  This paper presents a study on the soft-Dice loss, one of the most popular
loss functions in medical image segmentation, for situations where noise is
present in target labels. In particular, the set of optimal solutions are
characterized and sharp bounds on the volume bias of these solutions are
provided. It is further shown that a sequence of soft segmentations converging
to optimal soft-Dice also converges to optimal Dice when converted to hard
segmentations using thresholding. This is an important result because soft-Dice
is often used as a proxy for maximizing the Dice metric. Finally, experiments
confirming the theoretical results are provided.
