Abstract
Deep generative models, e.g., variational autoencoders and generative adversarial networks, result in latent representation of observed data. The low dimensionality of the latent space provides an ideal setting for analysing high-dimensional data that would otherwise often be infeasible to handle statistically. The linear Euclidean geometry of the high-dimensional data space pulls back to a nonlinear Riemannian geometry on latent space where classical linear statistical techniques are no longer applicable. We show how analysis of data in their latent space representation can be performed using techniques from the field of geometric statistics. Geometric statistics provide generalisations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood estimation of parametric distributions. Introduction to estimation procedures on latent space are considered, and the computational complexity of using geometric algorithms with high-dimensional data addressed by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold.
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Acknowledgment
The work presented in this paper was supported by the CSGB Centre for Stochastic Geometry and Advanced Bioimaging funded by a grant from the Villum foundation, the Villum Foundation grant 00022924, the Novo Nordisk Foundation grant NNF18OC0052000, and the NSF DMS grant number 1912030.
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Kühnel, L., Fletcher, T., Joshi, S., Sommer, S. (2021). Latent Space Geometric Statistics. In: Del Bimbo, A., et al. Pattern Recognition. ICPR International Workshops and Challenges. ICPR 2021. Lecture Notes in Computer Science(), vol 12666. Springer, Cham. https://doi.org/10.1007/978-3-030-68780-9_16
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