MIAS · Model Invariants in Algebraic Statistics

Model Invariants in Algebraic Statistics

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.

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