This paper describes an efficient algorithm for solving noisy linear inverse problems using pretrained diffusion models. Extending the paradigm of denoising diffusion implicit models (DDIM), we propose constrained diffusion implicit models (CDIM) that modify the diffusion updates to enforce a constraint upon the final output. For noiseless inverse problems, CDIM exactly satisfies the constraints; in the noisy case, we generalize CDIM to satisfy an exact constraint on the residual distribution of the noise. Experiments across a variety of tasks and metrics show strong performance of CDIM, with analogous inference acceleration to unconstrained DDIM: 10 to 50 times faster than previous conditional diffusion methods. We demonstrate the versatility of our approach on many problems including super-resolution, denoising, inpainting, deblurring, and 3D point cloud reconstruction.

This paper introduces an alternative approach to sampling from autoregressive models. Autoregressive models are typically sampled sequentially, according to the transition dynamics defined by the model. Instead, we propose a sampling procedure that initializes a sequence with white noise and follows a Markov chain defined by Langevin dynamics on the global log-likelihood of the sequence. This approach parallelizes the sampling process and generalizes to conditional sampling. Using an autoregressive model as a Bayesian prior, we can steer the output of a generative model using a conditional likelihood or constraints. We apply these techniques to autoregressive models in the visual and audio domains, with competitive results for audio source separation, super-resolution, and inpainting.

Given a multi-microphone recording of an unknown number of speakers talking concurrently, we simultaneously localize the sources and separate the individual speakers. At the core of our method is a deep network, in the waveform domain, which isolates sources within an angular region $\theta \pm w/2$, given an angle of interest $\theta$ and angular window size $w$. By exponentially decreasing $w$, we can perform a binary search to localize and separate all sources in logarithmic time. Our algorithm allows for an arbitrary number of potentially moving speakers at test time, including more speakers than seen during training. Experiments demonstrate state-of-the-art performance for both source separation and source localization, particularly in high levels of background noise.