Recently with the development of deep learning on data representation and generation, how to sampling on a data manifold becomes a crucial problem for research. In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method is able to generate high-quality uniform interpolations with the shortest path between two given data samples. Specifically, we use an autoencoder network to map data samples into the latent space and perform interpolation in the latent space via an interpolation network. We add prior geometric information to regularize our autoencoder for a flat latent embedding. The Riemannian metric on the data manifold is induced by the canonical metric in the Euclidean space in which the data manifold is isometrically immersed. Based on this defined Riemannian metric, we introduce a constant-speed loss and a minimizing geodesic loss to regularize the interpolation network to generate uniform interpolations along the learned geodesic on the manifold. We provide a theoretical analysis of our model and use image interpolation as an example to demonstrate the effectiveness of our method.

中文翻译：

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数据流形上均匀插值的测地线学习

最近随着深度学习在数据表示和生成方面的发展，如何对数据流形进行采样成为研究的关键问题。在本文中，我们提出了一种在数据流形中学习最小化测地线的方法。沿着学习到的测地线，我们的方法能够生成具有两个给定数据样本之间最短路径的高质量均匀插值。具体来说，我们使用自动编码器网络将数据样本映射到潜在空间，并通过插值网络在潜在空间中执行插值。我们添加了先验几何信息来规范我们的自动编码器以实现平面潜在嵌入。数据流形上的黎曼度量是由数据流形等距浸入的欧几里得空间中的规范度量引出的。基于这个定义的黎曼度量，我们引入了恒速损失和最小化测地线损失来规范插值网络，以沿流形上学习的测地线生成均匀插值。我们对我们的模型进行了理论分析，并以图像插值为例来证明我们方法的有效性。