Applying Ricci flow to high dimensional manifold learning

Abstract

In machine learning, a high dimensional data set such as the digital image of a human face is often viewed as a point set distributed on a differentiable manifold. In many cases, the intrinsic dimension of this manifold is low but the representation dimension of the data points is high. To ease data processing requirements, manifold learning (ML) techniques can be used to reduce a high dimensional manifold (HDM) to a low dimensional one while keeping the essential geometric properties, such as relative distances between points, unchanged. Traditional ML algorithms often assume that the local neighborhood of any point on an HDM is roughly equal to the tangent space at that point. This assumption leads to the disadvantage that the neighborhoods of points on the manifold, though they have a very different curvature, will be treated equally and will be projected to a lower dimensional space. The curvature is a different way of manifold processing, where traditional dimension reduction is ineffective at preserving the neighborhood. To overcome this obstacle, we perform an “operation” on the HDM using Ricci flow before a manifold’s dimension reduction. More precisely, with the Ricci flow, we transform each local neighborhood of the HDM to a constant curvature patch. The HDM, as a whole, is then transformed into a subset of a sphere with constant positive curvature. We compare the proposed algorithm with other traditional manifold learning algorithms. Experimental results have shown that the proposed method outperforms other ML algorithms with a better neighborhood preserving rate.

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Acknowledgements

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB1000902), National Natural Science Foundation of China (Grant Nos. 61472412, 61621003), Beijing Science and Technology Project, and Tsinghua-Tencent-AMSS-Joint Project.

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Correspondence to Yangyang Li.

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Li, Y., Lu, R. Applying Ricci flow to high dimensional manifold learning. Sci. China Inf. Sci. 62, 192101 (2019). https://doi.org/10.1007/s11432-018-9702-7

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Keywords

  • manifold learning
  • Ricci flow
  • Ricci curvature
  • dimension reduction
  • curvature estimation