SoftDTW-CUDA-Torch: Memory-Efficient GPU-Accelerated Soft Dynamic Time Warping for PyTorch

Dynamic Time Warping (DTW) is a common choice for measuring the discrepancy between time series. It proceeds by finding an optimal warping between timestamps through dynamic programming, which can be slow and is not differentiable. Consequently, several variants have been proposed. Among them, Soft-DTW proposes a smooth formulation of DTW, amenable to gradient computation, while Continuous-DTW (CDTW) introduces a continuous formulation through parametrizable warping functions. Yet, computational costs remain high for both methods and impractical for large datasets in modern machine learning pipelines. This paper revisits CDTW in several ways. After establishing its differentiability under mild conditions, we introduce FastCDTW, which is an efficient approximate solver with two different computational strategies: an exact closed-form scheme for piecewise-linear warping functions, and an unbiased Monte Carlo estimator for smooth ones, such as implicit neural networks. Both strategies provide gradients with respect to the warping function and the input time series, and the resulting algorithm is GPU-friendly. We demonstrate its benefits through its integration into more complex learning problems, including time series averaging, classification and deep domain adaptation.

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 aeon toolkit.

Lossless tensor compression could reduce memory usage during training, but training requires tensors to be repeatedly re-encoded. We introduce Losslessly Compressed Tensors (LCT), a GPU execution framework designed for this repeated encode-decode regime. LCT exploits the natural redundancy in floating point data to reduce storage for weights, activations and optimizer states. Highly optimized GPU kernels implement Huffman coding with precomputed codebooks for low compression and decompression overhead, essential for training. Our analysis characterizes LCT's storage costs and adaptation to magnitude shifts. LCT achieves a compression ratio of up to $1.455\times$, and is demonstrated in training and finetuning settings. On a 162M-parameter NanoGPT, LCT reduces memory usage by $24.6\%$ for a $7.2\%$ step-time increase. In Qwen3-4B fine-tuning, LCT saves up to $23.8\%$ of peak memory, enabling a $3.5\times$ larger context length in training on the same GPU.