@inproceedings{varifold,
  title={A general framework for curve and surface comparison and registration with oriented varifolds},
  author={Kaltenmark, Irene and Charlier, Benjamin and Charon, Nicolas},
  booktitle={Computer Vision and Pattern Recognition (CVPR)},
  year={2017}
}
@InProceedings{genevay,
  title = 	 {Learning Generative Models with Sinkhorn Divergences},
  author = 	 {Aude Genevay and Gabriel Peyre and Marco Cuturi},
  booktitle = 	 {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics},
  pages = 	 {1608--1617},
  year = 	 {2018},
  editor = 	 {Amos Storkey and Fernando Perez-Cruz},
  volume = 	 {84},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {09--11 Apr},
  publisher = 	 {PMLR},
}

@article{salimans2018improving,
  title={Improving {GANs} using optimal transport},
  author={Salimans, Tim and Zhang, Han and Radford, Alec and Metaxas, Dimitris},
  journal={arXiv preprint arXiv:1803.05573},
  year={2018}
}


@article{SanjabiSinkhGAN,
    title = "On the Convergence and Robustness of Training {GANs} with Regularized Optimal Transport", 
    author = "Maziar Sanjabi and Jimmy Ba and Meisam Razaviyayn and Jason D. Lee", 
  journal={arXiv preprint arXiv:1802.08249},
  year={2018}
}
@inproceedings{miccai,
  title={Optimal Transport for Diffeomorphic Registration},
  author={Feydy, Jean and Charlier, Benjamin and Vialard, Fran{\c{c}}ois-Xavier and Peyr{\'e}, Gabriel},
  booktitle={International Conference on Medical Image Computing and Computer-Assisted Intervention},
  pages={291--299},
  year={2017},
  organization={Springer}
}
@inproceedings{cuturi,
  title={Sinkhorn distances: Lightspeed computation of optimal transport},
  author={Cuturi, Marco},
  booktitle={Advances in neural information processing systems},
  pages={2292--2300},
  year={2013}
}
@article{peyre,
  title={Computational Optimal Transport},
  author={Peyr{\'e}, Gabriel and Cuturi, Marco},
  journal={arXiv preprint arXiv:1803.00567},
  year={2018}
}
@inproceedings{hausdorff,
  title={Mesh: Measuring errors between surfaces using the hausdorff distance},
  author={Aspert, Nicolas and Santa-Cruz, Diego and Ebrahimi, Touradj},
  booktitle={Multimedia and Expo, 2002. ICME'02. Proceedings. 2002 IEEE International Conference on},
  volume={1},
  pages={705--708},
  year={2002},
  organization={IEEE}
}
@article{energy,
  title={Energy statistics: A class of statistics based on distances},
  author={Sz{\'e}kely, G{\'a}bor J and Rizzo, Maria L},
  journal={Journal of statistical planning and inference},
  volume={143},
  number={8},
  pages={1249--1272},
  year={2013},
  publisher={Elsevier}
}
@article{spectral,
  title={Spectral log-demons: diffeomorphic image registration with very large deformations},
  author={Lombaert, Herve and Grady, Leo and Pennec, Xavier and Ayache, Nicholas and Cheriet, Farida},
  journal={International journal of computer vision},
  volume={107},
  number={3},
  pages={254--271},
  year={2014},
  publisher={Springer}
}
@misc{FVSP,
  title = {Sinkhorn entropies and divergences},
  author={Feydy, Jean and S\'ejourn\'e, Thibault and Vialard, Fran\c{c}ois-Xavier and Amari, Shun-ichi and Trouv\'{e}, Alain and Peyr\'{e}, Gabriel},
  howpublished = {to be published in July~2018},
}
@article{pytorch,
  title={Automatic differentiation in PyTorch},
  author={Paszke, Adam and Gross, Sam and Chintala, Soumith and Chanan, Gregory and Yang, Edward and DeVito, Zachary and Lin, Zeming and Desmaison, Alban and Antiga, Luca and Lerer, Adam},
  year={2017}
}
@article{slicer,
  title={3D Slicer as an image computing platform for the Quantitative Imaging Network},
  author={Fedorov, Andriy and Beichel, Reinhard and Kalpathy-Cramer, Jayashree and Finet, Julien and others},
  journal={Magnetic resonance imaging},
  volume={30},
  number={9},
  pages={1323--1341},
  year={2012},
  publisher={Elsevier}
}
@article{lenaic,
  title={Unbalanced optimal transport: Dynamic and Kantorovich formulations},
  author={Chizat, L{\'e}na{\"\i}c and Peyr{\'e}, Gabriel and Schmitzer, Bernhard and Vialard, Fran{\c{c}}ois-Xavier},
  journal={Journal of Functional Analysis},
  volume={274},
  number={11},
  pages={3090--3123},
  year={2018},
  publisher={Elsevier}
}
@Article{RamdasSinkhAsymptotics,
AUTHOR = {Ramdas, Aaditya and Trillos, Nicolas Garcia and Cuturi, Marco},
TITLE = {On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests},
JOURNAL = {Entropy},
VOLUME = {19},
YEAR = {2017},
NUMBER = {2},
}

@article{yuille,
  title={The invisible hand algorithm: Solving the assignment problem with statistical physics},
  author={Kosowsky, JJ and Yuille, Alan L},
  journal={Neural networks},
  volume={7},
  number={3},
  pages={477--490},
  year={1994},
  publisher={Pergamon}
}
@article{bernhard,
  title={Stabilized sparse scaling algorithms for entropy regularized transport problems},
  author={Schmitzer, Bernhard},
  journal={arXiv preprint arXiv:1610.06519},
  year={2016}
}

@misc{keops,
  year={2018},
  title = {Kernel Operations on the GPU, with autodiff, without memory overflows},
  author={Charlier, Benjamin and Feydy, Jean and Glaun\`es, Joan},
  howpublished = {\url{http://www.kernel-operations.io}},
  note = {Accessed: 2019-01-20}
}

@misc{irm_picture,
  year={2019},
  title = {IRM picture},
  author = {Ptrump16}, 
  note = {CC BY-SA 4.0}, 
  howpublished = {\url{https://commons.wikimedia.org/w/index.php?curid=64157788}},
}

@misc{mammography_picture,
  year={2019},
  title={Using AI to predict breast cancer and personalize care
},
  author = {Conner-Simons, Adam and Gordon, Rachel},
  note = {MIT CSAIL},
  howpublished = {\url{http://news.mit.edu/2019/using-ai-predict-breast-cancer-and-personalize-care-0507}},
}


@article {dmasif,
	author = {Sverrisson, Freyr and Feydy, Jean and Correia, Bruno E. and Bronstein, Michael M.},
	title = {Fast end-to-end learning on protein surfaces},
	elocation-id = {2020.12.28.424589},
	year = {2020},
	doi = {10.1101/2020.12.28.424589},
	publisher = {Cold Spring Harbor Laboratory},
	abstract = {Proteins{\textquoteright} biological functions are defined by the geometric and chemical structure of their 3D molecular surfaces. Recent works have shown that geometric deep learning can be used on mesh-based representations of proteins to identify potential functional sites, such as binding targets for potential drugs. Unfortunately though, the use of meshes as the underlying representation for protein structure has multiple drawbacks including the need to pre-compute the input features and mesh connectivities. This becomes a bottleneck for many important tasks in protein science.In this paper, we present a new framework for deep learning on protein structures that addresses these limitations. Among the key advantages of our method are the computation and sampling of the molecular surface on-the-fly from the underlying atomic point cloud and a novel efficient geometric convolutional layer. As a result, we are able to process large collections of proteins in an end-to-end fashion, taking as the sole input the raw 3D coordinates and chemical types of their atoms, eliminating the need for any hand-crafted pre-computed features.To showcase the performance of our approach, we test it on two tasks in the field of protein structural bioinformatics: the identification of interaction sites and the prediction of protein-protein interactions. On both tasks, we achieve state-of-the-art performance with much faster run times and fewer parameters than previous models. These results will considerably ease the deployment of deep learning methods in protein science and open the door for end-to-end differentiable approaches in protein modeling tasks such as function prediction and design.Competing Interest StatementThe authors have declared no competing interest.},
	URL = {https://www.biorxiv.org/content/early/2020/12/29/2020.12.28.424589},
	eprint = {https://www.biorxiv.org/content/early/2020/12/29/2020.12.28.424589.full.pdf},
	journal = {bioRxiv}
}



@article{agueh_2011,
author = {M. Agueh and G. Carlier},
title = {Barycenters in the {W}asserstein Space},
journal = {SIAM Journal on Mathematical Analysis},
volume = {43},
number = {2},
pages = {904--924},
year = {2011},
}

@misc{bus_picture,
  year={2018},
  title={Object Detection with 10 lines of code},
  author = {Olafenwa, Moses},
  note = {Towards Data Science},
  howpublished = {\url{https://towardsdatascience.com/object-detection-with-10-lines-of-code-d6cb4d86f606}},
}



@article{ecabert2011segmentation,
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  journal={Medical image analysis},
  volume={15},
  number={6},
  pages={863--876},
  year={2011},
  publisher={Elsevier}
}


@article{ledig2018structural,
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  journal={Scientific reports},
  volume={8},
  number={1},
  pages={11258},
  year={2018},
  publisher={Nature Publishing Group}
}



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  volume={94},
  pages={173--187},
  year={2014},
  publisher={Elsevier}
}


@article{wei2017mneuron,
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  author={Wei, Donglai and Zhou, Bolei and Torralba, Antonio and Freeman, William T},
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  year={2017}
}


@article{mallat2016understanding,
  title={Understanding deep convolutional networks},
  author={Mallat, St{\'e}phane},
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  volume={374},
  number={2065},
  pages={20150203},
  year={2016},
  publisher={The Royal Society Publishing}
}


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  pages={162--170},
  year={2011},
  organization={sn}
}


@article{nikulin2016exploring,
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  author={Nikulin, Yaroslav and Novak, Roman},
  journal={arXiv preprint arXiv:1602.07188},
  year={2016}
}



@inproceedings{ronneberger2015u,
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  booktitle={International Conference on Medical image computing and computer-assisted intervention},
  pages={234--241},
  year={2015},
  organization={Springer}
}



@inproceedings{stanford_dragon,
  title={A volumetric method for building complex models from range images},
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}

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  title={Transport par diff{\'e}omorphismes de points, de mesures et de courants pour la comparaison de formes et l’anatomie num{\'e}rique},
  author={Glaunes, Joan},
  journal={These de sciences, Universit{\'e} Paris},
  volume={13},
  year={2005}
}



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  volume={38},
  number={1},
  pages={95--113},
  year={2007},
  publisher={Elsevier}
}


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    AUTHOR = {Brenier, Y.},
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    VOLUME = {44},
      YEAR = {1991},
    NUMBER = {4},
     PAGES = {375--417},
}



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  title={On the translocation of masses},
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  pages={199--201},
  year={1942}
}


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  title={The {Hungarian} method for the assignment problem},
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  pages={83--97},
  year={1955},
  publisher={Wiley Online Library}
}



@article{bertsekas79auction,
  title={A Distributed Algorithm for the Assignment Problem},
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  year={1979}
}



@article{kosowsky1994invisible,
  title={The invisible hand algorithm: Solving the assignment problem with statistical physics},
  author={Kosowsky, Jeffrey J and Yuille, Alan L},
  journal={Neural networks},
  volume={7},
  number={3},
  pages={477--490},
  year={1994},
  publisher={Elsevier}
}


@article{gold1998new,
  title={New algorithms for 2D and 3D point matching: Pose estimation and correspondence},
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  journal={Pattern recognition},
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  year={1998},
  publisher={Elsevier}
}


@inproceedings{TPSRPM,
  title={A new algorithm for non-rigid point matching},
  author={Chui, Haili and Rangarajan, Anand},
  booktitle={Computer Vision and Pattern Recognition, 2000. Proceedings. IEEE Conference on},
  volume={2},
  pages={44--51},
  year={2000},
  organization={IEEE}
}


@inproceedings{CuturiSinkhorn,
  title={Sinkhorn distances: Lightspeed computation of optimal transport},
  author={Cuturi, Marco},
  booktitle={Advances in Neural Information Processing Systems},
  pages={2292--2300},
  year={2013}
}


@inproceedings{merigot2011multiscale,
  title={A multiscale approach to optimal transport},
  author={M{\'e}rigot, Quentin},
  booktitle={Computer Graphics Forum},
  volume={30},
  pages={1583--1592},
  year={2011},
  organization={Wiley Online Library}
}


@article{levy2015numerical,
  title={A numerical algorithm for L2 semi-discrete optimal transport in 3D},
  author={L{\'e}vy, Bruno},
  journal={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={49},
  number={6},
  pages={1693--1715},
  year={2015},
  publisher={EDP Sciences}
}



@article{schmitzer2019stabilized,
  title={Stabilized sparse scaling algorithms for entropy regularized transport problems},
  author={Schmitzer, Bernhard},
  journal={SIAM Journal on Scientific Computing},
  volume={41},
  number={3},
  pages={A1443--A1481},
  year={2019},
  publisher={SIAM}
}

@article{feydy2018interpolating,
  title={Interpolating between optimal transport and {MMD} using {Sinkhorn} divergences},
  author={Feydy, Jean and S{\'e}journ{\'e}, Thibault and Vialard, Fran{\c{c}}ois-Xavier and Amari, Shun-Ichi and Trouv{\'e}, Alain and Peyr{\'e}, Gabriel},
  journal={arXiv preprint arXiv:1810.08278},
  year={2018}
}


@inproceedings{degond2017alignment,
  title={Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations},
  author={Degond, Pierre and Frouvelle, Amic and Merino-Aceituno, Sara and Trescases, Ariane},
  booktitle={International workshop on Stochastic Dynamics out of Equilibrium},
  pages={28--66},
  year={2017},
  organization={Springer}
}

@article{degond2008continuum,
  title={Continuum limit of self-driven particles with orientation interaction},
  author={Degond, Pierre and Motsch, S{\'e}bastien},
  journal={Mathematical Models and Methods in Applied Sciences},
  volume={18},
  number={supp01},
  pages={1193--1215},
  year={2008},
  publisher={World Scientific}
}

@article{CMC,
  title={Collective Proposal Distributions for Nonlinear {MCMC} samplers: Mean-Field Theory and Fast Implementation},
  author={Clart{\'e}, Gr{\'e}goire and Diez, Antoine and Feydy, Jean},
  journal={arXiv preprint arXiv:1909.08988},
  year={2019}
}

@article{leclercq2018bayesian,
  title={Bayesian optimization for likelihood-free cosmological inference},
  author={Leclercq, Florent},
  journal={Physical Review D},
  volume={98},
  number={6},
  pages={063511},
  year={2018},
  publisher={APS}
}

@misc{scratchinghead,
  title = {Man scratching head},
  author={Dukesy68},
  howpublished = {Wikipedia, CC BY-SA v4.0},
}

@misc{mercator,
  title = {Earth maps},
  author={Strebe},
  howpublished = {Wikipedia, CC BY-SA v3.0},
}

@misc{umapprime,
  title = {What do numbers look like?},
  author={John Williamson},
  howpublished = {https://johnhw.github.io/umap\_primes/index.md.html},
}

@misc{celestialorbit,
  title = {Diagram illustrating and explaining various terms in relation to Orbits of Celestial bodies.},
  author={Lasunncty},
  howpublished = {Wikipedia, CC BY-SA v3.0}
}

@article{yaroslav,
  title={Exploring the neural algorithm of artistic style},
  author={Nikulin, Yaroslav and Novak, Roman},
  journal={arXiv preprint arXiv:1602.07188},
  year={2016}
}
@misc{nnviz,
  title = {mNeuron: A Matlab Plugin to Visualize Neurons from Deep Models},
  howpublished = {\url{http://vision03.csail.mit.edu/cnn_art/}},
  author={Wei, Donglai and Zhou, Bolei and Torralba, Antonio and Freeman, William},
  note = {Accessed: 2018-05-10}
}
@inproceedings{fooledai,
  title={Deep neural networks are easily fooled: High confidence predictions for unrecognizable images},
  author={Nguyen, Anh and Yosinski, Jason and Clune, Jeff},
  booktitle={Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition},
  pages={427--436},
  year={2015}
}
@article{mallatcnn,
  title={Understanding deep convolutional networks},
  author={Mallat, St{\'e}phane},
  journal={Phil. Trans. R. Soc. A},
  volume={374},
  number={2065},
  pages={20150203},
  year={2016},
  publisher={The Royal Society}
}
@misc{wikijpeg,
  title = {JPEG 2000 --- Wikipedia, The Free Encyclopedia},
  howpublished = {\url{http://en.wikipedia.org/wiki/JPEG_2000}},
  note = {Accessed: 2018-05-10},
  author={Damato, Allessio}
}
@inproceedings{hollandais,
  title={Speed sign detection and recognition by convolutional neural networks},
  author={Peemen, Maurice and Mesman, Bart and Corporaal, Henk},
  booktitle={Proceedings of the 8th International Automotive Congress},
  pages={162--170},
  year={2011}
}
@article{heart_raw,
  title={Segmentation of the heart and great vessels in CT images using
a model-based adaptation framework},
  author={Ecabert, Olivier and Peters, Jochen and Walker, Matthew},
  journal={Medical Image Analysis},
  number={15},
  pages={863-876},
  year={2011},
  publisher={Elsevier}
}
@article{heart_pop,
  title={A statistical model for quantification and prediction of cardiac remodelling: Application to tetralogy of fallot},
  author={Mansi, Tomaso},
  journal={IEEE transactions on medical imaging},
  year={2011},
}


@article{klingenberg2015analyzing,
  title={Analyzing fluctuating asymmetry with geometric morphometrics: concepts, methods, and applications},
  author={Klingenberg, Christian Peter},
  journal={Symmetry},
  volume={7},
  number={2},
  pages={843--934},
  year={2015},
  publisher={Multidisciplinary Digital Publishing Institute}
}



@inproceedings{kilian2007geometric,
  title={Geometric modeling in shape space},
  author={Kilian, Martin and Mitra, Niloy J and Pottmann, Helmut},
  booktitle={ACM Transactions on Graphics (TOG)},
  volume={26},
  pages={64},
  year={2007},
  organization={ACM}
}


@article{von2016optimized,
  title={Optimized subspaces for deformation-based modeling and shape interpolation},
  author={von Radziewsky, Philipp and Eisemann, Elmar and Seidel, Hans-Peter and Hildebrandt, Klaus},
  journal={Computers \& Graphics},
  volume={58},
  pages={128--138},
  year={2016},
  publisher={Elsevier}
}


@misc{withings,
  title = {About body mass index},
  howpublished = {\url{http://en.wikipedia.org/wiki/JPEG_2000www.withings.com}},
  author={Withings},
}

@article{fischl2012freesurfer,
  title={FreeSurfer},
  author={Fischl, Bruce},
  journal={Neuroimage},
  volume={62},
  number={2},
  pages={774--781},
  year={2012},
  publisher={Elsevier}
}

@article{peyre2011numerical,
  title={The numerical tours of signal processing-advanced computational signal and image processing},
  author={Peyr{\'e}, Gabriel},
  journal={IEEE Computing in Science and Engineering},
  volume={13},
  number={4},
  pages={94--97},
  year={2011}
}


@misc{konigsberg,
  title = {Bridges of Königsberg},
  howpublished = {https://www.britannica.com/topic/leisure/media/1/321794/68671},
  author={Encyclopædia Britannica},
}

@misc{albert_chern,
  title = {Sudanese Möbius band},
  howpublished = {https://cseweb.ucsd.edu/~alchern/gallery/},
  author={Albert Chern},
}

@misc{hyperbolic_salad,
  title = {Tessellations of the Hyperbolic Plane and M.C. Escher},
  howpublished = {https://web.colby.edu/thegeometricviewpoint/author/aredhunt/},
  author={Allyson Redhunt},
}

@article{sharp2021diffusion,
  author = {Sharp, Nicholas and Attaiki, Souhaib and Crane, Keenan and Ovsjanikov, Maks},
  title = {{DiffusionNet}: Discretization Agnostic Learning on Surfaces},
  journal = {ACM Trans. Graph.},
  volume = {01},
  number = {1},
  year = {2022},
  publisher = {ACM},
  address = {New York, NY, USA},
}

@article{topping2021understanding,
  title={Understanding over-squashing and bottlenecks on graphs via curvature},
  author={Topping, Jake and Di Giovanni, Francesco and Chamberlain, Benjamin Paul and Dong, Xiaowen and Bronstein, Michael M},
  journal={arXiv preprint arXiv:2111.14522},
  year={2021}
}


@misc{spurious_correlations,
  title = {Spurious correlations},
  howpublished = {https://www.tylervigen.com/spurious-correlations},
  author={Tyler Vigen},
}



@misc{spaghetti,
  year={2011},
  title = {Spaghetti picture},
  author = {Matsuyuki}, 
  note = {CC BY-SA 2.0}, 
  howpublished = {\url{https://commons.wikimedia.org/wiki/File:Aglio_e_olio.jpg}},
}


@article{song2022generation,
  title={Generation of tubular and membranous shape textures with curvature functionals},
  author={Song, Anna},
  journal={Journal of Mathematical Imaging and Vision},
  volume={64},
  number={1},
  pages={17--40},
  year={2022},
  publisher={Springer}
}

@incollection{gromov1987hyperbolic,
  title={Hyperbolic groups},
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@article{sreejith2016forman,
  title={Forman curvature for complex networks},
  author={Sreejith, RP and Mohanraj, Karthikeyan and Jost, J{\"u}rgen and Saucan, Emil and Samal, Areejit},
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  volume={2016},
  number={6},
  pages={063206},
  year={2016},
  publisher={IOP Publishing}
}
@article{samal2018comparative,
  title={Comparative analysis of two discretizations of {Ricci} curvature for complex networks},
  author={Samal, Areejit and Sreejith, RP and Gu, Jiao and Liu, Shiping and Saucan, Emil and Jost, J{\"u}rgen},
  journal={Scientific reports},
  volume={8},
  number={1},
  pages={1--16},
  year={2018},
  publisher={Nature Publishing Group}
}

@article{wang2022applying,
  title={Applying {Ollivier}-{Ricci} curvature to indicate the mismatch of travel demand and supply in urban transit network},
  author={Wang, Yaoli and Huang, Zhou and Yin, Ganmin and Li, Haifeng and Yang, Liu and Su, Yuelong and Liu, Yu and Shan, Xv},
  journal={International Journal of Applied Earth Observation and Geoinformation},
  volume={106},
  pages={102666},
  year={2022},
  publisher={Elsevier}
}


@article{bronstein2021geometric,
  title={Geometric deep learning: Grids, groups, graphs, geodesics, and gauges},
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