Time-series anomaly detection is critical for ensuring safety in high-stakes applications, where robustness is a fundamental requirement rather than a mere performance metric. Addressing the vulnerability of these systems to adversarial manipulation is therefore essential. Existing defenses are largely heuristic or provide certified robustness only under ℓp-norm constraints, which are incompatible with time-series data. In particular, ℓp-norm fails to capture the intrinsic temporal structure in time series, causing small temporal distortions to significantly alter the ℓp-norm measures. Instead, the similarity metric Dynamic Time Warping (DTW) is more suitable and widely adopted in the time-series domain, as DTW accounts for temporal alignment and remains robust to temporal variations. To date, however, there has been no certifiable robustness result in this metric that provides guarantees. In this work, we introduce the first DTW-certified robust defense in time-series anomaly detection by adapting the randomized smoothing paradigm. We develop this certificate by bridging the ℓp-norm to DTW distance through a lower-bound transformation.

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to a discrete response variable, like class labels. This task is fundamental in domains, including health monitoring, where the temporal structure of data is critical for accurate prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. To move beyond pairwise alignment, we propose a theoretical framework that learns mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between two time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function used to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms DTW on 60 of 86 datasets.

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.

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.

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Thefirsttwocolumns showresults for two different step-sizes, and the third one using the best step-size chosen retrospectively. The plots show the final ELBO achieved after trainingfor40000stepsvs. stepsizeused. Figure 11: VI using a diagonal Gaussian, with the best step-size chosen retrospectively. Bayesian logistic regression: We use a subset of700 rows of thea1a dataset. In this case the posterior p(z|x) has dimensionality d = 120.

Part qualification is crucial in additive manufacturing (AM) because it ensures that additively manufactured parts can be consistently produced and reliably used in critical applications. Part qualification aims at verifying that an additively manufactured part meets performance requirements; therefore, predicting the complex stress-strain behaviors of additively manufactured parts is critical. We develop a dynamic time warping (DTW)-transfer learning (TL) framework for additive manufacturing part qualification by transferring knowledge of the stress-strain behaviors of additively manufactured low-cost polymers to metals. Specifically, the framework employs DTW to select a polymer dataset as the source domain that is the most relevant to the target metal dataset. Using a long short-term memory (LSTM) model, four source polymers (i.e., Nylon, PLA, CF-ABS, and Resin) and three target metals (i.e., AlSi10Mg, Ti6Al4V, and carbon steel) that are fabricated by different AM techniques are utilized to demonstrate the effectiveness of the DTW-TL framework. Experimental results show that the DTW-TL framework identifies the closest match between polymers and metals to select one single polymer dataset as the source domain. The DTW-TL model achieves the lowest mean absolute percentage error of 12.41% and highest coefficient of determination of 0.96 when three metals are used as the target domain, respectively, outperforming the vanilla LSTM model without TL as well as the TL model pre-trained on four polymer datasets as the source domain.

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.

Another popular measure, known as Mahalanobis distance, can be viewed as the distorted Euclidean distance.