Orthogonality Matters: Invariant Time Series Representation for Out-of-distribution Classification

Previous works for time series classification tend to assume that both the training and testing sets originate from the same distribution. This oversimplification deviates from the complexity of reality and makes it challenging to generalize methods to out-of-distribution (OOD) time series data. Currently, there are limited works focusing on time series OOD generalization, and they typically disentangle time series into domain-agnostic and domain-specific features and design tasks to intensify the distinction between the two. However, previous models purportedly yielding domain-agnostic features continue to harbor domain-specific information, thereby diminishing their adaptability to OOD data. To address this gap, we introduce a novel model called Invariant Time Series Representation (ITSR). ITSR achieves a learnable orthogonal decomposition of time series using two sets of orthogonal axes. In detail, ITSR projects time series onto these two sets of axes separately and obtains mutually orthogonal invariant features and relevant features. ITSR theoretically ensures low similarity between these two features and further incorporates various tasks to optimize them. Furthermore, we explore the benefits of preserving orthogonality between invariant and relevant features for OOD time series classification in theory. The results on four real-world datasets underscore the superiority of ITSR over state-of-the-art methods and demonstrate the critical role of maintaining orthogonality between invariant and relevant features. Our code is available at https://github.com/CGCL-codes/ITSR.