OTW: Optimal Transport Warping for Time Series

Dynamic Time Warping (DTW) has become the pragmatic choice for measuring distance between time series. However, it suffers from unavoidable quadratic time complexity when the optimal alignment matrix needs to be computed exactly. This hinders its use in deep learning architectures, where layers involving DTW computations cause severe bottlenecks. To alleviate these issues, we introduce a new metric for time series data based on the Optimal Transport (OT) framework, called Optimal Transport Warping (OTW). OTW enjoys linear time/space complexity, is differentiable and can be parallelized. OTW enjoys a moderate sensitivity to time and shape distortions, making it ideal for time series. We show the efficacy and efficiency of OTW on 1-Nearest Neighbor Classification and Hierarchical Clustering, as well as in the case of using OTW instead of DTW in Deep Learning architectures.Comment: This is an extended version of an ICASSP 2023 accepted paper https://ieeexplore.ieee.org/document/1009591

Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging

Many graphics and vision problems can be expressed as non-linear least squares optimizations of objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance on modern GPUs in interactive applications. In this work, we propose a new language, Opt (available under http://optlang.org), for writing these objective functions over image- or graph-structured unknowns concisely and at a high level. Our compiler automatically transforms these specifications into state-of-the-art GPU solvers based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate different variations of the solver, so users can easily explore tradeoffs in numerical precision, matrix-free methods, and solver approaches. In our results, we implement a variety of real-world graphics and vision applications. Their energy functions are expressible in tens of lines of code, and produce highly-optimized GPU solver implementations. These solver have performance competitive with the best published hand-tuned, application-specific GPU solvers, and orders of magnitude beyond a general-purpose auto-generated solver