  Score-based diffusion models have significantly advanced high-dimensional
data generation across various domains, by learning a denoising oracle (or
score) from datasets. From a Bayesian perspective, they offer a realistic
modeling of data priors and facilitate solving inverse problems through
posterior sampling. Although many heuristic methods have been developed
recently for this purpose, they lack the quantitative guarantees needed in many
scientific applications.
  In this work, we introduce the \textit{tilted transport} technique, which
leverages the quadratic structure of the log-likelihood in linear inverse
problems in combination with the prior denoising oracle to transform the
original posterior sampling problem into a new `boosted' posterior that is
provably easier to sample from. We quantify the conditions under which this
boosted posterior is strongly log-concave, highlighting the dependencies on the
condition number of the measurement matrix and the signal-to-noise ratio. The
resulting posterior sampling scheme is shown to reach the computational
threshold predicted for sampling Ising models [Kunisky'23] with a direct
analysis, and is further validated on high-dimensional Gaussian mixture models
and scalar field $\varphi^4$ models.
