Abstract
In order to describe the nonlinear distribution of the image dataset, researchers propose a manifold assumption, called manifold learning (MAL). The geometry information based on a manifold is measured by the Riemannian metric, such as the geodesic distance. Thus, owing to mining the intrinsic geometric structure of the dataset, we need to learn the real Riemannian metric of the embedded manifold. By the Taylor expansion equation of the Riemannian metric, it clearly indicates that the Riemannian metric is relative to the Riemannian curvature. Based on it, we propose a new algorithm to learn the Riemannian metric by adding the curvature information into metric learning. By optimizing the objective function, we obtain a set of iterative equations. We call this model curvature flow. By employing this curvature flow, we obtain a Mahalanobis metric that approaches the Riemannian metric infinitely and can well uncover the intrinsic structure of the embedded manifold. In theory, we analyze several properties of our proposed method, e.g., the boundedness of the metric and the convergence of curvature flow. To show the effectiveness of our proposed method, we compare our algorithm with several traditional MAL algorithms on three real world datasets. The corresponding results indicate that our proposed method outperforms the other algorithms.
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References
Lin B, He X, Ye J. A geometric viewpoint of manifold learning. Appl Inform, 2015, 2: 3
Tenenbaum J B, Silva V D, Langford J C. A global geometric framework for nonlinear dimensionality reduction. Science, 2000, 290: 2319–2323
Roweis S T. Nonlinear dimensionality reduction by locally linear embedding. Science, 2000, 290: 2323–2326
Harandi M T, Sanderson C, Wiliem A, et al. Kernel analysis over Riemannian manifolds for visual recognition of actions, pedestrains and textures. In: Proceedings of IEEE Workshop on the Applications of Computer Vision, 2012
Silva V D, Tenenbaum J B. Global versus local methods in nonlinear dimensionality reduction. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2003. 16: 705–712
Belkin M, Niyogi P. Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2001. 14: 585–591
Donoho D L, Grimes C E. Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci USA, 2003, 100: 5591–5596
He X, Niyogi P. Locality preserving projections. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2003. 16: 153–160
Lee J M. Riemannian Manifolds: An Introduction to Curvature. New York: Springer, 1997
Hauberg S, Freifeld O, Black M J. A geometric take on metric learning. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2012. 2024–2032
Xing E P, Ng A Y, Jordan M I, et al. Distance metric learning with application to clustering with side-information. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2002. 15: 505–512
Davis J V, Kulis B, Jain P, et al. Information theoretic metric learning. In: Proceedings of the 24th International Conference on Machine Learning, 2007. 209–216
Wang J, Kalousisand A, Woznica A. Parameter local metric learning for nearest neighbor classification. In: Proceedings of Conference and Workshop on Neural Information Processing Systems, 2012
Saxena S, Verbeek J. Coordinated local metric learning. In: Proceedings of the 2015 IEEE International Conference on Computer Vision Workshop (ICCVW), 2015. 369–377
Kim K I, Tompkin J, Theobalt C. Curvature-aware regularization on riemannian sub-manifolds. In: Proceedings of IEEE International Conference on Computer Vision, 2013. 881–888
Hamilton R S. Three-manifolds with positive Ricci curvature. J Differ Geom, 1982, 17: 255–306
Xu W, Hancock E R, Wilson R C. Rectifying non-Euclidean similarity data using Ricci flow embedding. In: Proceedings of International Conference on Pattern Recognition, 2010. 3324–3327
Xu W, Hancock E R, Wilson R C. Ricci flow embedding for rectifying non-Euclidean dissimilarity data. Pattern Recognition, 2014, 47: 3709–3725
Li Y Y, Lu R Q. Applying Ricci flow to high dimensional manifold learning. Sci China Inf Sci, 2019, 62: 192101
Li Y Y, Lu R Q. Riemannian metric learning based on curvature flow. In: Proceedings of the 24th International Conference on Pattern Recognition, 2018
Guarrera D T, Johnson N G, Wolfe H F. The taylor expansion of a Riemannian metric. 2002. http://studylib.net/doc/13872779/the-taylor-expansion-of-ariemannian-metric
Jain P, Kulis B, Dhillon I S, et al. Online metric learning and fast similarity search. In: Proceedings of the 22nd Annual Conference on Neural Information Processing Systems, Vancouver, 2008. 761–768
Shalev-Shwartz S. Online learning and online convex optimization. Machine Learn, 2004, 56: 209–239
Mei J, Liu M, Karimi H R, et al. LogDet divergence based metric learning using triplet labels. In: Proceedings of ICML Workshop Divergence Learning, 2013. 1–9
Li Y Y. Curvature-aware manifold learning. Pattern Recogn, 2018, 83: 273–286
Acknowledgements
This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB1000902) and National Natural Science Foundation of China (Grant Nos. 61472412, 61621003).
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Li, Y., Lu, R. Curvature flow learning: algorithm and analysis. Sci. China Inf. Sci. 65, 192105 (2022). https://doi.org/10.1007/s11432-020-3068-7
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DOI: https://doi.org/10.1007/s11432-020-3068-7