  The Dice score and Jaccard index are commonly used metrics for the evaluation
of segmentation tasks in medical imaging. Convolutional neural networks trained
for image segmentation tasks are usually optimized for (weighted)
cross-entropy. This introduces an adverse discrepancy between the learning
optimization objective (the loss) and the end target metric. Recent works in
computer vision have proposed soft surrogates to alleviate this discrepancy and
directly optimize the desired metric, either through relaxations (soft-Dice,
soft-Jaccard) or submodular optimization (Lov\'asz-softmax). The aim of this
study is two-fold. First, we investigate the theoretical differences in a risk
minimization framework and question the existence of a weighted cross-entropy
loss with weights theoretically optimized to surrogate Dice or Jaccard. Second,
we empirically investigate the behavior of the aforementioned loss functions
w.r.t. evaluation with Dice score and Jaccard index on five medical
segmentation tasks. Through the application of relative approximation bounds,
we show that all surrogates are equivalent up to a multiplicative factor, and
that no optimal weighting of cross-entropy exists to approximate Dice or
Jaccard measures. We validate these findings empirically and show that, while
it is important to opt for one of the target metric surrogates rather than a
cross-entropy-based loss, the choice of the surrogate does not make a
statistical difference on a wide range of medical segmentation tasks.

Computer Vision and Pattern Recognition;Machine Learning;Image and Video Processing