Abstract
In this paper, we propose a practical approach for building \({{\,\mathrm{SU}\,}}(1,1)\) equivariant neural networks to process data with support within the Poincaré disk \(\mathbb {D}\). By leveraging on the transitive action of \({{\,\mathrm{SU}\,}}(1,1)\) on \(\mathbb {D}\), we define an equivariant convolution operator on \(\mathbb {D}\) and introduce a Helgason-Fourier analysis approach for its computation, that we compare with a conditional Monte-Carlo method. Finally, we illustrate the performance of such neural networks from both accuracy and robustness standpoints through the example of Toeplitz Hermitian Positive Definite (THPD) matrix classification in the context of radar clutter identification from the corresponding Doppler signals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barbaresco, F.: Radar micro-doppler signal encoding in Siegel unit poly-disk for machine learning in Fisher metric space. In: 2018 19th International Radar Symposium (IRS), pp. 1–10 (2018). https://doi.org/10.23919/IRS.2018.8448021
Bekkers, E.J.: B-spline CNNs on lie groups. In: International Conference on Learning Representations (2020). https://openreview.net/forum?id=H1gBhkBFDH
Bekkers, E.J., Lafarge, M.W., Veta, M., Eppenhof, K.A., Pluim, J.P., Duits, R.: Roto-translation covariant convolutional networks for medical image analysis (2018)
Borowka, S., Heinrich, G., Jahn, S., Jones, S., Kerner, M., Schlenk, J.: A GPU compatible Quasi-Monte Carlo integrator interfaced to pySecDec. Comput. Phys. Commun. 240, 120–137 (2019). https://doi.org/10.1016/j.cpc.2019.02.015
Brooks, D., Schwander, O., Barbaresco, F., Schneider, J., Cord, M.: A Hermitian positive definite neural network for micro-doppler complex covariance processing. In: 2019 International Radar Conference (RADAR), pp. 1–6 (2019). https://doi.org/10.1109/RADAR41533.2019.171277
Cabanes, Y., Barbaresco, F., Arnaudon, M., Bigot, J.: Toeplitz Hermitian positive definite matrix machine learning based on Fisher metric. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2019. LNCS, vol. 11712, pp. 261–270. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26980-7_27
Chakraborty, R., Banerjee, M., Vemuri, B.C.: H-CNNs: convolutional neural networks for Riemannian homogeneous spaces. arXiv preprint arXiv:1805.05487 1 (2018)
Cohen, T., Welling, M.: Group equivariant convolutional networks. In: Balcan, M.F., Weinberger, K.Q. (eds.) Proceedings of The 33rd International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 48, pp. 2990–2999. PMLR, New York, 20–22 June 2016. http://proceedings.mlr.press/v48/cohenc16.html
Cohen, T.S., Geiger, M., Köhler, J., Welling, M.: Spherical CNNs. CoRR abs/1801.10130 (2018)
Cohen, T.S., Geiger, M., Weiler, M.: A general theory of equivariant CNNs on homogeneous spaces. In: Wallach, H., Larochelle, H., Beygelzimer, A., Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 32, pp. 9145–9156. Curran Associates, Inc. (2019)
del Olmo, M.A., Gazeau, J.P.: Covariant integral quantization of the unit disk. J. Math. Phys. 61(2) (2020). https://doi.org/10.1063/1.5128066
Finzi, M., Stanton, S., Izmailov, P., Wilson, A.G.: Generalizing convolutional neural networks for equivariance to lie groups on arbitrary continuous data (2020)
Helgason, S.: Groups and geometric analysis (1984)
Kondor, R., Trivedi, S.: On the generalization of equivariance and convolution in neural networks to the action of compact groups. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 2747–2755. PMLR, Stockholmsmässan, 10–15 July 2018. http://proceedings.mlr.press/v80/kondor18a.html
Lecun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998). https://doi.org/10.1109/5.726791
Rieffel, M.A.: Lie group convolution algebras as deformation quantizations of linear poisson structures. Am. J. Math. 112(4), 657–685 (1990)
Wu, W., Qi, Z., Fuxin, L.: PointConv: deep convolutional networks on 3d point clouds. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Lagrave, PY., Cabanes, Y., Barbaresco, F. (2021). \({{\,\mathrm{SU}\,}}(1,1)\) Equivariant Neural Networks and Application to Robust Toeplitz Hermitian Positive Definite Matrix Classification. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_62
Download citation
DOI: https://doi.org/10.1007/978-3-030-80209-7_62
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-80208-0
Online ISBN: 978-3-030-80209-7
eBook Packages: Computer ScienceComputer Science (R0)