用隐式表示法在潜在空间中计算测地线路径，提升生成模型的几何分析能力
Geodesic Calculus on Implicitly Defined Latent Manifolds
- 将潜在流形视为嵌入空间中的隐式子流形，构建离散黎曼微积分工具
- 通过去噪目标学习投影，实现对不精确表示的鲁棒性，支持多种几何结构
- 可计算端点间的最短路径和指数映射，适合需要流形操作的研究者
自编码器的潜在流形提供数据的低维表示，可从几何角度研究。本文将这些潜在流形描述为某嵌入潜在空间中的隐式子流形，并基于此发展了近似经典几何算子的离散黎曼微积分工具，对实际中常见的隐式表示误差具有鲁棒性。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近似投影到潜在流形上的映射，该方法与底层自编码器无关，支持在潜在流形上使用不同的黎曼几何。框架特别支持计算连接给定端点的测地线路径及通过黎曼指数映射进行射击测地线。我们在多种在合成数据和真实数据上训练的自编码器上评估了该方法。
原文摘要 · Abstract (English)
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.
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