arXiv subjects

Johan Karlsson

Publications and source records attributed to Johan Karlsson.

At least 19 recordsLinked to original sources

arXiv2026

A proximal approach to the Schr\"odinger bridge problem with incomplete information and application to contamination tracking in water networks

In this work, we study a discrete Schr\"odinger bridge problem with partial marginal observations. A main difficulty compared to the classical Schr\"odinger bridge formulation is that our problem is not strictly convex and standard Sinkhorn-type methods cannot be directly applied. To address this issue, we propose a scalable computational method based on an entropic proximal scheme. Furthermore, we develop a framework for this problem that includes duality results, characterization of the optimal solutions, and an observability condition that determines when the optimal solution is unique. We validate the method on the problem of estimating contamination in a water distribution network, where the partial marginals correspond to measured pollutant concentrations at the sensor locations. The experiments were conducted on a laboratory-scale water distribution network.

math.OC↗

arXiv2026

Learning Sampled-data Control for Swarms via MeanFlow

Steering large-scale swarms with only limited control updates is often needed due to communication or computational constraints, yet most learning-based approaches do not account for this and instead model instantaneous velocity fields. As a result, the natural object for decision making is a finite-window control quantity rather than an infinitesimal one. To address this gap, we consider the recent machine learning framework MeanFlow and generalize it to the setting with general linear dynamic systems. This results in a new sampled-data learning framework that operates directly in control space and that can be applied for swarm steering. To this end, we learn the finite-horizon coefficient that parameterizes the minimum-energy control applied over each interval, and derive a differential identity that connects this quantity to a local bridge-induced supervision signal. This identity leads to a simple stop-gradient regression objective, allowing the interval coefficient field to be learned efficiently from bridge samples. The learned policy is deployed through sampled-data updates, guaranteeing that the resulting controller exactly respects the prescribed linear time-invariant dynamics and actuation channel. The resulting method enables few-step swarm steering at scale, while remaining consistent with the finite-window actuation structure of the underlying control system.

cs.LG↗

arXiv2026

Temporally Flexible Transport Scheduling on Networks with Departure-Arrival Constriction and Nodal Capacity Limits

We investigate the optimal transport (OT) problem over networks, wherein supply and demand are conceptualized as temporal marginals governing departure rates of particles from source nodes and arrival rates at sink nodes. This setting extends the classical OT framework, where all mass is conventionally assumed to depart at $t = 0$ and arrive at $t = t_f$. Our generalization accommodates departures and arrivals at specified times, referred as departure--arrival(DA) constraints. In particular, we impose nodal-temporal flux constraints at source and sink nodes, characterizing two distinct scenarios: (i) Independent DA constraints, where departure and arrival rates are prescribed independently, and (ii) Coupled DA constraints, where each particle's transportation time span is explicitly specified. We establish that OT with independent DA constraints admits a multi-marginal optimal transport formulation, while the coupled DA case aligns with the unequal-dimensional OT framework. For line graphs, we analyze the existence and uniqueness of the solution path. For general graphs, we use a constructive path-based reduction and optimize over a prescribed set of paths. From a computational perspective, we consider entropic regularization of the original problem to efficiently provide solutions based on multi-marginal Sinkhorn method, making use of the graphical structure of the cost to further improve scalability. Our numerical simulation further illustrates the linear convergence rate in terms of marginal violation.

math.OC↗

arXiv2026

Sound Field Estimation Using Optimal Transport Barycenters in the Presence of Phase Errors

This study introduces a novel approach for estimating plane-wave coefficients in sound field reconstruction, specifically addressing challenges posed by error-in-variable phase perturbations. Such systematic errors typically arise from sensor mis-calibration, including uncertainties in sensor positions and response characteristics, leading to measurement-induced phase shifts in plane wave coefficients. Traditional methods often result in biased estimates or non-convex solutions. To overcome these issues, we propose an optimal transport (OT) framework. This framework operates on a set of lifted non-negative measures that correspond to observation-dependent shifted coefficients relative to the unperturbed ones. By applying OT, the supports of the measures are transported toward an optimal average in the phase space, effectively morphing them into an indistinguishable state. This optimal average, known as barycenter, is linked to the estimated plane-wave coefficients using the same lifting rule. The framework addresses the ill-posed nature of the problem, due to the large number of plane waves, by adding a constant to the ground cost, ensuring the sparsity of the transport matrix. Convex consistency of the solution is maintained. Simulation results confirm that our proposed method provides more accurate coefficient estimations compared to baseline approaches in scenarios with both additive noise and phase perturbations.

eess.SP↗

arXiv2025

A convex approach for Markov chain estimation from aggregate data via inverse optimal transport

We address the problem of identifying the dynamical law governing the evolution of a population of indistinguishable particles, when only aggregate distributions at successive times are observed. Assuming a Markovian evolution on a discrete state space, the task reduces to estimating the underlying transition probability matrix from distributional data. We formulate this inverse problem within the framework of entropic optimal transport, as a joint optimization over the transition matrix and the transport plans connecting successive distributions. This formulation results in a convex optimization problem, and we propose an efficient iterative algorithm based on the entropic proximal method. We illustrate the accuracy and convergence of the method in two numerical setups, considering estimation from independent snapshots and estimation from a time series of aggregate observations, respectively.

math.OC↗

arXiv2025

Optimization of continuous-flow over traffic networks with fundamental diagram constraints

Optimal transport (OT) theory provides a principled framework for modeling mass movement in applications such as mobility, logistics, and economics. Classical formulations, however, generally ignore capacity limits that are intrinsic in applications, in particular in traffic flow problems. We address this limitation by incorporating fundamental diagrams into a dynamic continuous-flow OT model on graphs, thereby including empirical relations between local density and maximal flux. We adopt an Eulerian kinetic action on graphs that preserves displacement interpolation in direct analogy with the continuous theory. Momentum lives on edges and density on nodes, mirroring road-network semantics in which segments carry speed and intersections store mass. The resulting fundamental-diagram-constrained OT problem preserves mass conservation and admits a convex variational discretization, yielding optimal congestion-aware traffic flow over road networks. We establish the existence and uniqueness of the optimal flow with sources and sinks, and develop an efficient convex optimization method. Numerical studies begin with a single-lane line network and scale to a city-level road network.

math.OC↗

arXiv2025

Fundamental diagram constrained dynamic optimal transport via proximal splitting methods

Optimal transport has recently been brought forward as a tool for modeling and efficiently solving a variety of flow problems, such as origin-destination problems and multi-commodity flow problems. Although the framework has shown to be effective for many large scale flow problems, the formulations typically lack dynamic properties used in common traffic models, such as the Lighthill-Whitham-Richards model. In this work, we propose an optimal transport framework that includes dynamic constraints specified by the fundamental diagram for modeling macroscopic traffic flow. The problem is cast as a convex variant of dynamic optimal transport, with additional nonlinear temporal-spatial inequality constraints of momentum, modeled after the fundamental diagram from traffic theory. This constraint imposes a density-dependent upper bound on the admissible flux, capturing flow saturation and congestion effects, and thus leaves space for kinetic optimization. The formulation follows the Benamou-Brenier transportation rationale, whereby kinetic energy over density and momentum fields is optimized subject to the mass conservation law. We develop proximal splitting methods, namely the Douglas-Rachford and Chambolle-Pock algorithms, which exploit the separable structure of the constraint set and require only simple proximal operations, and can accommodate additional (time-varying) spatial restrictions or obstacles. Numerical experiments illustrate the impact of the constraint on transport behavior, including congestion-aware spreading, rerouting, and convergence. The framework establishes a connection between optimal transport and macroscopic traffic flow theory and provides a scalable, variational tool for modeling congestion-constricted (or saturation-aware) Wasserstein gradient flow.

math.OC↗

arXiv2025

Probabilistic Trajectory GOSPA: A Metric for Uncertainty-Aware Multi-Object Tracking Performance Evaluation

This paper presents a generalization of the trajectory general optimal sub-pattern assignment (GOSPA) metric for evaluating multi-object tracking algorithms that provide trajectory estimates with track-level uncertainties. This metric builds on the recently introduced probabilistic GOSPA metric to account for both the existence and state estimation uncertainties of individual object states. Similar to trajectory GOSPA (TGOSPA), it can be formulated as a multidimensional assignment problem, and its linear programming relaxation--also a valid metric--is computable in polynomial time. Additionally, this metric retains the interpretability of TGOSPA, and we show that its decomposition yields intuitive costs terms associated to expected localization error and existence probability mismatch error for properly detected objects, expected missed and false detection error, and track switch error. The effectiveness of the proposed metric is demonstrated through a simulation study.

eess.SP↗

arXiv2025

Task Allocation for Multi-agent Systems via Unequal-dimensional Optimal Transport

We consider a probabilistic model for large-scale task allocation problems for multi-agent systems, aiming to determine an optimal deployment strategy that minimizes the overall transport cost. Specifically, we assign transportation agents to delivery tasks with given pick-up and drop-off locations, pairing the spatial distribution of transport resources with the joint distribution of task origins and destinations. This aligns with the optimal mass transport framework where the problem and is in the unequal-dimensional setting. The task allocation problem can be thus seen as a linear programming problem that minimizes a quadratic transport cost functional, optimizing the energy of all transport units. The problem is motivated by time-sensitive medical deliveries using drones, such as emergency equipment and blood transport. In this paper, we establish the existence, uniqueness, and smoothness of the optimal solution, and illustrate its properties through numerical simulations.

eess.SY↗

arXiv2025

MultistageOT: Multistage optimal transport infers trajectories from a snapshot of single-cell data

Single-cell RNA-sequencing captures a temporal slice, or a snapshot, of a cell differentiation process. A major bioinformatical challenge is the inference of differentiation trajectories from a single snapshot, and methods that account for outlier cells that are unrelated to the differentiation process have yet to be established. We present MultistageOT: a Multistage Optimal Transport-based framework for trajectory inference in a snapshot (https://github.com/dahlinlab/MultistageOT). Application of optimal transport has proven successful for many single-cell tasks, but classical bimarginal optimal transport for trajectory inference fails to model temporal progression in a snapshot. Representing a novel generalization of optimal transport, MultistageOT addresses this major limitation by introducing a temporal dimension, allowing for high resolution modeling of intermediate differentiation stages. We challenge MultistageOT with snapshot data of cell differentiation, demonstrating effectiveness in pseudotime ordering, detection of outliers, and significantly improved fate prediction accuracy over state-of-the-art.

q-bio.QM↗

arXiv2024

Probabilistic GOSPA: A Metric for Performance Evaluation of Multi-Object Filters with Uncertainties

This paper presents a probabilistic generalization of the Generalized Optimal Sub-Pattern Assignment (GOSPA) metric, termed P-GOSPA. The GOSPA metric has been widely used to evaluate the distance between finite sets, particularly in multi-object estimation applications. The P-GOSPA extends GOSPA into the space of multi-Bernoulli densities, incorporating inherent uncertainty in probabilistic multi-object representations. Additionally, P-GOSPA retains the interpretability of GOSPA, such as its decomposition into localization, missed detection, and false detection errors in a sound and meaningful manner. Examples and simulations are provided to demonstrate the efficacy of the proposed P-GOSPA metric.

eess.SP↗

arXiv2024

A parallel framework for graphical optimal transport

We study multi-marginal optimal transport (MOT) problems where the underlying cost has a graphical structure. These graphical multi-marginal optimal transport problems have found applications in several domains including traffic flow control, barycenter and regression problems in the Wasserstein space, and Hidden Markov model inference problems. The MOT problem can be approached through two formulations: a single big MOT problem, or coupled minor OT problems. In this paper, we focus on the latter approach and demonstrate its efficiency gain from parallelization. For tree-structured MOT problems, we introduce a novel parallelizable algorithm that significantly reduces computational complexity. Additionally, we adapt this algorithm for general graphs, employing the modified junction trees to enable parallel updates. Our contributions, validated through numerical experiments, offer new avenues for MOT applications and establish benchmarks in computational efficiency.

math.OC↗

arXiv2024

Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport

We introduce an efficient computational framework for solving a class of multi-marginal martingale optimal transport problems, which includes many robust pricing problems of large financial interest. Such problems are typically computationally challenging due to the martingale constraint, however, by extending the state space we can identify them with problems that exhibit a certain sequential martingale structure. Our method exploits such structures in combination with entropic regularisation, enabling fast computation of optimal solutions and allowing us to solve problems with a large number of marginals. We demonstrate the method by using it for computing robust price bounds for different options, such as lookback options and Asian options.

q-fin.CP↗

arXiv2024

A cutting plane algorithm for globally solving low dimensional k-means clustering problems

Clustering is one of the most fundamental tools in data science and machine learning, and k-means clustering is one of the most common such methods. There is a variety of approximate algorithms for the k-means problem, but computing the globally optimal solution is in general NP-hard. In this paper we consider the k-means problem for instances with low dimensional data and formulate it as a structured concave assignment problem. This allows us to exploit the low dimensional structure and solve the problem to global optimality within reasonable time for large data sets with several clusters. The method builds on iteratively solving a small concave problem and a large linear programming problem. This gives a sequence of feasible solutions along with bounds which we show converges to zero optimality gap. The paper combines methods from global optimization theory to accelerate the procedure, and we provide numerical results on their performance.

math.OC↗

arXiv2023

Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces

This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm. The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning. The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points. We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.

math.OC↗

arXiv2023

Mean field type control with species dependent dynamics via structured tensor optimization

In this work we consider mean field type control problems with multiple species that have different dynamics. We formulate the discretized problem using a new type of entropy-regularized multimarginal optimal transport problems where the cost is a decomposable structured tensor. A novel algorithm for solving such problems is derived, using this structure and leveraging recent results in entropy-regularized optimal transport. The algorithm is then demonstrated on a numerical example in robot coordination problem for search and rescue, where three different types of robots are used to cover a given area at minimal cost.

math.OC↗

arXiv2022

Orthogonalization of data via Gromov-Wasserstein type feedback for clustering and visualization

In this paper we propose an adaptive approach for clustering and visualization of data by an orthogonalization process. Starting with the data points being represented by a Markov process using the diffusion map framework, the method adaptively increase the orthogonality of the clusters by applying a feedback mechanism inspired by the Gromov-Wasserstein distance. This mechanism iteratively increases the spectral gap and refines the orthogonality of the data to achieve a clustering with high specificity. By using the diffusion map framework and representing the relation between data points using transition probabilities, the method is robust with respect to both the underlying distance, noise in the data and random initialization. We prove that the method converges globally to a unique fixpoint for certain parameter values. We also propose a related approach where the transition probabilities in the Markov process are required to be doubly stochastic, in which case the method generates a minimizer to a nonconvex optimization problem. We apply the method on cryo-electron microscopy image data from biopharmaceutical manufacturing where we can confirm biologically relevant insights related to therapeutic efficacy. We consider an example with morphological variations of gene packaging and confirm that the method produces biologically meaningful clustering results consistent with human expert classification.

stat.ML↗

arXiv2021

Graph-structured tensor optimization for nonlinear density control and mean field games

In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems, which can be seen as a generalization of multi-marginal optimal transport problems with graph-structured costs, appear in many applications. Examples are unbalanced optimal transport and multi-species potential mean field games, where the latter is a class of nonlinear density control problems. The method we develop is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures it is possible to derive efficient methods for computing these projections, and here we specifically consider the graph structure that occurs in multi-species potential mean field games. We also illustrate the methodology on a numerical example from this problem class.

math.OC↗