流形扩散场

摘要

我们提出了流形扩散场（MDF），这是一种学习定义在黎曼流形上的连续函数生成模型的方法。借鉴了谱几何分析的见解，我们通过 Laplace-Beltrami 算子的特征函数在流形上定义了一个内在坐标系。MDF使用由一组多个输入-输出对形成的显式参数化来表示函数。我们的方法允许在流形上对连续函数进行采样，并且对流形的刚性和等距变换具有不变性。在多个数据集和流形上的实证结果表明，MDF能够比先前的方法更好地捕捉这些函数的分布，具有更好的多样性和保真度。

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.