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.

翻译：隐变量模型是从高维数据中学习低维流形的强大工具。然而，在处理具有约束的数据（如单位范数向量或对称正定矩阵）时，现有方法要么忽略了底层的几何约束，要么无法在隐空间中提供有意义的度量。为解决这些局限性，我们提出学习此类几何数据的黎曼隐表示。为此，我们估计由包裹高斯过程隐变量模型诱导的回拉度量，该模型明确考虑了数据的几何结构。这使我们能够在隐空间中定义具有几何意识的距离和最短路径概念，同时确保我们的模型仅将概率质量分配给数据流形。这推广了先前的工作，使我们能够处理多个领域的复杂任务，包括机器人运动合成和脑连接组分析。