Author(s):
- Miolane, Nina
- Mathe, Johan
- Donnat, Claire
- Jorda, Mikael
- Pennec, Xavier
Abstract:
We introduce geomstats, a python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. We provide efficient and extensively unit-tested implementations of these manifolds, together with useful Riemannian metrics and associated Exponential and Logarithm maps. The corresponding geodesic distances provide a range of intuitive choices of Machine Learning loss functions. We also give the corresponding Riemannian gradients. The operations implemented in geomstats are available with different computing backends such as numpy, tensorflow and keras. We have enabled GPU implementation and integrated geomstats manifold computations into keras deep learning framework. This paper also presents a review of manifolds in machine learning and an overview of the geomstats package with examples demonstrating its use for efficient and user-friendly Riemannian geometry.
Document:
https://arxiv.org/abs/1805.08308
References:
[1]Angulo, J., Velasco-Forero, S.: Morphological processing of univariate Gaussian distribution-valuedimages based on Poincaré upper-half plane representation. In: Nielsen, F. (ed.) Geometric Theory ofInformation, pp. 331–366. Signals and Communication Technology, Springer International Publishing(may 2014)
[2]Arnaudon, M., Barbaresco, F., Yang, L.: Riemannian Medians and Means With Applications to RadarSignal Processing. IEEE Journal of Selected Topics in Signal Processing 7(4), 595–604 (aug 2013)
[3]Arsigny, V.: Processing Data in {L}ie Groups: An Algebraic Approach. Application to Non-LinearRegistration and Diffusion Tensor MRI. Thèse de sciences (phd thesis), École polytechnique (nov 2006)
[4]Boisvert, J., Cheriet, F., Pennec, X., Labelle, H., Ayache, N.: Articulated Spine Models for 3D Reconstruc-tion from Partial Radiographic Data. IEEE Transactions on Bio-Medical Engineering 55(11), 2565–2574(nov 2008)
[5]Bronstein, M.M., Bruna, J., LeCun, Y., Szlam, A., Vandergheynst, P.: Geometric Deep Learning: Goingbeyond Euclidean data. IEEE Signal Processing Magazine 34(4), 18–42 (jul 2017)
[6]Boumal, N., Bamdev, M., Absil, P.-A., Sepulchre, R.: Manopt, a Matlab Toolbox for Optimization onManifolds. Journal of Machine Learning Research. 15, 1455—1459 (2014)
[7]Censi, A.: Pygeometry: library for handling various differentiable manifolds. (2010),https://github.com/AndreaCensi/geometry
[8]Cherian, A., Sra, S.: Positive Definite Matrices: Data Representation and Applications to Computer Vision.In: Algorithmic Advances in Riemannian Geometry and Applications. Springer (2016)
[9]Dodero, L., Minh, H.Q., Biagio, M.S., Murino, V., Sona, D.: Kernel-based classification for brainconnectivity graphs on the Riemannian manifold of positive definite matrices. In: 2015 IEEE 12thInternational Symposium on Biomedical Imaging (ISBI). pp. 42–45 (apr 2015)
[10]Faraki, M., Harandi, M.T., Porikli, F.: Material Classification on Symmetric Positive Definite Manifolds.In: 2015 IEEE Winter Conference on Applications of Computer Vision. pp. 749–756 (jan 2015)
[11]Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statisticsof shape. IEEE transactions on medical imaging 23(8), 995–1005 (2004)
[12]Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’institutHenri Poincaré 10(4), 215–310 (1948)
[13]Gao, Z., Wu, Y., Bu, X., Jia, Y.: Learning a Robust Representation via a Deep Network on SymmetricPositive Definite Manifolds. CoRR abs/1711.06540 (2017),http://arxiv.org/abs/1711.06540
[14]Grenander, U., Miller, M.: Pattern Theory: From Representation to Inference. Oxford University Press,Inc., New York, NY, USA (2007)
[15]Harandi, M.T., Hartley, R.I., Lovell, B.C., Sanderson, C.: Sparse Coding on Symmetric Positive DefiniteManifolds using Bregman Divergences. CoRR abs/1409.0083 (2014),http://arxiv.org/abs/1409.0083
[16]He, K., Zhang, X., Ren, S., Sun, J.: Deep Residual Learning for Image Recognition. CoRR abs/1512.03385(2015),http://arxiv.org/abs/1512.03385
[17]Hong, J., Vicory, J., Schulz, J., Styner, M., Marron, J.S., Pizer, S.: Non-Euclidean Classification ofMedically Imaged Objects via s-reps. Med Image Anal 31, 37–45 (2016)9
[18]Hou, B., Miolane, N., Khanal, B., Lee, M., Alansary, A., McDonagh, S., Hajnal, J., Ruecket, D., Glocker,B., Kainz, B.: Deep Pose Estimation for Image-Based Registration. Submitteed to MICCAI 2018. (2018)
[19]Huang, L., Liu, X., Lang, B., Yu, A.W., Li, B.: Orthogonal Weight Normalization: Solution to Optimizationover Multiple Dependent Stiefel Manifolds in Deep Neural Networks. CoRR abs/1709.06079 (2017)
[20]Ingalhalikar, M., Smith, A., Parker, D., Satterthwaite, T.D., Elliott, M.A., Ruparel, K., Hakonarson, H., Gur,R.E., Gur, R.C., Verma, R.: Sex differences in the structural connectome of the human brain. Proceedingsof the National Academy of Sciences 111(2), 823–828 (2014)
[21]Kendall, A., Grimes, M., Cipolla, R.: Convolutional networks for real-time 6-DOF camera relocalization.CoRR abs/1505.07427 (2015),http://arxiv.org/abs/1505.07427
[22] Kendall, D.G.: A Survey of the Statistical Theory of Shape. Statistical Science 4(2), pp. 87–99 (1989)
[23]Kent, J.T., Hamelryck, T.: Using the Fisher-Bingham distribution in stochastic models for protein structure.Quantitative Biology, Shape Analysis, and Wavelets pp. 57–60 (2005)
[24]Krizhevsky, A., Nair, V., Hinton, G.: CIFAR-10 (Canadian Institute for Advanced Research). bar (2010),http://www.cs.toronto.edu/{~}kriz/cifar.html
[25]Kühnel, L., Sommer, S.: Computational Anatomy in Theano. CoRR abs/1706.07690 (2017),http://arxiv.org/abs/1706.07690
[26]LeCun, Y., Cortes, C.: {MNIST} handwritten digit database. foo (2010),http://yann.lecun.com/exdb/mnist/
[27]Liu, W., Zhang, Y.M., Li, X., Yu, Z., Dai, B., Zhao, T., Song, L.: Deep Hyperspherical Learning. In:Advances in Neural Information Processing Systems. pp. 3953–3963 (2017)
[28]Mardia, K.V., Jupp, P.E.: Directional statistics. Wiley series in probability and statistics, Wiley (2000),https://books.google.com/books?id=zjPvAAAAMAAJ
[29]Mart\’\in ̃Abadi, Ashish ̃Agarwal, Paul ̃Barham, Eugene ̃Brevdo, Zhifeng ̃Chen, Craig ̃Citro,Greg ̃S. ̃Corrado, Andy ̃Davis, Jeffrey ̃Dean, Matthieu ̃Devin, Sanjay ̃Ghemawat, Ian ̃Goodfellow,Andrew ̃Harp, Geoffrey ̃Irving, Michael ̃Isard, Jia, Y., Rafal ̃Jozefowicz, Lukasz ̃Kaiser, Manju-nath ̃Kudlur, Josh ̃Levenberg, Dandelion ̃Mané, Rajat ̃Monga, Sherry ̃Moore, Derek ̃Murray, Chris ̃Olah,Mike ̃Schuster, Jonathon ̃Shlens, Benoit ̃Steiner, Ilya ̃Sutskever, Kunal ̃Talwar, Paul ̃Tucker, Vin-cent ̃Vanhoucke, Vijay ̃Vasudevan, Fernanda ̃Viégas, Oriol ̃Vinyals, Pete ̃Warden, Martin ̃Wattenberg,Martin ̃Wicke, Yuan ̃Yu, Xiaoqiang ̃Zheng: {TensorFlow}: Large-Scale Machine Learning on Heteroge-neous Systems (2015),https://www.tensorflow.org/
[30]Nickel, M., Kiela, D.: Poincaré Embeddings for Learning Hierarchical Representations. In: Guyon, I.,Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., Garnett, R. (eds.) Advances inNeural Information Processing Systems 30, pp. 6338–6347. Curran Associates, Inc. (2017)
[31] Oliphant, T.E.: Guide to NumPy. CreateSpace Independent Publishing Platform, USA, 2nd edn. (2015)
[32]Papadopoulos, F., Kitsak, M., Serrano, M.Á., Boguñá, M., Krioukov, D.: Popularity versus similarity ingrowing networks. Nature 489, 537 EP – (2012),http://dx.doi.org/10.1038/nature11459
[33]Pennec, X.: Intrinsic Statistics on {R}iemannian Manifolds: Basic Tools for Geometric Measurements.Journal of Mathematical Imaging and Vision 25(1), 127–154 (2006)
[34]Pennec, X., Fillard, P., Ayache, N.: A {R}iemannian Framework for Tensor Computing. InternationalJournal of Computer Vision 66(1), 41–66 (jan 2006)
[35] Postnikov, M.: Riemannian Geometry. Encyclopaedia of Mathem. Sciences, Springer (2001)
[36]Schaeben, H.: Towards statistics of crystal orientations in quantitative texture anaylsis. Journal of AppliedCrystallography 26(1), 112–121 (feb 1993),https://doi.org/10.1107/S0021889892009270
[37]Shinohara, Y., Masuko, T., Akamine, M.: Covariance clustering on Riemannian manifolds for acousticmodel compression. In: 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.pp. 4326–4329 (mar 2010)
[38]Sporns, O., Tononi, G., Kötter, R.: The human connectome: a structural description of the human brain.PLoS computational biology 1(4), e42 (2005)10
[39]Townsend, J., Koep, N., Weichwald, S.: Pymanopt: A Python Toolbox for Optimization on Manifoldsusing Automatic Differentiation. Journal of Machine Learning Research 17(137), 1–5 (2016),http://jmlr.org/papers/v17/16-177.html
[40]Wang, J., Zuo, X., Dai, Z., Xia, M., Zhao, Z., Zhao, X., Jia, J., Han, Y., He, Y.: Disrupted functional brainconnectome in individuals at risk for Alzheimer’s disease. Biological psychiatry 73(5), 472–481 (2013)
[41]Yuan, Y., Zhu, H., Lin, W., Marron, J.S.: Local polynomial regression for symmetric positive definitematrices. Journal of the Royal Statistical Society Series B 74(4), 697–719 (2012),https://econpapers.repec.org/RePEc:bla:jorssb:v:74:y:2012:i:4:p:697-71911