  Recent diffusion models provide a promising zero-shot solution to noisy
linear inverse problems without retraining for specific inverse problems. In
this paper, we reveal that recent methods can be uniformly interpreted as
employing a Gaussian approximation with hand-crafted isotropic covariance for
the intractable denoising posterior to approximate the conditional posterior
mean. Inspired by this finding, we propose to improve recent methods by using
more principled covariance determined by maximum likelihood estimation. To
achieve posterior covariance optimization without retraining, we provide
general plug-and-play solutions based on two approaches specifically designed
for leveraging pre-trained models with and without reverse covariance. We
further propose a scalable method for learning posterior covariance prediction
based on representation with orthonormal basis. Experimental results
demonstrate that the proposed methods significantly enhance reconstruction
performance without requiring hyperparameter tuning.
