Differentiable Representation of Warping Based on Lie Group Theory

Warping is a type of signal distortion where the sequential order is not changed. This operation is important for representing the alignment between series in signal processing. From the motivation of end-to-end optimization of series transformation processes using deep learning, some differentiable warping representations have been proposed. However, the conventional representations are not necessary and sufficient. That is, it cannot represent complex warping or contains non-warping. In this paper, we analyze the Lie group of warping, which is called General Warping group (GW), and investigate the representation of warping for continuous signals. We derive a complete matrix representation of warping that acts in the frequency domain. We show that this method can represent all warping without omission or error while remaining differentiable. Numerical experiments show that the warping curve can be optimized by gradient descent.