arXiv subjects

Hans Riess

Publications and source records attributed to Hans Riess.

At least 19 recordsLinked to original sources

arXiv2026

Multi-Agent System Identification with Nonlinear Sheaf Diffusion

Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels -- the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law.

eess.SY↗

arXiv2026

Quantale-Enriched Co-Design: Toward a Framework for Quantitative Heterogeneous System Design

Monotone co-design enables compositional engineering design by modeling components through feasibility relations between required resources and provided functionalities. However, its standard boolean formulation cannot natively represent quantitative criteria such as cost, confidence, or implementation choice. In practice, these quantities are often introduced through ad hoc scalarization or by augmenting the resource space, which obscures system structure and increases computational burden. We address this limitation by developing a quantale-enriched theory of co-design. We model resources and functionalities as quantale-enriched categories and design problems as quantale-enriched profunctors, thereby lifting co-design from boolean feasibility to general quantitative evaluation. We show that the fundamental operations of series, parallel, and feedback composition remain valid over arbitrary commutative quantales. We further introduce heterogeneous composition through change-of-base maps between quantales, enabling different subsystems to be evaluated in different local semantics and then composed in a common framework. The resulting theory unifies feasibility-, cost-, confidence-, and implementation-aware co-design within one compositional formalism. Numerical examples on a target-tracking system and a UAV delivery problem demonstrate the framework and highlight how native quantitative enrichment can avoid the architectural and computational drawbacks of boolean-only formulations.

eess.SY↗

arXiv2025

Asynchronous Nonlinear Sheaf Diffusion for Multi-Agent Coordination

Cellular sheaves and sheaf Laplacians provide a far-reaching generalization of graphs and graph Laplacians, resulting in a wide array of applications ranging from machine learning to multi-agent control. In the context of multi-agent systems, so called coordination sheaves provide a unifying formalism that models heterogeneous agents and coordination goals over undirected communication topologies, and applying sheaf diffusion drives agents to achieve their coordination goals. Existing literature on sheaf diffusion assumes that agents can communicate and compute updates synchronously, which is an unrealistic assumption in many scenarios where communication delays or heterogeneous agents with different compute capabilities cause disagreement among agents. To address these challenges, we introduce asynchronous nonlinear sheaf diffusion. Specifically, we show that under mild assumptions on the coordination sheaf and bounded delays in communication and computation, nonlinear sheaf diffusion converges to a minimizer of the Dirichlet energy of the coordination sheaf at a linear rate proportional to the delay bound. We further show that this linear convergence is attained from arbitrary initial conditions and the analysis depends on the spectrum of the sheaf Laplacian in a manner that generalizes the standard graph Laplacian case. We provide several numerical simulations to validate our theoretical results.

math.OC↗

arXiv2025

Distributed Multi-agent Coordination over Cellular Sheaves

Techniques for coordination of multi-agent systems are vast and varied, often utilizing purpose-built solvers or controllers with tight coupling to the types of systems involved or the coordination goal. In this paper, we introduce a general unified framework for heterogeneous multi-agent coordination using the language of cellular sheaves and nonlinear sheaf Laplacians, which are generalizations of graphs and graph Laplacians. Specifically, we introduce the concept of a nonlinear homological program encompassing a choice of cellular sheaf on an undirected graph, nonlinear edge potential functions, and constrained convex node objectives, which constitutes a standard form for a wide class of coordination problems. We use the alternating direction method of multipliers to derive a distributed optimization algorithm for solving these nonlinear homological programs. To demonstrate the applicability of this framework, we show how heterogeneous coordination goals including combinations of consensus, formation, and flocking can be formulated as nonlinear homological programs and provide numerical simulations showing the efficacy of our distributed solution algorithm.

math.OC↗

arXiv2025

Clearing Sections of Lattice Liability Networks

Modern financial networks involve complex obligations that transcend simple monetary debts: multiple currencies, prioritized claims, supply chain dependencies, and more. We present a mathematical framework that unifies and extends these scenarios by recasting the classical Eisenberg-Noe model of financial clearing in terms of lattice liability networks. Each node in the network carries a complete lattice of possible states, while edges encode nominal liabilities. Our framework generalizes the scalar-valued clearing vectors of the classical model to lattice-valued clearing sections, preserving the elegant fixed-point structure while dramatically expanding its descriptive power. Our main theorem establishes that such networks possess clearing sections that themselves form a complete lattice under the product order. This structure theorem enables tractable analysis of equilibria in diverse domains, including multi-currency financial systems, decentralized finance with automated market makers, supply chains with resource transformation, and permission networks with complex authorization structures. We further extend our framework to chain-complete lattices for term structure models and multivalued mappings for complex negotiation systems. Our results demonstrate how lattice theory provides a natural language for understanding complex network dynamics across multiple domains, creating a unified mathematical foundation for analyzing systemic risk, resource allocation, and network stability.

q-fin.MF↗

arXiv2025

Categorical Diffusion of Weighted Lattices

We introduce a categorical formalization of diffusion processes for network-structured data, motivated by applications in data science and information dynamics. At the heart of our construction is the Lawvere Laplacian, an endofunctor on a product category indexed by a graph and enriched in a quantale. This framework enables the systematic study of diffusion processes on network sheaves taking values in categories enriched in quantales, analogous to classical diffusion operators on metric spaces and vector spaces. Our main theoretical contribution extends Tarski's fixed point theorem to the quantale-enriched categorical setting, establishing that both prefix and suffix points of the Lawvere Laplacian form complete quantale-enriched categories. We develop a discrete-time distributed algorithm - harmonic flow - that provably converges to these fixed points, providing a constructive method for computing fuzzy global sections. This computational approach bridges sheaf-theoretic and dynamical perspectives on network diffusion, with applications ranging from discrete event systems to preference dynamics and path-finding problems.

math.CT↗

arXiv2024

Distributionally Robust Clustered Federated Learning: A Case Study in Healthcare

In this paper, we address the challenge of heterogeneous data distributions in cross-silo federated learning by introducing a novel algorithm, which we term Cross-silo Robust Clustered Federated Learning (CS-RCFL). Our approach leverages the Wasserstein distance to construct ambiguity sets around each client's empirical distribution that capture possible distribution shifts in the local data, enabling evaluation of worst-case model performance. We then propose a model-agnostic integer fractional program to determine the optimal distributionally robust clustering of clients into coalitions so that possible biases in the local models caused by statistically heterogeneous client datasets are avoided, and analyze our method for linear and logistic regression models. Finally, we discuss a federated learning protocol that ensures the privacy of client distributions, a critical consideration, for instance, when clients are healthcare institutions. We evaluate our algorithm on synthetic and real-world healthcare data.

cs.LG↗

arXiv2024

Network Preference Dynamics using Lattice Theory

Preferences, fundamental in all forms of strategic behavior and collective decision-making, in their raw form, are an abstract ordering on a set of alternatives. Agents, we assume, revise their preferences as they gain more information about other agents. Exploiting the ordered algebraic structure of preferences, we introduce a message-passing algorithm for heterogeneous agents distributed over a network to update their preferences based on aggregations of the preferences of their neighbors in a graph. We demonstrate the existence of equilibrium points of the resulting global dynamical system of local preference updates and provide a sufficient condition for trajectories to converge to equilibria: stable preferences. Finally, we present numerical simulations demonstrating our preliminary results.

cs.MA↗

arXiv2024

Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back

In this work we introduce a convolution operation over the tangent bundle of Riemann manifolds in terms of exponentials of the Connection Laplacian operator. We define tangent bundle filters and tangent bundle neural networks (TNNs) based on this convolution operation, which are novel continuous architectures operating on tangent bundle signals, i.e. vector fields over the manifolds. Tangent bundle filters admit a spectral representation that generalizes the ones of scalar manifold filters, graph filters and standard convolutional filters in continuous time. We then introduce a discretization procedure, both in the space and time domains, to make TNNs implementable, showing that their discrete counterpart is a novel principled variant of the very recently introduced sheaf neural networks. We formally prove that this discretized architecture converges to the underlying continuous TNN. Finally, we numerically evaluate the effectiveness of the proposed architecture on various learning tasks, both on synthetic and real data.

eess.SP↗

arXiv2024

Path Signatures and Graph Neural Networks for Slow Earthquake Analysis: Better Together?

The path signature, having enjoyed recent success in the machine learning community, is a theoretically-driven method for engineering features from irregular paths. On the other hand, graph neural networks (GNN), neural architectures for processing data on graphs, excel on tasks with irregular domains, such as sensor networks. In this paper, we introduce a novel approach, Path Signature Graph Convolutional Neural Networks (PS-GCNN), integrating path signatures into graph convolutional neural networks (GCNN), and leveraging the strengths of both path signatures, for feature extraction, and GCNNs, for handling spatial interactions. We apply our method to analyze slow earthquake sequences, also called slow slip events (SSE), utilizing data from GPS timeseries, with a case study on a GPS sensor network on the east coast of New Zealand's north island. We also establish benchmarks for our method on simulated stochastic differential equations, which model similar reaction-diffusion phenomenon. Our methodology shows promise for future advancement in earthquake prediction and sensor network analysis.

cs.LG↗

arXiv2023

Max-Plus Synchronization in Decentralized Trading Systems

We introduce a decentralized mechanism for pricing and exchanging alternatives constrained by transaction costs. We characterize the time-invariant solutions of a heat equation involving a (weighted) Tarski Laplacian operator, defined for max-plus matrix-weighted graphs, as approximate equilibria of the trading system. We study algebraic properties of the solution sets as well as convergence behavior of the dynamical system. We apply these tools to the "economic problem" of allocating scarce resources among competing uses. Our theory suggests differences in competitive equilibrium, bargaining, or cost-benefit analysis, depending on the context, are largely due to differences in the way that transaction costs are incorporated into the decision-making process. We present numerical simulations of the synchronization algorithm (RRAggU), demonstrating our theoretical findings.

cs.GT↗

arXiv2023

Lattice Theory in Multi-Agent Systems

In this thesis, we argue that (order-) lattice-based multi-agent information systems constitute a broad class of networked multi-agent systems in which relational data is passed between nodes. Mathematically modeled as lattice-valued sheaves, we initiate a discrete Hodge theory with a Laplace operator, analogous to the graph Laplacian and the graph connection Laplacian, acting on assignments of data to the nodes of a Tarski sheaf. The Hodge-Tarski theorem (the main theorem) relates the fixed point theory of this operator, called the Tarski Laplacian in deference to the Tarski Fixed Point Theorem, to the global sections (consistent global states) of the sheaf. We present novel applications to signal processing and multi-agent semantics and supply a plethora of examples throughout.

cs.MA↗

arXiv2022

Stable and Transferable Hyper-Graph Neural Networks

We introduce an architecture for processing signals supported on hypergraphs via graph neural networks (GNNs), which we call a Hyper-graph Expansion Neural Network (HENN), and provide the first bounds on the stability and transferability error of a hypergraph signal processing model. To do so, we provide a framework for bounding the stability and transferability error of GNNs across arbitrary graphs via spectral similarity. By bounding the difference between two graph shift operators (GSOs) in the positive semi-definite sense via their eigenvalue spectrum, we show that this error depends only on the properties of the GNN and the magnitude of spectral similarity of the GSOs. Moreover, we show that existing transferability results that assume the graphs are small perturbations of one another, or that the graphs are random and drawn from the same distribution or sampled from the same graphon can be recovered using our approach. Thus, both GNNs and our HENNs (trained using normalized Laplacians as graph shift operators) will be increasingly stable and transferable as the graphs become larger. Experimental results illustrate the importance of considering multiple graph representations in HENN, and show its superior performance when transferability is desired.

cs.LG↗

arXiv2022

Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back

In this work we introduce a convolution operation over the tangent bundle of Riemannian manifolds exploiting the Connection Laplacian operator. We use the convolution to define tangent bundle filters and tangent bundle neural networks (TNNs), novel continuous architectures operating on tangent bundle signals, i.e. vector fields over manifolds. We discretize TNNs both in space and time domains, showing that their discrete counterpart is a principled variant of the recently introduced Sheaf Neural Networks. We formally prove that this discrete architecture converges to the underlying continuous TNN. We numerically evaluate the effectiveness of the proposed architecture on a denoising task of a tangent vector field over the unit 2-sphere.

eess.SP↗

arXiv2022

Diffusion of Information on Networked Lattices by Gossip

We study time-dependent dynamics on a network of order lattices, where structure-preserving lattice maps are used to fuse lattice-valued data over vertices and edges. The principal contribution is a novel asynchronous Laplacian, generalizing the usual graph Laplacian, adapted to a network of heterogeneous lattices. The resulting gossip algorithm is shown to converge asymptotically to stable "harmonic" distributions of lattice data. This general theorem is applicable to several general problems, including lattice-valued consensus, Kripke semantics, and threat detection, all using asynchronous local update rules.

cs.MA↗

arXiv2022

Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions

Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from multiparameter persistence modules. We find that these show promise as an alternative to convolutions for the classification of multidimensional persistence modules.

math.AT↗

arXiv2021

A Temporal Logic-Based Hierarchical Network Connectivity Controller

In this paper, we consider networks of static sensors with integrated sensing and communication capabilities. The goal of the sensors is to propagate their collected information to every other agent in the network and possibly a human operator. Such a task requires constant communication among all agents which may result in collisions and congestion in wireless communication. To mitigate this issue, we impose locally non-interfering connectivity constraints that must be respected by every agent. We show that these constraints along with the requirement of propagating information in the network can be captured by a Linear Temporal Logic (LTL) framework. Existing temporal logic control synthesis algorithms can be used to design correct-by-construction communication schedules that satisfy the considered LTL formula. Nevertheless, such approaches are centralized and scale poorly with the size of the network. We propose a hierarchical LTL-based algorithm that designs communication schedules that determine which agents should communicate while maximizing network usage. We show that the proposed algorithm is complete and demonstrate its efficiency and scalability through analysis and numerical experiments.

eess.SY↗

arXiv2021

Cellular Sheaves of Lattices and the Tarski Laplacian

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications.

math.AT↗