2021/08/02 by Alexander Zeilmann, Zeilmann, Alexander, Stefania Petra +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #34C40 #62H35 #68T05 #68U10 #91A22 #Algorithm #Applied mathematics #Artificial intelligence #Artificial neural network #Automatic differentiation #Balanced flow #Cell Image Analysis Techniques #Computer science #FOS: Computer and information sciences #FOS: Mathematics #Flow (mathematics) #Function (biology) #Geometry #Gradient descent #Image (mathematics) #Iterative method #Krylov subspace #Machine Learning (cs.LG) #Mathematical analysis #Mathematical optimization #Mathematics #Medical Image Segmentation Techniques #Optimization and Control (math.OC) #Parameter space #Rank (graph theory) #Representation (politics) #Subspace topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2108.02571
openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We introduce a novel algorithm for estimating optimal parameters of linearized assignment flows for image labeling. An exact formula is derived for the parameter gradient of any loss function that is constrained by the linear system of ODEs determining the linearized assignment flow. We show how to efficiently evaluate this formula using a Krylov subspace and a low-rank approximation. This enables us to perform parameter learning by Riemannian gradient descent in the parameter space, without the need to backpropagate errors or to solve an adjoint equation. Experiments demonstrate that our method performs as good as highly-tuned machine learning software using automatic differentiation. Unlike methods employing automatic differentiation, our approach yields a low-dimensional representation of internal parameters and their dynamics which helps to understand how assignment flows and more generally neural networks work and perform.