Skip to main content

DLME: Deep Local-Flatness Manifold Embedding

  • Conference paper
  • First Online:
Computer Vision – ECCV 2022 (ECCV 2022)

Abstract

Manifold learning (ML) aims to seek low-dimensional embedding from high-dimensional data. The problem is challenging on real-world datasets, especially with under-sampling data, and we find that previous methods perform poorly in this case. Generally, ML methods first transform input data into a low-dimensional embedding space to maintain the data’s geometric structure and subsequently perform downstream tasks therein. The poor local connectivity of under-sampling data in the former step and inappropriate optimization objectives in the latter step leads to two problems: structural distortion and underconstrained embedding. This paper proposes a novel ML framework named Deep Local-flatness Manifold Embedding (DLME) to solve these problems. The proposed DLME constructs semantic manifolds by data augmentation and overcomes the structural distortion problem using a smoothness constrained based on a local flatness assumption about the manifold. To overcome the underconstrained embedding problem, we design a loss and theoretically demonstrate that it leads to a more suitable embedding based on the local flatness. Experiments on three types of datasets (toy, biological, and image) for various downstream tasks (classification, clustering, and visualization) show that our proposed DLME outperforms state-of-the-art ML and contrastive learning methods.

Z. Zang and S. Li—Equal contribution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrawal, R., Gehrke, J., Gunopulos, D., Raghavan, P.: Automatic subspace clustering of high dimensional data for data mining applications. In: Proceedings of the 1998 ACM SIGMOD International Conference on Management of Data, pp. 94–105 (1998)

    Google Scholar 

  2. Chang, J., Guo, Y., Wang, L., Meng, G., Xiang, S., Pan, C.: Deep discriminative clustering analysis. arXiv:1905.01681 [cs, stat] (2019)

  3. Chen, T., Kornblith, S., Norouzi, M., Hinton, G.: A simple framework for contrastive learning of visual representations. arXiv:2002.05709 [cs, stat] (2020)

  4. Cheng, B., Wu, W., Tao, D., Mei, S., Mao, T., Cheng, J.: Random cropping ensemble neural network for image classification in a robotic arm grasping system. IEEE Trans. Instrum. Meas. 69(9), 6795–6806 (2020)

    Article  Google Scholar 

  5. Do, K., Tran, T., Venkatesh, S.: Clustering by maximizing mutual information across views (2021)

    Google Scholar 

  6. Donoho, D.L., et al.: High-dimensional data analysis: the curses and blessings of dimensionality. AMS Math Challenges Lecture 1(2000), 32 (2000)

    Google Scholar 

  7. Duque, A.F., Morin, S., Wolf, G., Moon, K.: Extendable and invertible manifold learning with geometry regularized autoencoders. In: 2020 IEEE International Conference on Big Data (Big Data) (2020). https://doi.org/10.1109/bigdata50022.2020.9378049

  8. Flugge-Lotz, I.: Discontinuous Automatic Control. Princeton University Press (2015)

    Google Scholar 

  9. Flusser, J., Farokhi, S., Höschl, C., Suk, T., Zitova, B., Pedone, M.: Recognition of images degraded by Gaussian blur. IEEE Trans. Image Process. 25(2), 790–806 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gouk, H., Frank, E., Pfahringer, B., Cree, M.: Regularisation of neural networks by enforcing lipschitz continuity. arXiv:1804.04368 [cs, stat] (2018)

  11. Grill, J.B., et al.: Bootstrap your own latent: a new approach to self-supervised learning. arXiv:2006.07733 [cs, stat] (2020)

  12. Haeusser, P., Plapp, J., Golkov, V., Aljalbout, E., Cremers, D.: Associative deep clustering: training a classification network with no labels. In: Brox, T., Bruhn, A., Fritz, M. (eds.) GCPR 2018. LNCS, vol. 11269, pp. 18–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-12939-2_2

    Chapter  Google Scholar 

  13. Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85(3), 457–485 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, K., Fan, H., Wu, Y., Xie, S., Girshick, R.: Momentum contrast for unsupervised visual representation learning. arXiv:1911.05722 [cs] (2020)

  15. Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006). https://doi.org/10.1126/science.1127647

    Article  MathSciNet  MATH  Google Scholar 

  16. Hinton, G.E., Roweis, S.T.: Stochastic neighbor embedding. In: Advances in Neural Information Processing Systems, pp. 857–864 (2003)

    Google Scholar 

  17. Huang, J., Gong, S., Zhu, X.: Deep semantic clustering by partition confidence maximisation, pp. 8849–8858 (2020)

    Google Scholar 

  18. Kobak, D., Linderman, G.C.: UMAP does not preserve global structure any better than t-SNE when using the same initialization. BioRxiv (2019)

    Google Scholar 

  19. Kobak, D., Linderman, G.C.: Initialization is critical for preserving global data structure in both t-SNE and UMAP. Nat. Biotechnol. 39(2), 156–157 (2021)

    Article  Google Scholar 

  20. Kruskal, J.B.: Nonmetric multidimensional scaling: a numerical method. Psychometrika 29(2), 115–129 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, S., Liu, Z., Wu, D., Liu, Z., Li, S.Z.: Boosting discriminative visual representation learning with scenario-agnostic mixup. arXiv preprint arXiv:2111.15454 (2021)

  22. Li, S.Z., Zang, Z., Wu, L.: Deep manifold transformation for dimension reduction. arXiv preprint arXiv:2010.14831 (2020)

  23. Li, Y., Hu, P., Liu, Z., Peng, D., Zhou, J.T., Peng, X.: Contrastive clustering. AAAI2021 (2021). https://arxiv.org/abs/2009.09687

  24. Liu, Z., Li, S., Di Wu, Z.C., Wu, L., Guo, J., Li, S.Z.: AutoMix: unveiling the power of mixup (2021)

    Google Scholar 

  25. Liu, Z., Li, S., Wang, G., Tan, C., Wu, L., Li, S.Z.: Decoupled mixup for data-efficient learning. arXiv preprint arXiv:2203.10761 (2022)

  26. van der Maaten, L.: Learning a parametric embedding by preserving local structure. In: Artificial Intelligence and Statistics, pp. 384–391. PMLR (2009). https://proceedings.mlr.press/v5/maaten09a.html. ISSN 1938-7228

  27. van der Maaten, L., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9(Nov), 2579–2605 (2008)

    MATH  Google Scholar 

  28. McInnes, L., Healy, J., Melville, J.: UMAP: uniform manifold approximation and projection for dimension reduction. arXiv:1802.03426 [cs, stat] (2018). Version: 1

  29. Moon, K.R., van Dijk, D.: Visualizing structure and transitions in high dimensional biological data. Nat. Biotechnol. 37(12), 1482–1492 (2019)

    Article  Google Scholar 

  30. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 20–63 (1956)

    Google Scholar 

  31. Pers, T.H., Albrechtsen, A., Holst, C., Sørensen, T.I., Gerds, T.A.: The validation and assessment of machine learning: a game of prediction from high-dimensional data. PLoS One 4(8), e6287 (2009)

    Google Scholar 

  32. Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)

    Article  Google Scholar 

  33. Sainburg, T., McInnes, L., Gentner, T.Q.: Parametric UMAP embeddings for representation and semi-supervised learning. arXiv:2009.12981 [cs, q-bio, stat] (2021)

  34. Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Jpn. 9(4), 464–492 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  35. Szubert, B., Cole, J.E., Monaco, C., Drozdov, I.: Structure-preserving visualisation of high dimensional single-cell datasets. Sci. Rep. 9(1), 8914 (2019)

    Article  Google Scholar 

  36. Tenenbaum, J.B.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000). https://doi.org/10.1126/science.290.5500.2319. https://www.sciencemag.org/lookup/doi/10.1126/science.290.5500.2319

  37. Weng, L.: Contrastive representation learning. lilianweng.github.io (2021). https://lilianweng.github.io/posts/2021-05-31-contrastive/

  38. Wu, J., et al.: Deep comprehensive correlation mining for image clustering, pp. 8150–8159 (2019)

    Google Scholar 

  39. Wu, Z., Xiong, Y., Yu, S., Lin, D.: Unsupervised feature learning via non-parametric instance-level discrimination. arXiv:1805.01978 [cs] (2018)

  40. Zbontar, J., Jing, L., Misra, I., LeCun, Y., Deny, S.: Barlow twins: self-supervised learning via redundancy reduction. In: International Conference on Machine Learning, pp. 12310–12320. PMLR (2021)

    Google Scholar 

  41. Zhan, X., Xie, J., Liu, Z., Ong, Y.S., Loy, C.C.: Online deep clustering for unsupervised representation learning. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6688–6697 (2020)

    Google Scholar 

Download references

Acknowledgement

This work is supported by National Natural Science Foundation of China, named Geometric Deep Learning and Applications in Proteomics-Based Cancer Diagnosis (No. U21A20427). This work is supported by Alibaba Innovative Research (AIR) Programme.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stan Z. Li .

Editor information

Editors and Affiliations

1 Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 10644 KB)

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zang, Z. et al. (2022). DLME: Deep Local-Flatness Manifold Embedding. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M., Hassner, T. (eds) Computer Vision – ECCV 2022. ECCV 2022. Lecture Notes in Computer Science, vol 13681. Springer, Cham. https://doi.org/10.1007/978-3-031-19803-8_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-19803-8_34

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19802-1

  • Online ISBN: 978-3-031-19803-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics