2026
Lang, Quanjun; Wang, Xiong; Lu, Fei; Maggioni, Mauro
Learning multi-type heterogeneous interacting particle systems Unpublished
2026.
Abstract | Links | BibTeX | Tags: agent-based models, Artificial Intelligence, dynamical systems, interacting particle systems, inverse problems, Machine learning, stochastic systems
@unpublished{nokey,
title = {Learning multi-type heterogeneous interacting particle systems},
author = {Lang, Quanjun and Wang, Xiong and Lu, Fei and Maggioni, Mauro},
url = {https://arxiv.org/abs/2602.03954},
doi = {https://doi.org/10.48550/arXiv.2602.03954},
year = {2026},
date = {2026-02-05},
urldate = {2026-02-05},
abstract = {We propose a framework for the joint inference of network topology, multi-type interaction kernels, and latent type assignments in heterogeneous interacting particle systems from multi-trajectory data. This learning task is a challenging non-convex mixed-integer optimization problem, which we address through a novel three-stage approach. First, we leverage shared structure across agent interactions to recover a low-rank embedding of the system parameters via matrix sensing. Second, we identify discrete interaction types by clustering within the learned embedding. Third, we recover the network weight matrix and kernel coefficients through matrix factorization and a post-processing refinement. We provide theoretical guarantees with estimation error bounds under a Restricted Isometry Property (RIP) assumption and establish conditions for the exact recovery of interaction types based on cluster separability. Numerical experiments on synthetic datasets, including heterogeneous predator-prey systems, demonstrate that our method yields an accurate reconstruction of the underlying dynamics and is robust to noise.},
keywords = {agent-based models, Artificial Intelligence, dynamical systems, interacting particle systems, inverse problems, Machine learning, stochastic systems},
pubstate = {published},
tppubtype = {unpublished}
}
We propose a framework for the joint inference of network topology, multi-type interaction kernels, and latent type assignments in heterogeneous interacting particle systems from multi-trajectory data. This learning task is a challenging non-convex mixed-integer optimization problem, which we address through a novel three-stage approach. First, we leverage shared structure across agent interactions to recover a low-rank embedding of the system parameters via matrix sensing. Second, we identify discrete interaction types by clustering within the learned embedding. Third, we recover the network weight matrix and kernel coefficients through matrix factorization and a post-processing refinement. We provide theoretical guarantees with estimation error bounds under a Restricted Isometry Property (RIP) assumption and establish conditions for the exact recovery of interaction types based on cluster separability. Numerical experiments on synthetic datasets, including heterogeneous predator-prey systems, demonstrate that our method yields an accurate reconstruction of the underlying dynamics and is robust to noise.
2024
Kuemmerle, Christian; Maggioni, Mauro; Tang, Sui
Learning Transition Operators From Sparse Space-Time Samples Journal Article
In: IEEE Transactions on Information Theory, 2024.
Abstract | Links | BibTeX | Tags: dynamical systems, Machine learning, optimization, sparsity, statistics
@article{LearningTransitionOperators_1,
title = {Learning Transition Operators From Sparse Space-Time Samples},
author = {Christian Kuemmerle and Mauro Maggioni and Sui Tang},
url = {https://arxiv.org/abs/2212.00746
https://ieeexplore.ieee.org/document/10558780},
doi = {10.1109/TIT.2024.3413534},
year = {2024},
date = {2024-06-14},
urldate = {2024-06-14},
journal = {IEEE Transactions on Information Theory},
abstract = {We consider the nonlinear inverse problem of learning a transition operator A from partial observations at different times, in particular from sparse observations of entries of its powers A,A2,⋯,AT. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if A is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than (rnlog(nT)) space-time samples are sufficient to ensure accurate recovery of a rank-r operator A of size n×n. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order O(rnT) and per-iteration time complexity linear in n. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.},
keywords = {dynamical systems, Machine learning, optimization, sparsity, statistics},
pubstate = {published},
tppubtype = {article}
}
We consider the nonlinear inverse problem of learning a transition operator A from partial observations at different times, in particular from sparse observations of entries of its powers A,A2,⋯,AT. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if A is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than (rnlog(nT)) space-time samples are sufficient to ensure accurate recovery of a rank-r operator A of size n×n. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order O(rnT) and per-iteration time complexity linear in n. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.
2008
Coifman, Ronald R; Kevrekidis, Ioannis G; Lafon, Stephane; Maggioni, Mauro; Nadler, Boaz
Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems Journal Article
In: SIAM J.M.M.S., vol. 7, no. 2, pp. 842–864, 2008.
BibTeX | Tags: diffusion geometry, dynamical systems, Laplacian eigenfunctions, Machine learning, model reduction, stochastic systems
@article{CKLMN:DiffusionMapsReductionCoordinates,
title = {Diffusion Maps, reduction coordinates and low dimensional representation of stochastic systems},
author = {Ronald R Coifman and Ioannis G Kevrekidis and Stephane Lafon and Mauro Maggioni and Boaz Nadler},
year = {2008},
date = {2008-01-01},
journal = {SIAM J.M.M.S.},
volume = {7},
number = {2},
pages = {842--864},
keywords = {diffusion geometry, dynamical systems, Laplacian eigenfunctions, Machine learning, model reduction, stochastic systems},
pubstate = {published},
tppubtype = {article}
}
- Lectures at Summer School at Peking University, July 2017.
- PCMI Lectures, Summer 2016: Lecture 1, Lecture 2, Problems/discussion points
- Google Scholar
- Papers on the ArXiv
- Papers on MathsciNet
- Tutorials on diffusion geometry and multiscale analysis on graphs at the MRA Internet Program at IPAM: Part I and Part II.
- Diffusion Geometries, Diffusion Wavelets and Harmonic Analysis of large data sets, IPAM, Multiscale Geometric Analysis Program, Fall 2004.
- Diffusion Geometries, global and multiscale, IPAM, 2005.