  Metric learning has received conflicting assessments concerning its
suitability for solving instance segmentation tasks. It has been dismissed as
theoretically flawed due to the shift equivariance of the employed CNNs and
their respective inability to distinguish same-looking objects. Yet it has been
shown to yield state of the art results for a variety of tasks, and practical
issues have mainly been reported in the context of tile-and-stitch approaches,
where discontinuities at tile boundaries have been observed. To date, neither
of the reported issues have undergone thorough formal analysis. In our work, we
contribute a comprehensive formal analysis of the shift equivariance properties
of encoder-decoder-style CNNs, which yields a clear picture of what can and
cannot be achieved with metric learning in the face of same-looking objects. In
particular, we prove that a standard encoder-decoder network that takes
$d$-dimensional images as input, with $l$ pooling layers and pooling factor
$f$, has the capacity to distinguish at most $f^{dl}$ same-looking objects, and
we show that this upper limit can be reached. Furthermore, we show that to
avoid discontinuities in a tile-and-stitch approach, assuming standard batch
size 1, it is necessary to employ valid convolutions in combination with a
training output window size strictly greater than $f^l$, while at test-time it
is necessary to crop tiles to size $n\cdot f^l$ before stitching, with $n\geq
1$. We complement these theoretical findings by discussing a number of
insightful special cases for which we show empirical results on synthetic data.
