modalities such as fMRI, which have increased spatial precision. However, integrating these heterogeneous data sources remains a challenge, which limits a comprehensive understanding of brain function. We present the Spatiotemporal Alignment of Multimodal Brain Activity (SAMBA) framework, which bridges the spatial and temporal resolution gaps across modalities by learning a unified latent space free of modality-specific biases. SAMBA introduces a novel attention-based wavelet decomposition for spectral filtering of electrophysiological recordings, graph attention networks to model functional connectivity between functional brain units, and recurrent layers to capture temporal autocorrelations in brain signal. We show that the training of SAMBA, aside from achieving translation, also learns a rich representation of brain information processing. We showcase this classify external stimuli driving brain activity from the representation learned in hidden layers of SAMBA, paving the way for broad downstream applications in neuroscience research and clinical contexts.