To fix ideas, consider ECG recordings from healthy patients during rest.

Re Eq. (3): indeed, we defineθi only later (line 157); we'll fix this, thanks. As requested, we here add another evaluation using34 t-SNE visualization (Figure 1). We do not assume data is35 available at large scale (though wecan and do handle36 large data sets as well);e.g., UCR contains 85 different37 datasets, manyofwhich include onlyfewexemplars per-38 class('ECGFiveDays',Fig1.,mainpaper,hadonly 1039 samples per class). We believe our experiments section40 was extensive and thorough, and we refer the reviewer41 to our supmat which includes more analyses and results.42

In time-series analysis, nonlinear temporal misalignment remains a pivotal challenge that forestalls even simple averaging. Since its introduction, the Diffeomorphic Temporal Alignment Net (DTAN), which we first introduced (Weber et al., 2019) and further developed in (Weber & Freifeld, 2023), has proven itself as an effective solution for this problem (these conference papers are earlier partial versions of the current manuscript). DTAN predicts and applies diffeomorphic transformations in an input-dependent manner, thus facilitating the joint alignment (JA) and averaging of time-series ensembles in an unsupervised or a weakly-supervised manner. The inherent challenges of the weakly/unsupervised setting, particularly the risk of trivial solutions through excessive signal distortion, are mitigated using either one of two distinct strategies: 1) a regularization term for warps; 2) using the Inverse Consistency Averaging Error (ICAE). The latter is a novel, regularization-free approach which also facilitates the JA of variable-length signals. We also further extend our framework to incorporate multi-task learning (MT-DTAN), enabling simultaneous time-series alignment and classification. Additionally, we conduct a comprehensive evaluation of different backbone architectures, demonstrating their efficacy in time-series alignment tasks. Finally, we showcase the utility of our approach in enabling Principal Component Analysis (PCA) for misaligned time-series data. Extensive experiments across 128 UCR datasets validate the superiority of our approach over contemporary averaging methods, including both traditional and learning-based approaches, marking a significant advancement in the field of time-series analysis.