Dynamic Time Warping

We present a straightforward but effective method for repurposing existing contact-rich dexterous manipulation demonstrations. Starting from inputs of hand and object trajectories, our method outputs high-quality nonlinear trajectory warps that account for intermediate waypoints, environmental barriers, temporal shifts, and varied start/end configurations. Foundational to our method is the utilization of contact distributions, which we show allows us to reliably compute complex and high-dimensional dexterous hand trajectories following a simple object-centric warp specification pipeline. We evaluate our method across 12 variations sourced from 4 demonstrations in a publicly available dataset of human hand motion data, perform baseline comparisons, and demonstrate generalization of our approach to different manipulators. Results and code will be made available on publication.

Dynamic Time Warping (DTW) is the dominant approach for measuring similarity between time series, yet standard practice discards the optimal warping path after computing a single distance value, losing local alignment information most relevant to clinical diagnosis. We introduce DiaSeg, a framework that extracts diagonal segments from DTW paths with controlled breaks, characterizing each segment by five geometric features (effective length, interruption count, cost variation, temporal position, and path context), and enabling unsupervised pattern discovery without domain-specific feature engineering. Validated on 91 subjects across six clinical conditions (healthy aging, Parkinson's, Huntington's, ALS, brain tumor, and stroke), three findings emerge. First, diagonal segments form consistent unsupervised patterns (silhouette 0.33) aligned with biomechanical phase annotations, with label-based validation confirming near-perfect separation of healthy and pathological gait (ARI up to 0.986). Second, segments discriminate pathology at 69% (supervised) and 75% (patient-level clustering), with pathology manifesting through distributional shifts in segment length; combining segment and cycle-level features further improves classification to 91.7%. Third, while cycle-based methods achieve higher accuracy (91%), diagonal segments provide phase-specific interpretability unavailable in global representations, localizing where coordination breaks down within the gait cycle. DiaSeg thus transforms DTW from a black-box distance into a source of interpretable temporal features for neurodegenerative disease assessment.

Time-warping estimation is a fundamental problem in signal processing with applications in bioacoustics, radar, and biomedical analysis. This paper introduces a Time-Warping Estimation Trainable (TWET) model for estimating timewarping functions from a single observation. The proposed approach formulates time-warping estimation as a stationarization problem in the wavelet domain and leverages a hierarchical dilated convolutional architecture to estimate the time-warping functions. A differentiable stationarity criterion is introduced for end-to-end optimization. TWET is compared with existing approaches. Experimental results show improved deformation reconstruction accuracy together with significantly reduced computation time, making the framework compatible with low-latency applications.

Dynamic Time Warping (DTW)-based Nearest-Neighbor (NN) classifiers are effective for time-series classification but are vulnerable to mislabeled training samples and require numerous DTW computations during inference. We propose DTW-based Granular Ball Computing (DTW-GBC), which organizes temporally similar training samples into granular balls and performs classification at the granule level. We further develop two granular-ball construction strategies for DTW-GBC. Experiments on four benchmark datasets with symmetric label noise show that the two DTW-GBC variants generally mitigate the performance degradation caused by label noise while requiring substantially fewer comparisons than DTW-based 1-NN during inference. These findings suggest that DTW-GBC provides a favorable balance between classification robustness and inference efficiency.

In this study, we explore how pianists can customize Music Minus One (MMO) concerto accompaniments to match their playing style. Bypassing the need for a symbolic score, often not available digitally, we use three types of audio data: solo piano recordings, MMO orchestra-only recordings, and mixed recordings of both piano and orchestra (e.g., from YouTube). The mixed recording serves as an intermediary reference to align the solo and orchestra parts, with only the orchestral part being adjusted through time-scale modification to synchronize with the user's playing. The main challenge with estimating these alignments is the spectral mismatch between recordings containing different musical parts. Motivated by this application scenario, we introduce Dense-Sparse DTW, a variant of Dynamic Time Warping (DTW) that is designed to improve robustness of alignments to spectral mismatch by focusing on aligning a selected subset of audio frames containing prominent timing cues. We collect and annotate data from four piano concerto movements and establish a framework for generating and evaluating customized accompaniment recordings. On this benchmark, we show that Dense-Sparse DTW has better or comparable performance than more complex approaches based on source separation and spectral subtraction techniques.

L2 speech assessment has traditionally focused on phonetic assessment, leaving the scoring of suprasegmental features such as rhythm and intonation underexplored. Moreover, assessment methods often require training with labeled L2 speech data, making them difficult to apply in low-resource settings. We investigate whether DTW over self-supervised WavLM representations can provide a text-free framework for assessing phonetic accuracy, rhythm, and intonation in English and Japanese L2 speech. Results show that a basic DTW-based approach that compares learner speech to native templates exceeds human agreement on holistic and sentence-level phonetic scoring. For rhythm, we introduce methods that measure the degree of warping in the DTW alignment path; our best method approaches human-level performance. For intonation, we combine DTW distance over prosodic residuals with pitch and intensity features, but performance remains more modest on some tasks. Our results point to self-supervised representations as a promising, text-free basis for multi-aspect pronunciation assessment.

Dynamic Time Warping (DTW) is a cornerstone for time series classification, but its reliance on Euclidean distances fails to capture latent cross-channel correlations in complex multivariate data. We propose a hybrid Quantum Dynamic Time Warping (qDTW) architecture, replacing the classical distance metric with the parameterized geometry of a quantum Hilbert space. Through structural ablation on benchmarks up to $C=8$ spatial dimensions, we establish fundamental topological rules for quantum sequence alignment. We introduce a Unified Pre-Embedding Adjoint Ansatz that decouples trainable entanglement from classical data, eliminating the severe phase-scrambling and information bottlenecks inherent to traditional measurements. We demonstrate this decoupled architecture allows untrained quantum kernels to act as highly expressive baselines, while parameterized training effectively untangles deeply overlapping hyper-dimensional data. Furthermore, we identify a strict spatial-temporal expressivity tradeoff: temporal depth (data re-uploading) is necessary for dimensionally restricted univariate circuits, but applying it to wide multi-qubit registers triggers chaotic frequency-spectrum explosions and representation collapse. By navigating these topological hazards, our multivariate quantum architecture outperforms classical baselines, setting a new standard for integrating parameterized quantum circuits with dynamic programming

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.

Aligning structured data is a fundamental problem in computer vision and machine learning, underlying tasks such as time series analysis, human action recognition, and visual representation learning. Existing alignment methods, including Dynamic Time Warping (DTW) and its differentiable variants, rely on deterministic similarity measures and are therefore sensitive to heterogeneous and noisy features. In this work, we introduce uncertainty-aware alignment, a probabilistic framework that models pairwise correspondences with heteroscedastic uncertainty and performs structured matching along alignment paths. Our formulation, uncertainty-DTW (uDTW), assigns each correspondence a Normal distribution and parametrizes each alignment path by a Maximum Likelihood Estimate objective consisting of (i) a precision-weighted matching term that suppresses unreliable features, and (ii) a log-variance regularization that prevents degenerate solutions. This yields a probabilistic alignment mechanism that is robust to noise and interpretable, as uncertainty directly reflects the reliability of matches. We further generalize this framework from temporal sequences to tokenized visual representations, enabling structured matching over sets of visual tokens. The learned uncertainty can be interpreted as a reverse-attention: semantically relevant regions exhibit low uncertainty and dominate the alignment, while ambiguous/noisy regions have high uncertainty. This provides a connection between alignment, attention, and uncertainty modeling. We evaluate the proposed framework across diverse domains. The results demonstrate consistent improvements over state-of-the-art methods and show that learned uncertainty correlates with semantic importance. These findings establish uncertainty-aware alignment as a general, robust, and interpretable framework for learning from structured data.

Elastic distances like dynamic time warping (DTW) are central to time series machine learning because they compare sequences under local temporal misalignment. Soft-DTW is an adaptation of DTW that can be used as a gradient-based loss by replacing the hard minimum in its dynamic-programming recursion with a smooth relaxation. However, this approach does not directly extend to elastic distances whose transition costs depend on the local alignment context. Move-Split-Merge (MSM) is one such distance: it uses context-aware split and merge penalties and has often outperformed DTW in supervised and unsupervised time series machine learning tasks such as classification and clustering. We introduce Soft-MSM, a smooth relaxation of MSM and an elastic alignment loss with context-aware transition costs. Central to the formulation is a smooth gated surrogate for MSM's piecewise split/merge cost, which enables gradients through both the dynamic-programming recursion and the local transition structure. We derive the forward recursion, backward recursion, soft alignment matrix, closed-form gradient, limiting behaviour, and divergence-corrected formulation. Experiments on 112 UCR datasets show that Soft-MSM gives lower MSM barycentre loss than existing MSM barycentre methods, and yields significantly better clustering and nearest-centroid classification performance than Soft-DTW-based alternatives. An implementation is available in the open-source \texttt{aeon} toolkit.