  Many spectral unmixing methods rely on the non-negative decomposition of
spectral data onto a dictionary of spectral templates. In particular,
state-of-the-art music transcription systems decompose the spectrogram of the
input signal onto a dictionary of representative note spectra. The typical
measures of fit used to quantify the adequacy of the decomposition compare the
data and template entries frequency-wise. As such, small displacements of
energy from a frequency bin to another as well as variations of timber can
disproportionally harm the fit. We address these issues by means of optimal
transportation and propose a new measure of fit that treats the frequency
distributions of energy holistically as opposed to frequency-wise. Building on
the harmonic nature of sound, the new measure is invariant to shifts of energy
to harmonically-related frequencies, as well as to small and local
displacements of energy. Equipped with this new measure of fit, the dictionary
of note templates can be considerably simplified to a set of Dirac vectors
located at the target fundamental frequencies (musical pitch values). This in
turns gives ground to a very fast and simple decomposition algorithm that
achieves state-of-the-art performance on real musical data.
