Prior probability models are a central component of many image processing problems, but density estimation is notoriously difficult for high-dimensional signals such as photographic images. Deep neural networks have provided state-of-the-art solutions for problems such as denoising, which implicitly rely on a prior probability model of natural images. Here, we develop a robust and general methodology for making use of this implicit prior. We rely on a little-known statistical result due to Miyasawa (1961), who showed that the least-squares solution for removing additive Gaussian noise can be written directly in terms of the gradient of the log of the noisy signal density. We use this fact to develop a stochastic coarse-to-fine gradient ascent procedure for drawing high-probability samples from the implicit prior embedded within a CNN trained to perform blind (i.e., unknown noise level) least-squares denoising. A generalization of this algorithm to constrained sampling provides a method for using the implicit prior to solve any linear inverse problem, with no additional training. We demonstrate this general form of transfer learning in multiple applications, using the same algorithm to produce high-quality solutions for deblurring, super-resolution, inpainting, and compressive sensing.

Predicting the synthesizability of hypothetical crystals is challenging because of the wide range of parameters that govern materials synthesis. Yet, exploring the exponentially large space of novel crystals for any future application demands an accurate predictive capability for synthesis likelihood to avoid a haphazard trial-and-error. Typically, benchmarks of synthesizability are defined based on the energy of crystal structures. Here, we take an alternative approach to select features of synthesizability from the latent information embedded in crystalline materials. We represent the atomic structure of crystalline materials by three-dimensional pixel-wise images that are color-coded by their chemical attributes. The image representation of crystals enables the use of a convolutional encoder to learn the features of synthesizability hidden in structural and chemical arrangements of crystalline materials. Based on the presented model, we can accurately classify materials into synthesizable crystals versus crystal anomalies across a broad range of crystal structure types and chemical compositions. We illustrate the usefulness of the model by predicting the synthesizability of hypothetical crystals for battery electrode and thermoelectric applications.

We re-examine the problem of reconstructing a high-dimensional signal from a small set of linear measurements, in combination with image prior from a diffusion probabilistic model. Well-established methods for optimizing such measurements include principal component analysis (PCA), independent component analysis (ICA) and compressed sensing (CS), all of which rely on axis-or subspace-aligned statistical characterization. But many naturally occurring signals, including photographic images, contain richer statistical structure. To exploit such structure, we introduce a general method for obtaining an optimized set of linear measurements, assuming a Bayesian inverse solution that leverages the prior implicit in a neural network trained to perform denoising. We demonstrate that these measurements are distinct from those of PCA and CS, with significant improvements in minimizing squared reconstruction error. In addition, we show that optimizing the measurements for the SSIM perceptual loss leads to perceptually improved reconstruction. Our results highlight the importance of incorporating the specific statistical regularities of natural signals when designing effective linear measurements.

Deep neural networks can learn powerful prior probability models for images, as evidenced by the high-quality generations obtained with recent score-based diffusion methods. But the means by which these networks capture complex global statistical structure, apparently without suffering from the curse of dimensionality, remain a mystery. To study this, we incorporate diffusion methods into a multi-scale decomposition, reducing dimensionality by assuming a stationary local Markov model for wavelet coefficients conditioned on coarser-scale coefficients. We instantiate this model using convolutional neural networks (CNNs) with local receptive fields, which enforce both the stationarity and Markov properties. Global structures are captured using a CNN with receptive fields covering the entire (but small) low-pass image. We test this model on a dataset of face images, which are highly non-stationary and contain large-scale geometric structures. Remarkably, denoising, super-resolution, and image synthesis results all demonstrate that these structures can be captured with significantly smaller conditioning neighborhoods than required by a Markov model implemented in the pixel domain. Our results show that score estimation for large complex images can be reduced to low-dimensional Markov conditional models across scales, alleviating the curse of dimensionality.Deep neural networks (DNNs) have produced dramatic advances in synthesizing complex images and solving inverse problems, all of which rely (at least implicitly) on prior probability models. Of particular note is the recent development of "diffusion methods" (Sohl-Dickstein et al., 2015), in which a network trained for image denoising is incorporated into an iterative algorithm to draw samples from the prior (

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