  In neural network-based monaural speech separation techniques, it has been
recently common to evaluate the loss using the permutation invariant training
(PIT) loss. However, the ordinary PIT requires to try all $N!$ permutations
between $N$ ground truths and $N$ estimates. Since the factorial complexity
explodes very rapidly as $N$ increases, a PIT-based training works only when
the number of source signals is small, such as $N = 2$ or $3$. To overcome this
limitation, this paper proposes a SinkPIT, a novel variant of the PIT losses,
which is much more efficient than the ordinary PIT loss when $N$ is large. The
SinkPIT is based on Sinkhorn's matrix balancing algorithm, which efficiently
finds a doubly stochastic matrix which approximates the best permutation in a
differentiable manner. The author conducted an experiment to train a neural
network model to decompose a single-channel mixture into 10 sources using the
SinkPIT, and obtained promising results.
