On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

Abstract

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from [45], local and global stability properties of the model are identified under which such posterior distributions can be approximated in Wasserstein distance by suitable log-concave measures. This allows the use of fast gradient based sampling algorithms, for which convergence guarantees are established that scale polynomially in all relevant quantities (assuming 'warm' initialisation). The scope of the general theory is illustrated in a non-linear inverse problem from integral geometry for which new stability results are derived.

The Transport Oka-Grauert Principle for Simple Surfaces

“…In the context of Polarimetric Neutron Tomography, it has been of recent interest to rigorously study statistical algorithms for recovering a matrix field Φ from noisy measurements of C Φ [25,24]. In particular it was shown in [4], that if M is the Euclidean unit disk, then Φ can be recovered by a statistical algorithm in polynomial time, provided there is a suitable initialiser. Knowing the range of Φ → C Φ is a possible starting point to construct a computable intitialiser -we hope to address this in furture work.…”

On some information-theoretic aspects of non-linear statistical inverse problems

“…Local curvature of G . The quantitative nature of (47) in Theorem 5 is compatible with 'gradient stability conditions' employed in [25,3] to establish polynomial time posterior computation time bounds for gradient based Langevin MCMC schemes. Specifically, arguing as in Lemma 4.7 in [25], for a neighbourhood B of θ 0 one can deduce local average 'curvature' inf…”

“…While the success of this approach has become evident empirically, an objective mathematical framework that allows to give rigorous statistical and computational guarantees for such algorithms in non-linear problems has only emerged more recently. The types of results obtained so far include statistical consistency and contraction rate results for posterior distributions and their means, see [20,1,13] and also [23,21,22,14,16], as well as computational guarantees for MCMC based sampling schemes [15,25,3].…”

Stability estimate for the broken non-abelian X-ray transform in Minkowski space

“…To do so, we follow the approach first laid out in [MNP21]. Specifically, we will use Theorem 5.1 in [BN21] as it only requires checking a nice set of conditions.…”

“…Hence, we can choose γ = α−4 α−3 for α ≥ 5 by taking α = k + 3. We can finally apply Theorem 5.1 in [BN21] to get the following estimate regarding the concentration of the posterior distribution around the real parameter A obtained through noisy samples of S A as the number of samples goes to infinity. Theorem 4.2.…”