# Differentiable Divergences Between Time Series

arXiv: https://arxiv.org/abs/2010.08354 · HTML: https://mlsanity.com/paper/2010.08354/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2020.

## Built on by (most cited first, 28 of 28)

- [ScanGAN360: A Generative Model of Realistic Scanpaths for 360circ Images](https://mlsanity.com/paper/2103.13922/) (2021)
- [Learning with Algorithmic Supervision via Continuous Relaxations](https://mlsanity.com/paper/2110.05651/) (2021)
- [Uncertainty-DTW for Time Series and Sequences](https://mlsanity.com/paper/2211.00005/) (2022)
- [Learning Temporal Point Processes for Efficient Retrieval of Continuous Time Event Sequences](https://mlsanity.com/paper/2202.11485/) (2022)
- [Deep Time Series Forecasting with Shape and Temporal Criteria](https://mlsanity.com/paper/2104.04610/) (2021)
- [Closed-Form Diffeomorphic Transformations for Time Series Alignment](https://mlsanity.com/paper/2206.08107/) (2022)
- [A Convolutional-Attentional Neural Framework for Structure-Aware Performance-Score Synchronization](https://mlsanity.com/paper/2204.08822/) (2022)
- [Deep Declarative Dynamic Time Warping for End-to-End Learning of Alignment Paths](https://mlsanity.com/paper/2303.10778/) (2023)
- [SCORE: Self-supervised Correspondence Fine-tuning for Improved Content Representations](https://mlsanity.com/paper/2403.06260/) (2024)
- [Diffeomorphic Temporal Alignment Nets for Time-series Joint Alignment and Averaging](https://mlsanity.com/paper/2502.06591/) (2025)
- [An Empirical Study of Graph-Based Approaches for Semi-Supervised Time Series Classification](https://mlsanity.com/paper/2104.08153/) (2021)
- [OTW: Optimal Transport Warping for Time Series](https://mlsanity.com/paper/2306.00620/) (2023)
- [Revisiting the Entropy Semiring for Neural Speech Recognition](https://mlsanity.com/paper/2312.10087/) (2023)
- [LASER: Learning by Aligning Self-supervised Representations of Speech for Improving Content-related Tasks](https://mlsanity.com/paper/2406.09153/) (2024)
- [Time Series Forecasting Models Copy the Past: How to Mitigate](https://mlsanity.com/paper/2207.13441/) (2022)
- [Retrieving Continuous Time Event Sequences using Neural Temporal Point Processes with Learnable Hashing](https://mlsanity.com/paper/2307.09613/) (2023)
- [Measuring Time-Series Dataset Similarity using Wasserstein Distance](https://mlsanity.com/paper/2507.22189/) (2025)
- [Measuring dissimilarity with diffeomorphism invariance](https://mlsanity.com/paper/2202.05614/) (2022)
- [Efficient Forecasting of Large Scale Hierarchical Time Series via Multilevel Clustering](https://mlsanity.com/paper/2205.14104/) (2022)
- [DPP-TTS: Diversifying prosodic features of speech via determinantal point processes](https://mlsanity.com/paper/2310.14663/) (2023)
- [STAER: Temporal Aligned Rehearsal for Continual Spiking Neural Network](https://mlsanity.com/paper/2601.20870/) (2026)
- [Position-invariant Fine-tuning of Speech Enhancement Models with Self-supervised Speech Representations](https://mlsanity.com/paper/2601.21084/) (2026)
- [SoftDTW-CUDA-Torch: Memory-Efficient GPU-Accelerated Soft Dynamic Time Warping for PyTorch](https://mlsanity.com/paper/2602.17206/) (2026)
- [Deep Time-Series Forecasting in 10 Years: A Survey](https://mlsanity.com/paper/2603.19899/) (2026)
- [Soft-MSM: Differentiable Context-Aware Elastic Alignment for Time Series](https://mlsanity.com/paper/2605.00069/) (2026)
- [Distill Locally, Schedule Globally: Flow Maps for Few-Step Text-to-Speech](https://mlsanity.com/paper/2609.36324/) (2026)
- [Fast Gradient-Based Approximation of Continuous Dynamic Time Warping](https://mlsanity.com/paper/2610.13129/) (2026)
- [Deterministic Prediction as Global Velocity: Flow Matching for Probabilistic Time-Series Forecast](https://mlsanity.com/paper/2610.18621/) (2026)

## Abstract (arXiv)

Computing the discrepancy between time series of variable sizes is notoriously challenging. While dynamic time warping (DTW) is popularly used for this purpose, it is not differentiable everywhere and is known to lead to bad local optima when used as a "loss". Soft-DTW addresses these issues, but it is not a positive definite divergence: due to the bias introduced by entropic regularization, it can be negative and it is not minimized when the time series are equal. We propose in this paper a new divergence, dubbed soft-DTW divergence, which aims to correct these issues. We study its properties; in particular, under conditions on the ground cost, we show that it is a valid divergence: it is non-negative and minimized if and only if the two time series are equal. We also propose a new "sharp" variant by further removing entropic bias. We showcase our divergences on time series averaging and demonstrate significant accuracy improvements compared to both DTW and soft-DTW on 84 time series classification datasets.