  Nonnegative Matrix Factorization (NMF) is a versatile and powerful tool for
discovering latent structures in data matrices, with many variations proposed
in the literature. Recently, Leplat et al.\@ (2019) introduced a minimum-volume
NMF for the identifiable recovery of rank-deficient matrices in the presence of
noise. The performance of their formulation, however, requires the selection of
a tuning parameter whose optimal value depends on the unknown noise level. In
this work, we propose an alternative formulation of minimum-volume NMF inspired
by the square-root lasso and its tuning-free properties. Our formulation also
requires the selection of a tuning parameter, but its optimal value does not
depend on the noise level. To fit our NMF model, we propose a
majorization-minimization (MM) algorithm that comes with global convergence
guarantees. We show empirically that the optimal choice of our tuning parameter
is insensitive to the noise level in the data.
