Smooth Interpolation on Series of Measures

Abstract

In this thesis, we explore concepts related to interpolation between series of measures with a focus on trajectory inference. In many application scenarios, we seek to model continuous phenomena with sequences of discrete data. This task is particularly important when working with time-series data, where we have access to snapshots of a process at discrete time points and wish to infer behavior at unmeasured time steps. Due to realities of obtaining measurements, it is infeasible to measure the same samples multiple times. In the field of biology, for example, the development of singlecell sequencing methods has enabled study of cell development at an unprecedented resolution, but cells are destroyed when measured. As a result, there is no direct correspondence between data at different time steps, rendering learning about the evolution of a single cell over time difficult. Previous methods have focused primarily on the case of two time steps, and also suffer from a number of issues, ranging from expressiveness to the quality of their predictions. We present a model based on continuous normalizing flows which simultaneously interpolates within and across time steps. Our model’s trajectories have a number of desirable geometric properties such as smoothness and continuity. We also provide an extension of our model, linking it to previous work on measure-valued splines, and suggest modifications to increase model expressiveness.