no code implementations • 13 Feb 2024 • Florian Beier, Hancheng Bi, Clément Sarrazin, Bernhard Schmitzer, Gabriele Steidl
In this paper, we are concerned with estimating the joint probability of random variables $X$ and $Y$, given $N$ independent observation blocks $(\boldsymbol{x}^i,\boldsymbol{y}^i)$, $i=1,\ldots, N$, each of $M$ samples $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M$, where $\sigma^i$ denotes an unknown permutation of i. i. d.
no code implementations • 14 Nov 2023 • Keaton Hamm, Caroline Moosmüller, Bernhard Schmitzer, Matthew Thorpe
This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures on a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $W$.
no code implementations • 9 Mar 2021 • Mauro Bonafini, Massimo Fornasier, Bernhard Schmitzer
We prove convergence of minimizing solutions obtained from a finite number of observations to a mean field limit and the minimal value provides a quantitative error bound on the data-driven evolutions.
Optimization and Control
1 code implementation • 17 Feb 2021 • Tianji Cai, Junyi Cheng, Bernhard Schmitzer, Matthew Thorpe
Working with the local linearization and the corresponding embeddings allows for the advantages of the Euclidean setting, such as faster computations and a plethora of data analysis tools, whilst still enjoying approximately the descriptive power of the Hellinger--Kantorovich metric.
Optimization and Control
no code implementations • 20 Feb 2019 • Bernhard Schmitzer, Klaus P. Schäfers, Benedikt Wirth
In contrast to conventional PET reconstruction our method combines the information from all detected events not only to reconstruct the dynamic evolution of the radionuclide distribution, but also to improve the reconstruction at each single time point by enforcing temporal consistency.
no code implementations • 8 Jan 2017 • Bernhard Schmitzer, Benedikt Wirth
We propose a unifying framework for generalising the Wasserstein-1 metric to a discrepancy measure between nonnegative measures of different mass.
4 code implementations • 20 Oct 2016 • Bernhard Schmitzer
Scaling algorithms for entropic transport-type problems have become a very popular numerical method, encompassing Wasserstein barycenters, multi-marginal problems, gradient flows and unbalanced transport.
Optimization and Control Computational Engineering, Finance, and Science Numerical Analysis
3 code implementations • 20 Jul 2016 • Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard
This article introduces a new class of fast algorithms to approximate variational problems involving unbalanced optimal transport.
Optimization and Control 65K10
no code implementations • 16 Mar 2016 • Freddie Åström, Stefania Petra, Bernhard Schmitzer, Christoph Schnörr
We introduce a novel geometric approach to the image labeling problem.
1 code implementation • 21 Aug 2015 • Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, François-Xavier Vialard
These distances are defined by two equivalent alternative formulations: (i) a "fluid dynamic" formulation defining the distance as a geodesic distance over the space of measures (ii) a static "Kantorovich" formulation where the distance is the minimum of an optimization program over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures.
Optimization and Control
1 code implementation • 22 Jun 2015 • Lenaic Chizat, Bernhard Schmitzer, Gabriel Peyré, François-Xavier Vialard
This metric interpolates between the quadratic Wasserstein and the Fisher-Rao metrics and generalizes optimal transport to measures with different masses.
Analysis of PDEs
no code implementations • 15 Jul 2014 • Bernhard Schmitzer, Christoph Schnörr
While the overall functional is non-convex, non-convexity is confined to a low-dimensional variable.
no code implementations • 9 Sep 2013 • Bernhard Schmitzer, Christoph Schnörr
Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions.