  Many datasets are obtained as a resolution trade-off between two adversarial
dimensions; for example between the frequency and the temporal resolutions for
the spectrogram of an audio signal, and between the number of wavelengths and
the spatial resolution for a hyper/multi-spectral image. To perform blind
source separation using observations with different resolutions, a standard
approach is to use coupled nonnegative matrix factorizations (NMF). Most
previous works have focused on the least squares error measure, which is the
$\beta$-divergence for $\beta = 2$. In this paper, we formulate this
multi-resolution NMF problem for any $\beta$-divergence, and propose an
algorithm based on multiplicative updates (MU). We show on numerical
experiments that the MU are able to obtain high resolutions in both dimensions
on two applications: (1) blind unmixing of audio spectrograms: to the best of
our knowledge, this is the first time a coupled NMF model is used in this
context, and (2) the fusion of hyperspectral and multispectral images: we show
that the MU compete favorable with state-of-the-art algorithms in particular in
the presence of non-Gaussian noise.
