The Itakura–Saito distance (or Itakura–Saito divergence) is a measure of the difference between an original
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
and an approximation
of that spectrum. Although it is not a perceptual measure, it is intended to reflect perceptual (dis)
similarity
Similarity may refer to:
In mathematics and computing
* Similarity (geometry), the property of sharing the same shape
* Matrix similarity, a relation between matrices
* Similarity measure, a function that quantifies the similarity of two objects
* ...
. It was proposed by
Fumitada Itakura is a Japanese scientist. He did pioneering work in statistical signal processing, and its application to speech analysis, synthesis and coding, including the development of the linear predictive coding (LPC) and line spectral pairs (LSP) meth ...
and
Shuzo Saito in the 1960s while they were with
NTT.
The distance is defined as:
:
The Itakura–Saito distance is a
Bregman divergence
In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. ...
generated by minus the logarithmic function, but is not a true
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
since it is not symmetric and it does not fulfil triangle inequality.
In
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix is factorized into (usually) two matrices and , with the property that ...
, 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
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
.
The Itakura-Saito distance is the Bregman divergence associated with the Gamma exponential family where the information divergence of one distribution in the family from another element in the family is given by the Itakura-Saito divergence of the mean value of the first distribution from the mean value of the second distribution.
See also
*
Log-spectral distance The log-spectral distance (LSD), also referred to as log-spectral distortion or root mean square log-spectral distance, is a distance measure between two spectra. The log-spectral distance between spectra P\left(\omega\right) and \hat\left(\omega\r ...
References
{{DEFAULTSORT:Itakura-Saito distance
Signal processing