  We investigate the use of persistent homology, a tool from topological data
analysis, as a means to detect and quantitatively describe center vortices in
$\mathrm{SU}(2)$ lattice gauge theory in a gauge-invariant manner. We provide
evidence for the sensitivity of our method to vortices by detecting a vortex
explicitly inserted using twisted boundary conditions in the deconfined phase.
This inspires the definition of a new phase indicator for the deconfinement
phase transition. We also construct a phase indicator without reference to
twisted boundary conditions using a simple $k$-nearest neighbours classifier.
Finite-size scaling analyses of both persistence-based indicators yield
accurate estimates of the critical $\beta$ and critical exponent of correlation
length $\nu$ of the deconfinement phase transition.
