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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.15946 |
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| _version_ | 1866915489512423424 |
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| author | Cassel, Jonas Boll, Bastian Petra, Stefania Albers, Peter Schnörr, Christoph |
| author_facet | Cassel, Jonas Boll, Bastian Petra, Stefania Albers, Peter Schnörr, Christoph |
| contents | This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors.
The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance.
Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15946 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sigma Flows for Image and Data Labeling and Learning Structured Prediction Cassel, Jonas Boll, Bastian Petra, Stefania Albers, Peter Schnörr, Christoph Dynamical Systems Computer Vision and Pattern Recognition Machine Learning 53B12, 35R01, 35R02, 62H35, 68U10, 68T05, 68T07 This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning. |
| title | Sigma Flows for Image and Data Labeling and Learning Structured Prediction |
| topic | Dynamical Systems Computer Vision and Pattern Recognition Machine Learning 53B12, 35R01, 35R02, 62H35, 68U10, 68T05, 68T07 |
| url | https://arxiv.org/abs/2408.15946 |