Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Point Cloud Classification via Deep Set Linearized Optimal Transport (2401.01460v1)

Published 2 Jan 2024 in cs.LG and stat.ML

Abstract: We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein space while constructing a classifier to distinguish between various classes of point clouds. Our approach is motivated by the observation that $L2-$distances between optimal transport maps for distinct point clouds, originating from a shared fixed reference distribution, provide an approximation of the Wasserstein-2 distance between these point clouds, under certain assumptions. To learn approximations of these transport maps, we employ input convex neural networks (ICNNs) and establish that, under specific conditions, Euclidean distances between samples from these ICNNs closely mirror Wasserstein-2 distances between the true distributions. Additionally, we train a discriminator network that attaches weights these samples and creates a permutation invariant classifier to differentiate between different classes of point clouds. We showcase the advantages of our algorithm over the standard deep set approach through experiments on a flow cytometry dataset with a limited number of labeled point clouds.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (43)
  1. Partitioning signal classes using transport transforms for data analysis and machine learning. Sampl. Theory Signal Process. Data Anal., 19(6), 2021.
  2. Luís A. Alexandre. 3d Object Recognition Using Convolutional Neural Networks with Transfer Learning Between Input Channels. In Emanuele Menegatti, Nathan Michael, Karsten Berns, and Hiroaki Yamaguchi, editors, Intelligent Autonomous Systems 13, pages 889–898, Cham, 2016. Springer International Publishing.
  3. Input convex neural networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 146–155. PMLR, 06–11 Aug 2017.
  4. Detecting and visualizing cell phenotype differences from microscopy images using transport-based morphometry. Proceedings of the National Academy of Sciences, 111(9):3448–3453, 2014.
  5. Performance of histogram descriptors for the classification of 3d laser range data in urban environments. In 2012 IEEE International Conference on Robotics and Automation, pages 4391–4398, 2012.
  6. R.J. Berman. Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Found Comput Math, 21:1099–1140, 2021.
  7. Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics, 44(4):375–417, 1991.
  8. Automated identification of stratifying signatures in cellular subpopulations. Proceedings of the National Academy of Sciences, 111(26):E2770–E2777, 2014.
  9. Optimal control via neural networks: A convex approach. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019.
  10. Two-sample statistics based on anisotropic kernels. Information and Inference: A Journal of the IMA, 9(3):677–719, 12 2019.
  11. The manifold scattering transform for high-dimensional point cloud data. In Topological, Algebraic and Geometric Learning Workshops 2022, pages 67–78. PMLR, 2022.
  12. Linearized wasserstein dimensionality reduction with approximation guarantees. arXiv preprint: 2302.07373, 2023.
  13. People mover’s distance: Class level geometry using fast pairwise data adaptive transportation costs. Applied and Computational Harmonic Analysis, 47(1):248–257, 2019.
  14. Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In C.J. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 26. Curran Associates, Inc., 2013.
  15. Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, 2021.
  16. Recognizing objects in range data using regional point descriptors. In Tomáš Pajdla and Jiří Matas, editors, Computer Vision - ECCV 2004, pages 224–237, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.
  17. Nicola Gigli. On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proceedings of the Edinburgh Mathematical Society, 54(2):401–409, 2011.
  18. Shape-based recognition of 3d point clouds in urban environments. In 2009 IEEE 12th International Conference on Computer Vision, pages 2154–2161, 2009.
  19. Fast semantic segmentation of RGB-D scenes with GPU-accelerated deep neural networks. In Carsten Lutz and Michael Thielscher, editors, KI 2014: Advances in Artificial Intelligence, pages 80–85, Cham, 2014. Springer International Publishing.
  20. Supervised learning of sheared distributions using linearized optimal transport. Sampling Theory, Signal Processing, and Data Analysis, 21(1), 2023.
  21. Wasserstein-2 generative networks. In International Conference on Learning Representations, 2021.
  22. Do neural optimal transport solvers work? A continuous wasserstein-2 benchmark. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 14593–14605. Curran Associates, Inc., 2021.
  23. Optimal transport mapping via input convex neural networks. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 6672–6681. PMLR, 13–18 Jul 2020.
  24. VoxNet: A 3d convolutional neural network for real-time object recognition. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 922–928, 2015.
  25. Quantitative stability of optimal transport maps and linearization of the 2-wasserstein space. In Silvia Chiappa and Roberto Calandra, editors, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine Learning Research, pages 3186–3196. PMLR, 26–28 Aug 2020.
  26. Linear optimal transport embedding: provable wasserstein classification for certain rigid transformations and perturbations. Information and Inference: A Journal of the IMA, 12(1):363–389, 2023.
  27. H. Moravec and A. Elfes. High resolution maps from wide angle sonar. In Proceedings. 1985 IEEE International Conference on Robotics and Automation, volume 2, pages 116–121, 1985.
  28. Kernel mean embedding of distributions: A review and beyond. 2017.
  29. Hypothesis testing with efficient method of moments estimation. International Economic Review, 28(3):777–787, 1987.
  30. Matching 3d models with shape distributions. In Proceedings International Conference on Shape Modeling and Applications, pages 154–166, 2001.
  31. The cumulative distribution transform and linear pattern classification. Applied and Computational Harmonic Analysis, 45(3):616 – 641, 2018.
  32. Convolutional-recursive deep learning for 3d object classification. In F. Pereira, C.J. Burges, L. Bottou, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 25. Curran Associates, Inc., 2012.
  33. Towards 3d object recognition via classification of arbitrary object tracks. In 2011 IEEE International Conference on Robotics and Automation, pages 4034–4041, 2011.
  34. C. Villani. Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2008.
  35. C. Villani and American Mathematical Society. Topics in Optimal Transportation. Graduate studies in mathematics. American Mathematical Society, 2003.
  36. An optimal transportation approach for nuclear structure-based pathology. IEEE Trans Med Imaging, 30(3):621–631, 2011.
  37. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. IJCV, 101(2):254–269, 2013.
  38. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In Advances in Neural Information Processing Systems, pages 82–90, 2016.
  39. 3d ShapeNets: A deep representation for volumetric shapes. In 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 1912–1920, 06 2015.
  40. Deep sets. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
  41. Understanding bag-of-words model: A statistical framework. International Journal of Machine Learning and Cybernetics, 1:43–52, 12 2010.
  42. Detection of differentially abundant cell subpopulations in scrna-seq data. Proceedings of the National Academy of Sciences, 118(22):e2100293118, 2021.
  43. Xingyu Zhou. On the fenchel duality between strong convexity and lipschitz continuous gradient. arXiv:1803.06573, 2018.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now