Diffusion models have recently shown considerable potential in solving Bayesian inverse problems when used as priors. However, sampling from the resulting denoising posterior distributions remains a challenge as it involves intractable terms. To tackle this issue, state-of-the-art approaches formulate the problem as that of sampling from a surrogate diffusion model targeting the posterior and decompose its scores into two terms: the prior score and an intractable guidance term. While the former is replaced by the pre-trained score of the considered diffusion model, the guidance term has to be estimated. In this paper, we propose a novel approach that utilises a decomposition of the transitions which, in contrast to previous methods, allows a trade-off between the complexity of the intractable guidance term and that of the prior transitions. We validate the proposed approach through extensive experiments on linear and nonlinear inverse problems, including challenging cases with latent diffusion models as priors, and demonstrate its effectiveness in reconstructing electrocardiogram (ECG) from partial measurements for accurate cardiac diagnosis.

Recent works in Variational Inference have examined alternative criteria to the commonly used exclusive Kullback-Leibler divergence. Encouraging empirical results have been obtained with the family of alpha-divergences, but few works have focused on the asymptotic properties of the proposed algorithms, especially as the number of iterations goes to infinity. In this paper, we study a procedure that ensures a monotonic decrease in the alpha-divergence. We provide sufficient conditions to guarantee its convergence to a local minimizer of the alpha-divergence at a geometric rate when the variational family belongs to the class of exponential models. The sample-based version of this ideal procedure involves biased gradient estimators, thus hindering any theoretical study. We propose an alternative unbiased algorithm, we prove its almost sure convergence to a local minimizer of the alpha-divergence, and a law of the iterated logarithm. Our results are exemplified with toy and real-data experiments.

In this paper, we introduce a novel family of iterative algorithms which carry out $\alpha$-divergence minimisation in a Variational Inference context. They do so by ensuring a systematic decrease at each step in the $\alpha$-divergence between the variational and the posterior distributions. In its most general form, the variational distribution is a mixture model and our framework allows us to simultaneously optimise the weights and components parameters of this mixture model. Our approach permits us to build on various methods previously proposed for $\alpha$-divergence minimisation such as Gradient or Power Descent schemes and we also shed a new light on an integrated Expectation Maximization algorithm. Lastly, we provide empirical evidence that our methodology yields improved results on several multimodal target distributions and on a real data example.

This paper focuses on $\alpha$-divergence minimisation methods for Variational Inference. More precisely, we are interested in algorithms optimising the mixture weights of any given mixture model, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all $\alpha \neq 1$, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when $\alpha <1$. Since the $\alpha$-divergence recovers the widely-used forward Kullback-Leibler when $\alpha \to 1$, we then extend the Power Descent to the case $\alpha = 1$ and show that we obtain an Entropic Mirror Descent. This leads us to investigate the link between Power Descent and Entropic Mirror Descent: first-order approximations allow us to introduce the R\'{e}nyi Descent, a novel algorithm for which we prove an $O(1/N)$ convergence rate. Lastly, we compare numerically the behavior of the unbiased Power Descent and of the biased R\'{e}nyi Descent and we discuss the potential advantages of one algorithm over the other.