arXiv subjects

Jonas Cassel

Publications and source records attributed to Jonas Cassel.

6 recordsLinked to original sources

arXiv2026

Generative Modeling of Discrete Data Using Geometric Latent Subspaces

We propose a geometric latent-subspace framework for generative modeling of discrete data. Specifically, we introduce latent subspaces in the exponential parameter space of product manifolds of categorical distributions as a novel approach to learning low-dimensional representations of high-dimensional discrete data. The resulting low-dimensional latent space captures statistical dependencies and removes redundant degrees of freedom among the categorical variables. We equip the parameter domain with a Riemannian geometry such that the latent subspace and induced data manifold are related isometrically, enabling consistent flow matching. Exploiting this structure, we propose a geometry-aware dimensionality reduction objective, called geometric PCA (GPCA), which we formulate as a regularized cross-entropy minimization that encourages small Riemannian distances between the data and their reconstructions. In particular, under the induced geometry, geodesics correspond to straight lines in the latent parameter space, allowing flow matching to be performed directly in reduced coordinates. Empirical results show that low-dimensional latent representations suffice to accurately model high-dimensional discrete data and enable substantially more computationally efficient flow matching.

stat.ML↗

arXiv2025

Yang-Mills Meets Data

Gauge symmetric methods for data representation and analysis utilize tools from the differential geometry of vector bundles in order to achieve consistent data processing architectures with respect to local symmetry and equivariance. In this work, we elaborate concepts of geometric gauge theory for data science. Motivated by lattice gauge theory, we focus on discrete descriptions of vector bundles for data representation and analysis, with clear relations to the established mathematical bundle formalism. Our approach unifies various existing approaches to data processing via vector bundles, within the framework of gauge theory. We provide geometric insights into gauge symmetric heat kernel operators that are closely related to graph connection Laplacians, and into their data transformation properties in terms of the non-trivial nullspace of the corresponding gauged Laplacians. In particular, we utilize a discrete Yang-Mills energy in order to characterize significant properties of the heat kernel in terms of associated synchronization problems.

math.DG↗

arXiv2025

Sigma Flows for Image and Data Labeling and Learning Structured Prediction

This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.

math.DS↗

arXiv2025

Riemannian Patch Assignment Gradient Flows

This paper introduces patch assignment flows for metric data labeling on graphs. Labelings are determined by regularizing initial local labelings through the dynamic interaction of both labels and label assignments across the graph, entirely encoded by a dictionary of competing labeled patches and mediated by patch assignment variables. Maximal consistency of patch assignments is achieved by geometric numerical integration of a Riemannian ascent flow, as critical point of a Lagrangian action functional. Experiments illustrate properties of the approach, including uncertainty quantification of label assignments.

cs.CV↗

arXiv2024

A Geometric Embedding Approach to Multiple Games and Multiple Populations

This paper studies a meta-simplex concept and geometric embedding framework for multi-population replicator dynamics. Central results are two embedding theorems which constitute a formal reduction of multi-population replicator dynamics to single-population ones. In conjunction with a robust mathematical formalism, this provides a toolset for analyzing complex multi-population models. Our framework provides a unifying perspective on different population dynamics in the literature which in particular enables to establish a formal link between multi-population and multi-game dynamics.

math.DS↗

arXiv2023

Quantum State Assignment Flows

This paper introduces assignment flows for density matrices as state spaces for representing and analyzing data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions which can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorial probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment flows can also be characterized as Riemannian gradient flows with respect to a non-local non-convex potential, after proper reparametrization and under mild conditions on the underlying weight function. This weight function generates the parameters of the layers of a neural network, corresponding to and generated by each step of the geometric integration scheme. Numerical results indicates and illustrate the potential of the novel approach for data representation and analysis, including the representation of correlations of data across the graph by entanglement and tensorization.

math.DS↗