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| The current focus is on the **//assignment manifold//** and image labeling, and on learning from image assignments in large-scale unsupervised scenarios, within the mathematical frameworks of information geometry and regularised optimal transport. A novel smooth dynamical system evolving on a statistical manifold, called **//assignment flow//**, forms the basis of our work. | The current focus is on the **//assignment manifold//** and image labeling, and on learning from image assignments in large-scale unsupervised scenarios, within the mathematical frameworks of information geometry and regularised optimal transport. A novel smooth dynamical system evolving on a statistical manifold, called **//assignment flow//**, forms the basis of our work. | ||
| - | **Current work.** We conduct a comprehensive study of //geometric integration// techniques, including automatic step size adaption, for numerically computing the assignment flow in a stable, efficient and parameter-free way. Based on this, we study how weights for geometric diffusion can be learned from data, by applying optimal control to the assignment flow. This enables to attach a semantic meaning to such weights, a property that is missing in current models of artificial neural networks. | + | **Current work.** We conduct a comprehensive study of //geometric integration// techniques, including automatic step size adaption, for numerically computing the assignment flow in a stable, efficient and parameter-free way. |
| + | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Zeilmann2018aa.pdf|Geometric Numerical Integration of the Assignment Flow, preprint: arXiv:1810.06970]] | ||
| + | Based on this, we study how weights for geometric diffusion can be learned from data, by applying optimal control to the assignment flow. This enables to attach a semantic meaning to such weights, a property that is missing in current models of artificial neural networks. | ||
| **Recent work.** We extended the assignment flow to //unsupervised// scenarios, where label evolution on a feature manifold is simultaneously performed together with label assignment to given data - see the | **Recent work.** We extended the assignment flow to //unsupervised// scenarios, where label evolution on a feature manifold is simultaneously performed together with label assignment to given data - see the | ||
| - | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/gcpr2018.pdf|preliminary announcement at the GCPR 2018]]. | + | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, preliminary announcement: GCPR 2018]]. |
| This paper sketches a special instance of a more general framework, the //unsupervised assignment flow//, to be introduced in a forthcoming report. | This paper sketches a special instance of a more general framework, the //unsupervised assignment flow//, to be introduced in a forthcoming report. | ||
| We applied our approach to solve in a novel way the //MAP labeling problem// based on a given graphical model by smoothly combining a geometric reformulation of the local polytope relaxation with rounding to an integral solution. A key ingredient are local `//Wasserstein messages//' that couple local assignment measures along edges. | We applied our approach to solve in a novel way the //MAP labeling problem// based on a given graphical model by smoothly combining a geometric reformulation of the local polytope relaxation with rounding to an integral solution. A key ingredient are local `//Wasserstein messages//' that couple local assignment measures along edges. | ||
| - | * [[https://epubs.siam.org/doi/abs/10.1137/17M1150669|SIAM J. on Imaging Science, 11/2 (2018) 1317--1362]] | + | * [[https://epubs.siam.org/doi/abs/10.1137/17M1150669|Image Labeling Based on Graphical Models Using Wasserstein Messages and Geometric Assignment, SIAM J. on Imaging Science, 11/2 (2018) 1317--1362]] |
| Kick-off paper that introduces the basic approach: | Kick-off paper that introduces the basic approach: | ||
| - | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Astroem2017.pdf|J. Math. Imag. Vision 58/2 (2017) 211--238]] | + | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Astroem2017.pdf|Image Labeling by Assignment., J. Math. Imag. Vision 58/2 (2017) 211--238]] |
| * [[http://www-rech.telecom-lille.fr/diff-cv2016/|Proceedings DIFF-CVML'16; Grenander best paper award]] | * [[http://www-rech.telecom-lille.fr/diff-cv2016/|Proceedings DIFF-CVML'16; Grenander best paper award]] | ||
| * [[https://ipa.iwr.uni-heidelberg.de/dokuwiki/Papers/Astroem2016d.pdf|Proceedings ECCV'16]] | * [[https://ipa.iwr.uni-heidelberg.de/dokuwiki/Papers/Astroem2016d.pdf|Proceedings ECCV'16]] | ||