学习带几何约束的数据的黎曼潜空间表示，提升建模精度与可解释性。
Riemann$^2$: Learning Riemannian Submanifolds from Riemannian Data
- 基于包裹高斯过程潜变量模型，估计数据诱导的拉回度量。
- 在潜空间中定义几何感知的距离与最短路径，保持数据流形结构。
- 适用于机器人运动合成、脑连接组分析等复杂任务，适合几何数据研究者。
潜在变量模型是从高维数据中学习低维流形的强大工具。然而，面对单位范数向量或对称正定矩阵等受限数据时，现有方法要么忽略底层几何约束，要么无法在潜空间提供有意义的度量。为此，我们提出学习此类几何数据的黎曼潜表示。通过估计由包裹高斯过程潜变量模型诱导的拉回度量，显式考虑数据几何结构。这使得我们能在潜空间中定义几何感知的距离与最短路径，同时确保模型仅在数据流形上分配概率质量。该方法推广了先前工作，可处理机器人运动合成与脑连接组分析等多种复杂任务。
原文摘要 · Abstract (English)
Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches ignore the underlying geometric constraints or fail to provide meaningful metrics in the latent space. To address these limitations, we propose to learn Riemannian latent representations of such geometric data. To do so, we estimate the pullback metric induced by a Wrapped Gaussian Process Latent Variable Model, which explicitly accounts for the data geometry. This enables us to define geometry-aware notions of distance and shortest paths in the latent space, while ensuring that our model only assigns probability mass to the data manifold. This generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.
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