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--- research [2018/10/17 12:17]
+++ research [2019/10/24 11:52]
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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.
-**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.
+**Mathematical aspects.** The assignment flow evolves non-locally for any data given on a graph. Geometric and variational aspects, extensions to continuous domains, scale separation and models of knowledge representation across the scales are investigated.
-  * [[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
+A preliminary extension from graphs to the continuous domain in the `zero-scale limit' (local interaction only) reveals the interplay between the underlying geometry and variational aspects.
-  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, preliminary announcement: GCPR 2018]].
+  * [[https://arxiv.org/abs/1910.07287|Continuous-Domain Assignment Flow, preprint arXiv:1910.07287]].
-This paper sketches a special instance of a more general framework, the //unsupervised assignment flow//, to be introduced in a forthcoming report.
+A a more classical //additive// variational reformulation provides a smooth geometric version of the continuous cut approach.
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Savarino2019aa.pdf|A Variational Perspective on the Assignment Flow, SSVM 2019]].
-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.
+**Parameter learning.** We study how weights for geometric diffusion that parametrize the adaptivity of the assignment flow can be learned from data. Symplectic integration ensures the commutativity of discretisation and optimisation operations. Results reveal the steerability of the assignment flow and its potential for pattern //formation//.
+  * [[https://arxiv.org/abs/1910.09976|Learning Adaptive Regularization for Image Labeling Using Geometric Assignment, preprint arXiv:1910.09976]]
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Huhnerbein2019aa.pdf|Conference version, SSVM 2019]].
-  * [[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]]
+**Unsupervised label learning.** Our recent work concerns the emergence of labels in a completely unsupervised way by data //self//-assignment. The resulting unsupervised assignment flow has connections to low-rank matrix factorisation and discrete optimal mass transport that are explored in our current work.
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Zisler2019aa.pdf|Unsupervised Labeling by Geometric and Spatially Regularized Self-Assignment, SSVM 2019]].
-Kick-off paper that introduces the basic approach:
+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. The following papers introduce the corresponding //unsupervised assignment flow//.
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Zern2019aa.pdf|Unsupervised Assignment Flow: Label Learning on Feature Manifolds by Spatially Regularized Geometric Assignment, preprint arXiv:1904.10863]]
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, GCPR 2018]].
+**Geometric numerical integration.** We conducted 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://iopscience.iop.org/article/10.1088/1361-6420/ab2772|Geometric Numerical Integration of the Assignment Flow, Inverse Problems, 2019]]
+  * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Zeilmann2018aa.pdf|preprint: arXiv:1810.06970]]
+**Evaluation of discrete graphical models.** 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|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:
   * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Astroem2017.pdf|Image Labeling by Assignment., J. Math. Imag. Vision 58/2 (2017) 211--238]]