  The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes the
notion of linearly disentangled representations, but there is currently no
metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and
to compare to previous understandings of disentanglement. We propose
$\mathcal{D}_\mathrm{LSBD}$, a mathematically sound metric to quantify LSBD,
and provide a practical implementation for $\mathrm{SO}(2)$ groups.
Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to
learn LSBD representations. We demonstrate the utility of our metric by showing
that (1) common VAE-based disentanglement methods don't learn LSBD
representations, (2) LSBD-VAE as well as other recent methods can learn LSBD
representations, needing only limited supervision on transformations, and (3)
various desirable properties expressed by existing disentanglement metrics are
also achieved by LSBD representations.
