  The nature of the transition to collective motion in assemblies of aligning
self-propelled particles remains a long-standing matter of debate. In this
article, we focus on dry active matter and show that weak fluctuations suffice
to generically turn second-order mean-field transitions into a `discontinuous'
coexistence scenario. Our theory shows how fluctuations induce a
density-dependence of the polar-field mass, even when this effect is absent at
mean-field level. In turn, this dependency on density triggers a feedback loop
between ordering and advection that ultimately leads to an inhomogeneous
transition to collective motion and the emergence of non-linear travelling
`flocks'. Importantly, we show that such a fluctuation-induced first order
transition is present in both metric models, in which particles align with
neighbors within a finite distance, and in topological ones, in which alignment
is not based on relative distances. We compute analytically the noise-induced
renormalization of the polar-field mass using stochastic calculus, which we
further back up by a one-loop field-theoretical analysis. Finally, we confirm
our analytical predictions by numerical simulations of fluctuating
hydrodynamics as well as of topological microscopic models with either
$k$-nearest neighbors or Voronoi alignment.
