  We present a topological approach to the input-output relations of photonic
driven-dissipative lattices acting as directional amplifiers. Our theory relies
on a mapping from the optical non-Hermitian coupling matrix to an effective
topological insulator Hamiltonian. This mapping is based on the singular value
decomposition of non-Hermitian coupling matrices, whose inverse matrix
determines the linear input-output response of the system. In topologically
non-trivial regimes, the input-output response of the lattice is dominated by
singular vectors with zero singular values that are the equivalent of
zero-energy states in topological insulators, leading to directional
amplification of a coherent input signal. In such topological amplification
regime, our theoretical framework allows us to fully characterize the
amplification properties of the quantum device such as gain, bandwidth, added
noise, and noise-to-signal ratio. We exemplify our ideas in a one-dimensional
non-reciprocal photonic lattice, for which we derive fully analytical
predictions. We show that the directional amplification is near quantum-limited
with a gain growing exponentially with system size, $N$, while the
noise-to-signal ratio is suppressed as $1/\sqrt{N}$. This points out to
interesting applications of our theory for quantum signal amplification and
single-photon detection.
