Interpolating between points is a problem connected simultaneously with finding geodesics and study of generative models. In the case of geodesics, we search for the curves with the shortest length, while in the case of generative models we typically apply linear interpolation in the latent space. However, this interpolation uses implicitly the fact that Gaussian is unimodal. Thus the problem of interpolating in the case when the latent density is non-Gaussian is an open problem. In this paper, we present a general and unified approach to interpolation, which simultaneously allows us to search for geodesics and interpolating curves in latent space in the case of arbitrary density. Our results have a strong theoretical background based on the introduced quality measure of an interpolating curve. In particular, we show that maximising the quality measure of the curve can be equivalently understood as a search of geodesic for a certain redefinition of the Riemannian metric on the space. We provide examples in three important cases. First, we show that our approach can be easily applied to finding geodesics on manifolds. Next, we focus our attention in finding interpolations in pre-trained generative models. We show that our model effectively works in the case of arbitrary density. Moreover, we can interpolate in the subset of the space consisting of data possessing a given feature. The last case is focused on finding interpolation in the space of chemical compounds.

翻译：在点之间进行插值是一个同时与寻找测地线和生成模型研究相关的问题。在测地线情形中，我们寻找长度最短的曲线，而在生成模型情形中，我们通常对潜空间应用线性插值。然而，这种插值隐式地利用了高斯分布是单峰的这一事实。因此，当潜密度为非高斯分布时，插值问题仍然是一个开放问题。本文提出了一种通用且统一的插值方法，该方法允许我们在任意密度情形下同时寻找潜空间中的测地线和插值曲线。我们的结果基于引入的插值曲线质量测度具有扎实的理论基础。特别地，我们证明了最大化曲线质量测度等价于在黎曼度量的某种重新定义下寻找测地线。我们在三种重要情形下提供了示例。首先，我们展示了该方法可轻松应用于流形上的测地线寻找。其次，我们专注于在预训练生成模型中寻找插值路径，证明了该方法在任意密度情形下均有效。此外，我们能够在由具有给定特征的数据组成的子空间中进行插值。最后一种情形聚焦于化学化合物空间中的插值路径寻找。