A Closed-form Gradient for the 1D Earth Mover’s Distance for Spectral Deep Learning on Biological Data

Abstract

Spectral analysis is performed in several domains of physiological monitoring (e.g. respiratory analysis [1], EEG [2], ECG [3]). Regression models in the spectral domain enable several applications, often through the use of Power Spectral Density (PSD). Within machine learning frameworks, PSD is commonly treated as a probability distribution and learned using the KüllbackLeibler (KL) divergence. However, KL compares each bin independently. The Earth Mover’s Distance (EMD) is a natural metric to compare distributions, but has seen limited use due to its computational cost. Nevertheless, for one dimensional distributions (e.g. PSD) the EMD can be computed efficiently, and we derive a closed-form solution for its gradient. We enforce the gradient to preserve the `1 norm of the original distribution. We evaluate on a data set of 81 sleep laboratory patients, predict breathing rate, and compare EMD as a loss against KL divergence and Mean Squared Error.

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Cite this paper

@inproceedings{Martinez2016ACG, title={A Closed-form Gradient for the 1D Earth Mover’s Distance for Spectral Deep Learning on Biological Data}, author={Manuel Martinez and Makarand Tapaswi and Rainer Stiefelhagen}, year={2016} }