Fundamentals of Computing Continuous Dynamic Time Warping in 2D under Different Norms
This work addresses the fundamental computational limitations of CDTW for shape matching and time series analysis, providing theoretical foundations and algorithms for practitioners.
The paper shows that Continuous Dynamic Time Warping (CDTW) cannot be computed exactly under the Euclidean 2-norm using algebraic operations, and provides an exact algorithm for CDTW under norms that approximate the 2-norm, with results generalizing to any norm and related measures.
Continuous Dynamic Time Warping (CDTW) measures the similarity of polygonal curves robustly to outliers and to sampling rates, but the design and analysis of CDTW algorithms face multiple challenges. We show that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and we give an exact algorithm for CDTW under norms approximating the 2-norm. The latter result relies on technical fundamentals that we establish, and which generalise to any norm and to related measures such as the partial Fréchet similarity.