arXiv subjects

Adam Gosztolai

Publications and source records attributed to Adam Gosztolai.

11 recordsLinked to original sources

arXiv2026

Riemannian Metric Matching for Scalable Geometric Modeling of Distributions

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carr\'e du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carr\'e du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

cs.LG↗

arXiv2026

Diffusion Processes on Implicit Manifolds

High-dimensional data are often assumed to lie on lower-dimensional manifolds. We study how to construct diffusion processes on this data manifold using only point cloud samples and without access to charts, projections, or other geometric primitives. Here, we introduce Implicit Manifold-valued Diffusions (IMDs), a data-driven mathematical formalism for defining stochastic differential equations in the original high-dimensional space that describe drifting Brownian particles evolving intrinsically on the underlying manifold. Our construction hinges on approximating the corresponding infinitesimal generator of the diffusion process using a proximity graph over the data and using the carr\'e-du-champ of the generator, which encodes the local tangent spaces of the manifold and lifts the intrinsic process into ambient coordinates. We show that as the number of samples grows, our discrete diffusion process converges in law on the space of probability paths to its smooth manifold counterpart. We further present an Euler-Maruyama scheme for the numerical integration of IMDs. We validate our framework using numerical experiments on synthetic manifolds and the MNIST data manifold, showing that IMDs remain confined over the manifold and enable its guided exploration. Our work provides the mathematical foundation and practical implementations of diffusion processes on data manifolds, opening new avenues for manifold-aware sampling, exploration, and generative modeling.

cs.LG↗

arXiv2025

Carr\'e du champ flow matching: better quality-generalisation tradeoff in generative models

Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carr\'e du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.

cs.LG↗

arXiv2023

Implicit Gaussian process representation of vector fields over arbitrary latent manifolds

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

cs.LG↗

arXiv2023

Interpretable statistical representations of neural population dynamics and geometry

The dynamics of neuron populations commonly evolve on low-dimensional manifolds. Thus, we need methods that learn the dynamical processes over neural manifolds to infer interpretable and consistent latent representations. We introduce a representation learning method, MARBLE, that decomposes on-manifold dynamics into local flow fields and maps them into a common latent space using unsupervised geometric deep learning. In simulated non-linear dynamical systems, recurrent neural networks, and experimental single-neuron recordings from primates and rodents, we discover emergent low-dimensional latent representations that parametrise high-dimensional neural dynamics during gain modulation, decision-making, and changes in the internal state. These representations are consistent across neural networks and animals, enabling the robust comparison of cognitive computations. Extensive benchmarking demonstrates state-of-the-art within- and across-animal decoding accuracy of MARBLE compared with current representation learning approaches, with minimal user input. Our results suggest that manifold structure provides a powerful inductive bias to develop powerful decoding algorithms and assimilate data across experiments.

cs.LG↗

arXiv2023

PyGenStability: Multiscale community detection with generalized Markov Stability

We present PyGenStability, a general-use Python software package that provides a suite of analysis and visualisation tools for unsupervised multiscale community detection in graphs. PyGenStability finds optimized partitions of a graph at different levels of resolution by maximizing the generalized Markov Stability quality function with the Louvain or Leiden algorithms. The package includes automatic detection of robust graph partitions and allows the flexibility to choose quality functions for weighted undirected, directed and signed graphs, and to include other user-defined quality functions.

cs.SI↗

arXiv2021

Connecting the dots in ethology: applying network theory to understand neural and animal collectives

A major goal shared by neuroscience and collective behavior is to understand how dynamic interactions between individual elements give rise to behaviors in populations of neurons and animals, respectively. This goal has recently become within reach thanks to techniques providing access to the connectivity and activity of neuronal ensembles as well as to behaviors among animal collectives. The next challenge using these datasets is to unravel network mechanisms generating population behaviors. This is aided by network theory, a field that studies structure-function relationships in interconnected systems. Here we review studies that have taken a network view on modern datasets to provide unique insights into individual and collective animal behaviors. Specifically, we focus on how analyzing signal propagation, controllability, symmetry, and geometry of networks can tame the complexity of collective system dynamics. These studies illustrate the potential of network theory to accelerate our understanding of behavior across ethological scales.

q-bio.QM↗

arXiv2021

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

Describing networks geometrically through low-dimensional latent metric spaces has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, latent space embeddings are limited to specific classes of networks because incompatible metric spaces generally result in information loss. Here, we study arbitrary networks geometrically by defining a dynamic edge curvature measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps are robust to large fluctuations in node degrees, encoding communities until the phase transition of detectability, where spectral and node-clustering methods fail. Using this insight, we derive geometric modularity to find multiscale communities based on deviations from constant network curvature in generative and real-world networks, significantly outperforming most previous methods. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.

physics.soc-ph↗

arXiv2020

Cellular memory enhances bacterial chemotactic navigation in rugged environments

The response of microbes to external signals is mediated by biochemical networks with intrinsic time scales. These time scales give rise to a memory that impacts cellular behaviour. Here we study theoretically the role of cellular memory in Escherichia coli chemotaxis. Using an agent-based model, we show that cells with memory navigating rugged chemoattractant landscapes can enhance their drift speed by extracting information from environmental correlations. Maximal advantage is achieved when the memory is comparable to the time scale of fluctuations as perceived during swimming. We derive an analytical approximation for the drift velocity in rugged landscapes that explains the enhanced velocity, and recovers standard Keller-Segel gradient-sensing results in the limits when memory and fluctuation time scales are well separated. Our numerics also show that cellular memory can induce bet-hedging at the population level resulting in long-lived multi-modal distributions in heterogeneous landscapes.

physics.bio-ph↗

arXiv2018

Collective search with finite perception: transient dynamics and search efficiency

Motile organisms often use finite spatial perception of their surroundings to navigate and search their habitats. Yet standard models of search are usually based on purely local sensory information. To model how a finite perceptual horizon affects ecological search, we propose a framework for optimal navigation that combines concepts from random walks and optimal control theory. We show that, while local strategies are optimal on asymptotically long and short search times, finite perception yields faster convergence and increased search efficiency over transient time scales relevant in biological systems. The benefit of the finite horizon can be maintained by the searchers tuning their response sensitivity to the length scale of the stimulant in the environment, and is enhanced when the agents interact as a result of increased consensus within subpopulations. Our framework sheds light on the role of spatial perception and transients in search movement and collective sensing of the environment.

physics.bio-ph↗

arXiv2017

GlnK facilitates the dynamic regulation of bacterial nitrogen assimilation

Ammonium assimilation in E. coli is regulated by two paralogous proteins (GlnB and GlnK), which orchestrate interactions with regulators of gene expression, transport proteins and metabolic pathways. Yet how they conjointly modulate the activity of glutamine synthetase (GS), the key enzyme for nitrogen assimilation, is poorly understood. We combine experiments and theory to study the dynamic roles of GlnB and GlnK during nitrogen starvation and upshift. We measure time-resolved in vivo concentrations of metabolites, total and post-translationally modified proteins, and develop a concise biochemical model of GlnB and GlnK that incorporates competition for active and allosteric sites, as well as functional sequestration of GlnK. The model predicts the responses of GS, GlnB and GlnK under time-varying external ammonium level in the wild type and two genetic knock-outs. Our results show that GlnK is tightly regulated under nitrogen-rich conditions, yet it is expressed during ammonium run-out and starvation. This suggests a role for GlnK as a buffer of nitrogen shock after starvation, and provides a further functional link between nitrogen and carbon metabolisms.

q-bio.SC↗