# Itakura–Saito distance

The Itakura–Saito distance (or Itakura–Saito divergence) is a measure of the difference between an original spectrum ${\displaystyle P(\omega )}$ and an approximation ${\displaystyle {\hat {P}}(\omega )}$ of that spectrum. Although it is not a perceptual measure it is intended to reflect perceptual (dis)similarity. It was proposed by Fumitada Itakura and Shuzo Saito in the 1960s while they were with NTT.[1]

The distance is defined as:[2]

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

The Itakura–Saito distance is a Bregman divergence, but is not a true metric since it is not symmetric[3] and it does not fulfil triangle inequality.

In Non-negative matrix factorization the Itakura-Saito divergence can be used as a measure of the quality of the factorization: this implies a meaningful statistical model of the components and can be solved through an iterative method.[4]