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1. Reduced resonance schemes and Chen ranks

   https://doi.org/10.1515/crelle-2024-0051  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Suciu, Alexander I (July 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts.We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme.We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups. 

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2. The Lauritzen-Chen Likelihood For Graphical Models

   Ilya Shpitser (April 2023, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics)

   Francisco Ruiz, Jennifer Dy (Ed.)

   Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric set- tings, likelihoods specified via clique potential functions are generally not known to be congenial (jointly well-specified) or non-redundant. Congenial and non-redundant DAG likelihoods are far simpler to specify in both parametric and semi-parametric settings by modeling Markov factors in the DAG factorization. However, DAG likelihoods specified in this way are not guaranteed to coincide in distinct DAGs within the same Markov equivalence class. This complicates likelihoods based model selection procedures for DAGs by “sneaking in” potentially un- warranted assumptions about edge orientations. In this paper we link a density function decomposition due to Chen with the clique factorization of MRFs described by Lauritzen to provide a general likelihood for MRF models. The proposed likelihood is composed of variationally independent, and non-redundant closed form functionals of the observed data distribution, and is sufficiently general to apply to arbitrary parametric and semi-parametric models. We use an extension of our developments to give a general likelihood for DAG models that is guaranteed to coincide for all members of a Markov equivalence class. Our results have direct applications for model selection and semi-parametric inference. 

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3. Nonlinear Matched Filters via Chen–Fliess Series

   Gray, W. Steven (December 2020, 2020 59th IEEE Conference on Decision and Control)

   null (Ed.)

   A class of nonlinear matched filters is introduced suitable for detection problems using Chen–Fliess functional series. Such series can be viewed as a noncommutative analogue of Taylor series. They are written in terms of a weighted sum of iterated integrals indexed by words over a noncommuting set of symbols. The primary goal is to identify within this class the set of filters which maximizes the signal-to-noise ratio at a given time instant in order to provide a statistic for detecting a known signal. 

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4. Generating series for networks of Chen–Fliess series

   https://doi.org/10.1016/j.sysconle.2020.104827  

   Gray, W. Steven; Ebrahimi-Fard, Kurusch (January 2021, Systems & Control Letters)

   null (Ed.)
5. Extending the Execution Time of Chen-Fliess Series

   https://doi.org/10.1109/CISS64860.2025.10944683  

   Boudaghi, Farnaz; Espinosa, Luis_A Duffaut (March 2025, IEEE)