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research [2019/10/24 11:37]
ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces]
research [2019/10/24 11:52] (current)
ipa [Variational Image Analysis on Manifolds and Metric Measure Spaces]
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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.
  
-**Mathematical aspects.** The assignment flow evolves non-locally for any data given on a graph. ​Variational ​aspects, extensions to continuous domains ​and scale separation are investigated. A preliminary ​step concerns ​a more classical //​additive//​ variational ​formulation that provides a smooth geometric version of the continuous cut approach.+**Mathematical aspects.** The assignment flow evolves non-locally for any data given on a graph. ​Geometric and variational ​aspects, extensions to continuous domainsscale separation ​and models of knowledge representation across the scales ​are investigated. ​ 
 + 
 +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://​arxiv.org/​abs/​1910.07287|Continuous-Domain Assignment Flow, preprint arXiv:​1910.07287]]. 
 +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]].   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Savarino2019aa.pdf|A Variational Perspective on the Assignment Flow, SSVM 2019]].
  
-**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. ​We currently investigate this approach in connection with more general objective functions+**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, preprintarXiv:​1910.09976]]+  * [[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://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​Huhnerbein2019aa.pdf|Conference version, SSVM 2019]].
  
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 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//. 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, preprintarXiv:​1904.10863]]+  * [[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]].   * [[https://​ipa.math.uni-heidelberg.de/​dokuwiki/​Papers/​gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, GCPR 2018]].