Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data.

翻译：自编码器的潜在流形提供了数据的低维表示，可从几何角度进行研究。我们提出将这些潜在流形描述为某环境潜在空间中的隐式子流形。基于此，我们开发了一套离散黎曼演算工具，用于近似经典几何算子。这些工具对实际例子中常出现的隐式表示不准确性具有鲁棒性。为获得合适的隐式表示，我们提出通过最小化去噪目标来学习潜在流形上的近似投影。该方法独立于底层自编码器，并支持在潜在流形上使用不同的黎曼几何结构。该框架特别能够在潜在流形上计算连接给定端点的测地线路径，并通过黎曼指数映射实现测地线射击。我们在合成数据和真实数据上训练的各种自编码器上评估了该方法。