  Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap
and extreme sensitivity to boundary conditions. Due to the NH skin effect, the
separation between edge and bulk states is blurred and the (conventional)
bulk-boundary correspondence is lost. Here, we restore the bulk-boundary
correspondence for the most paradigmatic class of NH Hamiltonians, namely those
with one complex band and without symmetries. We obtain the desired NH
Hamiltonian from the (mean-field) unconditional evolution of driven-dissipative
cavity arrays, in which NH terms -- in the form of non-reciprocal hopping
amplitudes, gain and loss -- are explicitly modeled via coupling to (engineered
and non-engineered) reservoirs. This approach removes the arbitrariness in the
definition of the topological invariant, as point-gapped spectra differing by a
complex-energy shift are not treated as equivalent; the origin of the complex
plane provides a common reference (base point) for the evaluation of the
topological invariant. This implies that topologically non-trivial Hamiltonians
are only a strict subset of those with a point gap and that the NH skin effect
does not have a topological origin. We analyze the NH Hamiltonians so obtained
via the singular value decomposition, which allows to express the NH
bulk-boundary correspondence in the following simple form: an integer value
$\nu$ of the topological invariant defined in the bulk corresponds to $\vert
\nu\vert$ singular vectors exponentially localized at the system edge under
open boundary conditions, in which the sign of $\nu$ determines which edge.
Non-trivial topology manifests as directional amplification of a coherent input
with gain exponential in system size. Our work solves an outstanding problem in
the theory of NH topological phases and opens up new avenues in topological
photonics.
