  Linear dimensionality reduction methods are commonly used to extract
low-dimensional structure from high-dimensional data. However, popular methods
disregard temporal structure, rendering them prone to extracting noise rather
than meaningful dynamics when applied to time series data. At the same time,
many successful unsupervised learning methods for temporal, sequential and
spatial data extract features which are predictive of their surrounding
context. Combining these approaches, we introduce Dynamical Components Analysis
(DCA), a linear dimensionality reduction method which discovers a subspace of
high-dimensional time series data with maximal predictive information, defined
as the mutual information between the past and future. We test DCA on synthetic
examples and demonstrate its superior ability to extract dynamical structure
compared to commonly used linear methods. We also apply DCA to several
real-world datasets, showing that the dimensions extracted by DCA are more
useful than those extracted by other methods for predicting future states and
decoding auxiliary variables. Overall, DCA robustly extracts dynamical
structure in noisy, high-dimensional data while retaining the computational
efficiency and geometric interpretability of linear dimensionality reduction
methods.
