Spatiotemporal (ST) image data are increasingly common and often high-dimensional (high-D). Modeling ST data can be a challenge due to the plethora of independent and interacting processes which may or may not contribute to the measurements. Characterization can be considered the complement to modeling by helping guide assumptions about generative processes and their representation in the data. Dimensionality reduction (DR) is a frequently implemented type of characterization designed to mitigate the "curse of dimensionality" on high-D signals. For decades, Principal Component (PC) and Empirical Orthogonal Function (EOF) analysis has been used as a linear, invertible approach to DR and ST analysis. Recent years have seen the additional development of a suite of nonlinear DR algorithms, frequently categorized as "manifold learning". Here, we explore the idea of joint characterization of ST data manifolds using PCs/EOFs alongside two nonlinear DR approaches: Laplacian Eigenmaps (LE) and t-distributed stochastic neighbor embedding (t-SNE). Starting with a synthetic example and progressing to global, regional, and field scale ST datasets spanning roughly 5 orders of magnitude in space and 2 in time, we show these three DR approaches can yield complementary information about ST manifold topology. Compared to the relatively diffuse TFS produced by PCs/EOFs, the nonlinear approaches yield more compact manifolds with decreased ambiguity in temporal endmembers (LE) and/or in spatiotemporal clustering (t-SNE). These properties are compensated by the greater interpretability, significantly lower computational demand and diminished sensitivity to spatial aliasing for PCs/EOFs than LE or t-SNE. Taken together, we find joint characterization using the three complementary DR approaches capable of greater insight into generative ST processes than possible using any single approach alone.
Joint Manifold Learning and Density Estimation Using Normalizing Flows
University of Tehran（德黑兰大学）
基于流形假设，真实世界的数据通常位于低维流形上，而作为基于可能性的生成模型的归一化流由于其结构约束而无法找到该流形。因此，出现了一个有趣的问题：$\textit{"Can we find sub-manifold(s) of data in normalizing flows and estimate the density of the data on the sub-manifold(s)?"}$。在本文中，我们介绍了两种方法，即每像素惩罚对数似然和分层训练，回答上述问题。我们提出了一种单步方法，用于联合流形学习和密度估计，通过将流归一化为流形和非流形部分而获得的变换空间进行分离。这是通过用于学习数据的子流形的每像素惩罚似然函数完成的。规范化流假设转换后的数据是高斯化的，但这种强加的假设不一定是真的，特别是在高维中。为了解决这个问题，采用分层训练方法来改善子流形上的密度估计。结果验证了所提出的方法的优越性，同时流形学习和密度估计使用归一化流生成的图像质量和似然性。
