  Identifying a low-dimensional informed parameter subspace offers a viable
path to alleviating the dimensionality challenge in the sampled-based solution
to large-scale Bayesian inverse problems. This paper introduces a novel
gradient-based dimension reduction method in which the informed subspace does
not depend on the data. This permits an online-offline computational strategy
where the expensive low-dimensional structure of the problem is detected in an
offline phase, meaning before observing the data. This strategy is particularly
relevant for multiple inversion problems as the same informed subspace can be
reused. The proposed approach allows controlling the approximation error (in
expectation over the data) of the posterior distribution. We also present
sampling strategies that exploit the informed subspace to draw efficiently
samples from the exact posterior distribution. The method is successfully
illustrated on two numerical examples: a PDE-based inverse problem with a
Gaussian process prior and a tomography problem with Poisson data and a
Besov-$\mathcal{B}^2_{11}$ prior.
