Time-series analysis is confounded by nonlinear time warping of the data. Traditional methods for joint alignment do not generalize: after aligning a given signal ensemble, they lack a mechanism, that does not require solving a new optimization problem, to align previously-unseen signals. In the multi-class case, they must also first classify the test data before aligning it. Here we propose the Diffeomorphic Temporal alignment Net (DTAN), a learning-based method for time-series joint alignment. Via flexible temporal transformer layers, DTAN learns and applies an input-dependent nonlinear time warping to its input signal.

Dynamic time warping (DTW) is a fundamental technique in time series analysis for comparing one curve to another using a flexible time-warping function. However, it was designed to compare a single pair of curves. In many applications, such as in metabolomics and image series analysis, alignment is simultaneously needed for multiple pairs. Because the underlying warping functions are often related, independent application of DTW to each pair is a sub-optimal solution. Yet, it is largely unknown how to efficiently conduct a joint alignment with all warping functions simultaneously considered, since any given warping function is constrained by the others and dynamic programming cannot be applied.

Where the constraint ensures that the solution is unique.

Time-series analysis is confounded by nonlinear time warping of the data.

We develop the strategy of linear component operations that integrates dynamic programming technique and the push-relabel approach.

Time-series analysis is confounded by nonlinear time warping of the data. Traditional methods for joint alignment do not generalize: after aligning a given signal ensemble, they lack a mechanism, that does not require solving a new optimization problem, to align previously-unseen signals. In the multi-class case, they must also first classify the test data before aligning it. Here we propose the Diffeomorphic Temporal alignment Net (DTAN), a learning-based method for time-series joint alignment. Via flexible temporal transformer layers, DTAN learns and applies an input-dependent nonlinear time warping to its input signal.

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.

Technical quality: The discussion of the proofs in the paper is very brief, making the technical sections quite dense. The supplemental material fills in the details and the proofs appear to be correct. One issue that could be addressed more clearly in the paper and the supplemental is the need for infinite weight reverse edges in the dual graph. This is currently not explained at all in the main paper other than stating these edges are important and later empirically demonstrating that not including them leads to violations of DTW warping constraints. The hyper-parameter tuning section was very brief and the algorithm is not clear.

Multiple time series data occur in many real applications and the alignment among them is usually a fundamental step of data analysis. Frequently, these multiple time series are inter-dependent, which provides extra information for the alignment task and this information cannot be well utilized in the conventional pairwise alignment methods. Recently, the joint alignment was modeled as a max-flow problem, in which both the profile similarity between the aligned time series and the distance between adjacent warping functions are jointly optimized. In this report, we present BIdirectional pushing with Linear Component Operations (BILCO), a novel algorithm that solves the joint alignment max-flow problems efficiently and exactly. We develop the strategy of linear component operations that integrates dynamic programming technique and the push-relabel approach.

Time-series analysis is confounded by nonlinear time warping of the data. Traditional methods for joint alignment do not generalize: after aligning a given signal ensemble, they lack a mechanism, that does not require solving a new optimization problem, to align previously-unseen signals. In the multi-class case, they must also first classify the test data before aligning it. Here we propose the Diffeomorphic Temporal alignment Net (DTAN), a learning-based method for time-series joint alignment. Via flexible temporal transformer layers, DTAN learns and applies an input-dependent nonlinear time warping to its input signal.