Periodic Reporting for period 1 - MIAS (Model Invariants in Algebraic Statistics)
Periodo di rendicontazione: 2023-09-01 al 2025-08-31
Model Invariants in Algebraic Statistics is a research project in mathematical statistics and computer vision. The project’s unifying theme is the use of algebraic techniques to advance the state of the art in these disciplines.
Mathematical statistics is the application of mathematical concepts to data analysis. This project focuses on statistical models, which are tools for encoding specific hypotheses that may be formulated about a data set. The aim is to increase the theoretical understanding of these models, develop new models, and provide algorithms for working with these models.
Computer vision is the algorithmic analysis of digital images. Within this area, this project addresses the structure-from-motion (SfM) problem, which is the task of reconstructing a 3D scene from a collection of 2D images taken from different angles of that scene. The aim is to develop a mathematical model for rolling shutter cameras, classify its minimal SfM problems, and solve these algorithmically.
The project has met its goals. On the statistical side, progress was made in the classification of discrete and Gaussian statistical models with maximum likelihood degree one. Furthermore, a systematic way to combine discrete data models from different sources was developed. On the computer vision side, the project’s goals were accomplished within the newly-introduced framework of “Order-One” rolling shutter camera models.
Algebraic methods and algebraic model invariants have played an important role in all aspects of the project. From this project’s results, we conclude that algebra and geometry can help advance the state of the art in applied disciplines.
Mathematical statistics is the application of mathematical concepts to data analysis. This project focuses on statistical models, which are tools for encoding specific hypotheses that may be formulated about a data set. The aim is to increase the theoretical understanding of these models, develop new models, and provide algorithms for working with these models.
Computer vision is the algorithmic analysis of digital images. Within this area, this project addresses the structure-from-motion (SfM) problem, which is the task of reconstructing a 3D scene from a collection of 2D images taken from different angles of that scene. The aim is to develop a mathematical model for rolling shutter cameras, classify its minimal SfM problems, and solve these algorithmically.
The project has met its goals. On the statistical side, progress was made in the classification of discrete and Gaussian statistical models with maximum likelihood degree one. Furthermore, a systematic way to combine discrete data models from different sources was developed. On the computer vision side, the project’s goals were accomplished within the newly-introduced framework of “Order-One” rolling shutter camera models.
Algebraic methods and algebraic model invariants have played an important role in all aspects of the project. From this project’s results, we conclude that algebra and geometry can help advance the state of the art in applied disciplines.
1) Research and implementation of novel algebraic algorithms for computer vision. The research focused on modeling SfM problems with rolling shutter cameras. Classifying and identifying minimal problems for basic rolling-shutter camera models was an important part of this work. Ready-to-run solver and classifier code was developed. This work resulted in the publication
Marvin Anas Hahn, Kathlén Kohn, Orlando Marigliano, and Tomas Pajdla. Order-One Rolling Shutter Cameras. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2025), 27007–27016
which was designated as “Highlight” by the CVPR community. Its methodology received positive feedback from conference attendees.
2) Research and manuscript revision for Gaussian statistical models with maximum likelihood degree one. A new description of these models in terms of partial differential equations was developed. As a consequence, important obstacles to achieving a full classification of such models were identified. Notably, a complete classification would entail resolving a long-standing open problem in rational algebraic geometry. This work led to the publication
Carlos Améndola, Lukas Gustafsson, Kathlén Kohn, Orlando Marigliano and Anna Seigal. Differential equations for Gaussian statistical models with rational maximum likelihood estimator. SIAM Journal on Applied Algebra and Geometry 8 (2024), 465–492.
3) Exploration of new research directions in statistical modeling, focusing on algebraic approaches to compositional data. This line of inquiry was not developed further, but this work enhanced the research group’s understanding of the structural limitations of compositional models.
4) Training, communication and dissemination activities, research visit, conference travel. The researcher attended a MSCA-focused workshop at University of Genoa and participated in four European conferences in statistics, applied algebra, and computer vision, delivering two conference talks. A research visit was made at TU Munich to explore new research directions.
5) Development of research ideas on Gaussian non-exponential families. Formulation of a research plan involving these ideas, to be implemented later in the researcher’s career.
6) Revision of a manuscript on the classification of discrete statistical models with maximum likelihood degree one, resulting in the publication
Arthur Bik and Orlando Marigliano. Classifying one-dimensional discrete models with maximum likelihood degree one. Advances in Applied Mathematics 170 (2025), 102928.
This publication inspired subsequent work by C. Améndola, V. D. Nguyen, J. Oldekop (ArXiv: 2507.18686).
7) Training, communication and dissemination activities, conference travel. The researcher and supervisor held an algebraic statistics seminar at University of Genoa. The researcher attended a career development workshop and scientific conference on algebraic statistics in Munich.
8) Research on statistical modeling and the development of new methods for the analysis of categorical data. The theory of Markov combinations for these data was developed, providing several ways to combine modeling hypotheses and evidence from multiple sources. The research manuscript
Orlando Marigliano and Eva Riccomagno. Markov combinations of discrete statistical models. ArXiv:2509.18983 2025
was drafted and submitted to leading statistics journals. A comprehensive treatment of discrete Markov combinations and their sampling, algorithmic, and theoretical aspects was achieved.
Marvin Anas Hahn, Kathlén Kohn, Orlando Marigliano, and Tomas Pajdla. Order-One Rolling Shutter Cameras. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2025), 27007–27016
which was designated as “Highlight” by the CVPR community. Its methodology received positive feedback from conference attendees.
2) Research and manuscript revision for Gaussian statistical models with maximum likelihood degree one. A new description of these models in terms of partial differential equations was developed. As a consequence, important obstacles to achieving a full classification of such models were identified. Notably, a complete classification would entail resolving a long-standing open problem in rational algebraic geometry. This work led to the publication
Carlos Améndola, Lukas Gustafsson, Kathlén Kohn, Orlando Marigliano and Anna Seigal. Differential equations for Gaussian statistical models with rational maximum likelihood estimator. SIAM Journal on Applied Algebra and Geometry 8 (2024), 465–492.
3) Exploration of new research directions in statistical modeling, focusing on algebraic approaches to compositional data. This line of inquiry was not developed further, but this work enhanced the research group’s understanding of the structural limitations of compositional models.
4) Training, communication and dissemination activities, research visit, conference travel. The researcher attended a MSCA-focused workshop at University of Genoa and participated in four European conferences in statistics, applied algebra, and computer vision, delivering two conference talks. A research visit was made at TU Munich to explore new research directions.
5) Development of research ideas on Gaussian non-exponential families. Formulation of a research plan involving these ideas, to be implemented later in the researcher’s career.
6) Revision of a manuscript on the classification of discrete statistical models with maximum likelihood degree one, resulting in the publication
Arthur Bik and Orlando Marigliano. Classifying one-dimensional discrete models with maximum likelihood degree one. Advances in Applied Mathematics 170 (2025), 102928.
This publication inspired subsequent work by C. Améndola, V. D. Nguyen, J. Oldekop (ArXiv: 2507.18686).
7) Training, communication and dissemination activities, conference travel. The researcher and supervisor held an algebraic statistics seminar at University of Genoa. The researcher attended a career development workshop and scientific conference on algebraic statistics in Munich.
8) Research on statistical modeling and the development of new methods for the analysis of categorical data. The theory of Markov combinations for these data was developed, providing several ways to combine modeling hypotheses and evidence from multiple sources. The research manuscript
Orlando Marigliano and Eva Riccomagno. Markov combinations of discrete statistical models. ArXiv:2509.18983 2025
was drafted and submitted to leading statistics journals. A comprehensive treatment of discrete Markov combinations and their sampling, algorithmic, and theoretical aspects was achieved.
This project’s results include contributions towards the classification of discrete and Gaussian statistical models with rational maximum likelihood estimator, the development of the Markov Combinations framework for combining discrete data, and the full classification of minimal problems for order-one rolling shutter cameras.
The first expected impact of these results is in their respective scientific communities: the algebraic statistics, mathematical statistics, and computer vision communities. In this area, the project has already achieved publications in several important venues, as well as the inspiration for further developments undertaken by colleagues in these communities.
From a socio-economic perspective, the project’s impact on the statistical field may transfer to the economic or societal fields if the data analysis techniques developed by the project gain wider adoption. In particular, the Markov Combinations results may see adoption by professionals working with categorical data, e.g. psychometric scientists. In the computer vision field, impact may be reflected in either a direct implementation of our minimal problem algorithms to enhance the SfM pipeline for rolling shutter cameras, or in the improvement and generalization of the Order-One camera model.
The wider societal impact of the project lies in the widespread nature of the models used. Discrete variables are used e.g. for analysing questionnaire data. This is a standard data collection technique in many socially-relevant fields. Gaussian models are the standard way to encode continuous data with a bell distribution, which again commonly arises in nature and industry. Markov combinations may be used to combine two related data sets in a systematic way. Rolling shutter cameras are ubiquitous, being the type of camera present in almost all smartphone models.
Most generally, this project has helped meet the need for the analysis of ever-growing sets of discrete, continuous, and image data via transparent, explainable, and reproducible techniques.
The first expected impact of these results is in their respective scientific communities: the algebraic statistics, mathematical statistics, and computer vision communities. In this area, the project has already achieved publications in several important venues, as well as the inspiration for further developments undertaken by colleagues in these communities.
From a socio-economic perspective, the project’s impact on the statistical field may transfer to the economic or societal fields if the data analysis techniques developed by the project gain wider adoption. In particular, the Markov Combinations results may see adoption by professionals working with categorical data, e.g. psychometric scientists. In the computer vision field, impact may be reflected in either a direct implementation of our minimal problem algorithms to enhance the SfM pipeline for rolling shutter cameras, or in the improvement and generalization of the Order-One camera model.
The wider societal impact of the project lies in the widespread nature of the models used. Discrete variables are used e.g. for analysing questionnaire data. This is a standard data collection technique in many socially-relevant fields. Gaussian models are the standard way to encode continuous data with a bell distribution, which again commonly arises in nature and industry. Markov combinations may be used to combine two related data sets in a systematic way. Rolling shutter cameras are ubiquitous, being the type of camera present in almost all smartphone models.
Most generally, this project has helped meet the need for the analysis of ever-growing sets of discrete, continuous, and image data via transparent, explainable, and reproducible techniques.