流形擴散場

摘要

我們提出了流形擴散場（MDF），這是一種學習在黎曼流形上定義的連續函數生成模型的方法。借鑒了光譜幾何分析的見解，我們通過 Laplace-Beltrami 操作子的特徵函數在流形上定義了一個內在坐標系統。MDF使用由多個輸入-輸出對形成的明確參數化來表示函數。我們的方法允許在流形上對連續函數進行抽樣，並且對流形的剛性和等度變換具有不變性。對多個數據集和流形的實證結果表明，MDF能夠比以前的方法更好地捕捉這些函數的分佈，具有更好的多樣性和保真度。

English

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.