Diffeomorphic Temporal Alignment Nets for Time-series Joint Alignment and Averaging

TL;DR

This work introduces DTAN, a diffeomorphic temporal alignment network that learns input-dependent warps to jointly align and average time-series in unsupervised or weakly supervised settings. It leverages CPAB-based 1D diffeomorphisms within a Temporal Transformer Net, and introduces ICAE as a regularization-free objective for robust alignment, along with a recurrent variant (RDTAN) and a multi-task version (MT-DTAN) that incorporates classification. The approach achieves state-of-the-art Nearest Centroid Classification (NCC) performance across 128 UCR datasets, while enabling fast inference, variable-length handling, and improved PCA-based dimensionality reduction on aligned data. These results demonstrate scalable, generalizable JA for time-series, with practical impact on preprocessing, classification, and downstream analyses. The findings emphasize the value of diffeomorphic warps and learned alignment in enabling robust statistical analyses of misaligned time-series in real-world datasets.

Abstract

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.

Figures (16)

• Figure 1: An illustration of the joint-alignment problem in ECG data. The data shown is test data. Top: temporal misalignment between ECG signals and its effect on the sample mean (ECGFiveDays Dataset). Bottom: joint-alignment prediction by DTAN at test time.
• Figure 2: The benefits of joint alignment for dimensionality reduction, evaluated on the Trace dataset Dau:2019:ucr using Principal Component Analysis. The top panel shows the cumulative explained variance as a function of the number of Principal Components (PCs). The middle-top and middle-bottom panels depict the first 3 PCs of the original and DTAN-aligned data, respectively. The bottom panel illustrates the reconstruction of the original and (inverse warped) aligned data using the first 6 PCs.
• Figure 3: DTAN joint alignment demonstrated on a class of the "Trace" dataset Chen:UCR:Archive:2015 with a simple 1D ConvNet backbone which was used in Shapira:NIPS:2019:DTAN. Signals are denoted in gray and their average in blue. Each Convolution layer is followed by a ReLU, Batch Normalization, and a Max-Pooling layer. The final Fully-Connected layer (fc) predicts the warping parameters, $\btheta$, of the CPA velocity fields, $v^{\btheta}$, which is then integrated to form a CPAB warp, $T^{\btheta}$. The latter, in turn, is applied to the input signal ($u$) to create the output, $u\circ T^{\btheta}=v$. The loss consists of the empirical within-class variance ($\mathcal{L}_{\mathrm{data}}$) and a regularization term on $\btheta$ ($\mathcal{L}_{\mathrm{reg}}$).
• Figure 4: CPAB warps for different partitions of $\Omega\in{\{8, 16,32\}}$. Top: Continuous Piecewise-Affine (CPA) velocity fields. Bottom: The resulting CPAB warp, obtained via integration of $v^\btheta$.
• Figure 5: The effect of the smoothness prior on the predicted warps in the ECG200 dataset. Left: no prior. Right: $\lambda_{\sigma}=.01, \lambda_{smooth}=.01$. Color indicates class label.

Theorems & Definitions (1)

• Definition 1