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| ====== Research ====== | ====== Research ====== | ||
| - | ===== Variational Image Analysis on Manifolds and Metric Measure Spaces ===== | + | ===== Assignment Flows: Dynamical Systems on Riemannian Manifolds for Data Analysis ===== |
| - | **Scope.** We exploit basic statistical manifolds to devise variational models of low-level image analysis that exhibit favourable properties in comparison to established convex and non-convex models: smoothness, probabilistic interpretation, efficiently converging parallel and sparse Riemannian numerical updates that scale up to large problem sizes. | + | This project is funded by the DFG within the [[https://www.foundationsofdl.de|Priority Programme on the Theoretical Foundations of Deep Learning]] |
| - | 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. | + | **Scope.** We study product spaces of elementary Riemannian manifolds for the context-sensitive analysis of data observed in any metric space. State spaces interact dynamically by geometric averaging and locally according to the adjacency structure of an underlying graph. The corresponding interaction parameters are learned from data. Geometric integration of the resulting continuous-time flow generates layers of a neural network. Our approach enables to study dynamical relations of inference and learning in neural networks from a geometric viewpoint, along with a probabilistic interpretation of contextual decision making. From the numerical point of view, the approach copes with high dimensions and large problem sizes. |
| - | **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.** Information geometry, coupled and regularized information transport, geometric mechanics on manifolds and variational principles, geometric numerical integration, statistical performance characterization using PAC-Bayesian analysis. |
| - | * [[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.** [[https://ipa.math.uni-heidelberg.de/publications|download]] |
| - | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/gcpr2018.pdf|Unsupervised Label Learning on Manifolds by Spatially Regularized Geometric Assignment, preliminary announcement: GCPR 2018]]. | + | * extension to generative assignment flows for discrete joint distributions (arXiv:2402.07846, 2024) |
| - | This paper sketches a special instance of a more general framework, the //unsupervised assignment flow//, to be introduced in a forthcoming report. | + | * geometric embedding approach to multiple games and populations (arXiv::2401.05918, 2024) |
| - | + | * quantum state assignment flows (Entropy, 2023) | |
| - | 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. | + | * geometric mechanics of assignment flows (Information Geometry, 2023) |
| - | + | * novel PAC-Bayes bound for structured prediction (NeurIPS, 2023) | |
| - | * [[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]] | + | * self-certifying classification by linearized deep assignment flows (PAMM, 2023) |
| - | + | * non-local graph PDE for structured labeling based on assignment flows (SIAM SIIMS, 2023) | |
| - | Kick-off paper that introduces the basic approach: | + | * learning linearized assignment flows (JMIV, 2023) |
| - | + | * convergence and stability of assignment flows (Information Geometry, 2022) | |
| - | * [[https://ipa.math.uni-heidelberg.de/dokuwiki/Papers/Astroem2017.pdf|Image Labeling by Assignment., J. Math. Imag. Vision 58/2 (2017) 211--238]] | + | * continuous-domain assignment flows (Europ. J. Appl. Math., 2021) |
| - | * [[http://www-rech.telecom-lille.fr/diff-cv2016/|Proceedings DIFF-CVML'16; Grenander best paper award]] | + | * order-constrained 3D OCT segmentation using assignment flows (IJCV, 2021) |
| - | * [[https://ipa.iwr.uni-heidelberg.de/dokuwiki/Papers/Astroem2016d.pdf|Proceedings ECCV'16]] | + | * self-assignment flows (SIAM SIIMS, 2020) |
| - | ===== Estimating Vehicle Ego-Motion and Piecewise Planar Scene Structure from Optical Flow in a Continuous Framework ===== | + | * unsupervised assignment flows (JMIV, 2020) |
| - | + | * geometric integration of assignment flows (Inverse Problems, 2020) | |
| - | We propose a variational approach for estimating egomotion and structure of a static scene from a pair of images recorded by a single moving camera. In our approach the scene structure is described by a set of 3D planar surfaces, which are linked to a SLIC superpixel decomposition of the image domain. The continuously parametrized planes are determined along with the extrinsic camera parameters by jointly minimizing a non-convex smooth objective function, that comprises a data term based on the pre-calculated optical flow between the input images and suitable priors on the scene variables. | + | * assignment flows for labeling: introduction (Handbook: Var. Meth. Nonl. Geom. Data, 2020) |
| - | + | * assignment flows for metric data labeling (JMIV, 2017) | |
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| - | **Researchers**: Andreas Neufeld, Johannes Berger, Florian Becker, Frank Lenzen, Christoph Schnörr | + | |
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| - | [[research:hflow:start|Details]] | + | |
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| - | ===== Minimum Energy Filtering on Lie Groups with Application to Structure and Motion Estimation from Monocular Videos ===== | + | |
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| - | We investigate Minimum Energy Filters on Lie Groups in order to reliably estimate camera motion relative to a static scene from noisy data. In addition to properly taking into account the geometry of the state space, we also deal with nonlinearities of the observation equation. A long-term objective concerns the estimation of accelerated camera motion in connection with scene depth from monocular videos. | + | |
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| - | **Researchers**: | + | |
| - | Johannes Berger, Andreas Neufeld, Florian Becker, Frank Lenzen, Christoph Schnörr | + | |
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| - | Details: [[https://ipa.iwr.uni-heidelberg.de/dokuwiki/Papers/Berger2015a.pdf|Paper]] | + | |
| - | ===== Partial Optimality in MAP-MRF ===== | + | |
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| - | We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. For this task we devise a novel pruning strategy that utilizes standard MAP-solvers as subroutine. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method’s runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem. | + | |
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| - | **Researchers**: Paul Swoboda, Bogdan Savchynskyy, Alexander Shekhovtsov, Jörg Hendrik Kappes, Christoph Schnörr\\ | + | |
| - | Details: [[https://ipa.iwr.uni-heidelberg.de/dokuwiki/Papers/Swoboda2016.pdf|Paper]] | + | |