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Cited by 16Neural Networks and ApplicationsTopological and Geometric Data AnalysisGaussian Processes and Bayesian Inference

Learning Weighted Submanifolds With Variational Autoencoders and Riemannian Variational Autoencoders

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Miolane, Nina, Holmes, Susan·2020·2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
DOI 10.1109/cvpr42600.2020.01451
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Abstract

Manifold-valued data naturally arises in medical imaging. In cognitive neuroscience for instance, brain connectomes base the analysis of coactivation patterns between different brain regions on the analysis of the correlations of their functional Magnetic Resonance Imaging (fMRI) time series - an object thus constrained by construction to belong to the manifold of symmetric positive definite matrices. One of the challenges that naturally arises in these studies consists in finding a lower-dimensional subspace for representing such manifold-valued and typically high-dimensional data. Traditional techniques, like principal component analysis, are ill-adapted to tackle non-Euclidean spaces and may fail to achieve a lower-dimensional representation of the data - thus potentially pointing to the absence of lower-dimensional representation of the data. However, these techniques are restricted in that: (i) they do not leverage the assumption that the connectomes belong on a pre-specified manifold, therefore discarding information; (ii) they can only fit a linear subspace to the data. In this paper, we are interested in variants to learn potentially highly curved submanifolds of manifold-valued data. Motivated by the brain connectomes example, we investigate a latent variable generative model, which has the added benefit of providing us with uncertainty estimates - a crucial quantity in the medical applications we are considering. While latent variable models have been proposed to learn linear and nonlinear spaces for Euclidean data, or geodesic subspaces for manifold data, no intrinsic latent variable model exists to learn non-geodesic subspaces for manifold data. This paper fills this gap and formulates a Riemannian variational autoencoder with an intrinsic generative model of manifold-valued data. We evaluate its performances on synthetic and real datasets, by introducing the formalism of weighted Riemannian submanifolds.

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  • In 2020, Lee and Lee [648] used GraphCNN and GAN with an Uncertainty-Aware Mesh Decoder for 3D face reconstruction on CASIA-Webface, COMA, 300W-LP, CelebA, and AFLW2000-3D.

    “Lee and Lee [648] | | 2020 | CASIA- Webface, COMA, 300W-LP, CelebA, AFLW2000- 3D | 3D face reconstruc- tion | CASIA-Webface: 494k images, COMA: 20,000 different meshes for 12 various subjects, 300W-LP: 60k | N/A | GraphCNN and GAN | Uncertainty- Aware Mesh Dec”

    From A review of uncertainty quantification in deep learning: Techniques, applications and challenges (2021) · cites this paper as [648]

    Section 2.2.3 introduces a Riemannian variational autoencoder, not a GraphCNN/GAN mesh decoder. Section 7 reports experiments on brain connectomes from the Human Connectome Project, rather than 3D face reconstruction on the listed datasets. No page numbers are available in our copy. (LLM, conf=0.88)

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Research works printed but not linked: 9

  1. Nikolai Chernov. Circular and linear regression : fitting circles and lines by least squares. Monographs on statistics and applied probability. CRC Press/Taylor & Francis, Boca Raton, 2011. 6
  2. Trevor Cox and Michael Cox. Multidimensional Scaling. Springer h edition, 2000. 2
  3. Diederik P. Kingma and Max Welling. Auto-Encoding Vari- ational Bayes. In Proceedings of the 2nd International Con- ference on Learning Representations (ICLR), 2014. 2, 3
  4. N. Miolane, A. Le Brigant, B. Hou, C. Donnat, M. Jorda, J. Mathe, X. Pennec, and S. Holmes. Geomstats: a python module for computations and statistics on manifolds. Sub- mitted to JMLR, 2019. 6
  5. Victor M. Panaretos, Tung Pham, and Zhigang Yao. Princi- pal flows. Journal of the American Statistical Association, 109(505):424-436, 2014. 2
  6. Mikhail Postnikov. Riemannian Geometry. Encyclopaedia of Mathem. Sciences. Springer, 2001. 6
  7. Danilo J Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic Backpropagation and Approximate Inference in Deep Generative Models. In Proceedings of the 31st Inter- national Conference on Machine Learning, 2014. 3
  8. Michael E Tipping and Christopher M Bishop. Probabilis- tic Principal Component Analysis. Source: Journal of the Royal Statistical Society. Series B (Statistical Methodology), 61(3):611-622, 1999. 2, 6
  9. Miaomiao Zhang and P. Thomas Fletcher. Probabilistic prin- cipal geodesic analysis. Advances in Neural Information Processing Systems, pages 1-9, 2013. 2, 4

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Related

Authors
Miolane, N., & Holmes, S.
Affiliations
Stanford University (US)
ORCID
Venue
2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
Publisher
IEEE
Published
2020-06
Pages
14491-14499
ISSN
1063-6919 (ISSN-L)
DOI
10.1109/cvpr42600.2020.01451
OpenAlex
W2990349966
Cited by
16
References
25
Citations by year
2026: 1 · 2025: 1 · 2024: 2 · 2023: 2 · 2022: 5 · 2021: 3 · 2020: 2
FWCI
1.13
Type
conference paper
Language
en
Open access
Licence
Locations
  • IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
  • Physical Sciences › Computer Science › Artificial Intelligence › Gaussian Processes and Bayesian Inference
Keywords
manifold-valued data, brain connectome, symmetric positive definite matrices, uncertainty estimation, dimensionality reduction, non-Euclidean data

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