We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using a simple alternating scheme. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.

翻译：我们提出了一种基于最优传输的模型，用于从公共黎曼流形上演化概率测度的横截面样本中学习度量张量。我们通过神经网络将度量参数化为空间变化的矩阵场，并采用简单的交替优化方案高效地求解模型目标函数。利用该学习度量，可对概率测度进行非线性插值并计算流形上的测地线。实验表明，该方法学习到的度量在仅需少量额外横截面数据的情况下，能够显著提升单细胞RNA测序数据与候鸟迁徙数据的轨迹推断质量。