# Itakura–Saito distance

The Itakura–Saito distance is a measure of the perceptual difference between an original spectrum $P(\omega)$ and an approximation $\hat{P}(\omega)$ of that spectrum. It was proposed by Fumitada Itakura and Shuzo Saito in the 1970s while they were with NTT.

The distance is defined as:[1]

$D_{IS}(P(\omega),\hat{P}(\omega))=\frac{1}{2\pi}\int_{-\pi}^{\pi} \left[ \frac{P(\omega)}{\hat{P}(\omega)}-\log \frac{P(\omega)}{\hat{P}(\omega)} - 1 \right] \, d\omega$

The Itakura–Saito distance is a Bregman divergence, but is not a true metric since it is not symmetric[2] and it does not fulfil triangle inequality.