The Star Geometry of Critic-Based Regularizer Learning

Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.

Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching

Generative models based on flow matching have attracted significant attention for their simplicity and superior performance in high-resolution image synthesis. By leveraging the instantaneous change-of-variables formula, one can directly compute image likelihoods from a learned flow, making them enticing candidates as priors for downstream tasks such as inverse problems. In particular, a natural approach would be to incorporate such image probabilities in a maximum-a-posteriori (MAP) estimation problem. A major obstacle, however, lies in the slow computation of the log-likelihood, as it requires backpropagating through an ODE solver, which can be prohibitively slow for high-dimensional problems. In this work, we propose an iterative algorithm to approximate the MAP estimator efficiently to solve a variety of linear inverse problems. Our algorithm is mathematically justified by the observation that the MAP objective can be approximated by a sum of $N$ ``local MAP'' objectives, where $N$ is the number of function evaluations. By leveraging Tweedie's formula, we show that we can perform gradient steps to sequentially optimize these objectives. We validate our approach for various linear inverse problems, such as super-resolution, deblurring, inpainting, and compressed sensing, and demonstrate that we can outperform other methods based on flow matching.

Score-Based Diffusion Models for Photoacoustic Tomography Image Reconstruction

Photoacoustic tomography (PAT) is a rapidly-evolving medical imaging modality that combines optical absorption contrast with ultrasound imaging depth. One challenge in PAT is image reconstruction with inadequate acoustic signals due to limited sensor coverage or due to the density of the transducer array. Such cases call for solving an ill-posed inverse reconstruction problem. In this work, we use score-based diffusion models to solve the inverse problem of reconstructing an image from limited PAT measurements. The proposed approach allows us to incorporate an expressive prior learned by a diffusion model on simulated vessel structures while still being robust to varying transducer sparsity conditions.

Ill-Posed Image Reconstruction Without an Image Prior

We consider solving ill-posed imaging inverse problems without access to an image prior or ground-truth examples. An overarching challenge in these inverse problems is that an infinite number of images, including many that are implausible, are consistent with the observed measurements. Thus, image priors are required to reduce the space of possible solutions to more desireable reconstructions. However, in many applications it is difficult or potentially impossible to obtain example images to construct an image prior. Hence inaccurate priors are often used, which inevitably result in biased solutions. Rather than solving an inverse problem using priors that encode the spatial structure of any one image, we propose to solve a set of inverse problems jointly by incorporating prior constraints on the collective structure of the underlying images. The key assumption of our work is that the underlying images we aim to reconstruct share common, low-dimensional structure. We show that such a set of inverse problems can be solved simultaneously without the use of a spatial image prior by instead inferring a shared image generator with a low-dimensional latent space. The parameters of the generator and latent embeddings are found by maximizing a proxy for the Evidence Lower Bound (ELBO). Once identified, the generator and latent embeddings can be combined to provide reconstructed images for each inverse problem. The framework we propose can handle general forward model corruptions, and we show that measurements derived from only a small number of ground-truth images ($\leqslant 150$) are sufficient for "prior-free" image reconstruction. We demonstrate our approach on a variety of convex and non-convex inverse problems, ranging from denoising, phase retrieval, and black hole video reconstruction.

Image Reconstruction without Explicit Priors

We consider solving ill-posed imaging inverse problems without access to an explicit image prior or ground-truth examples. An overarching challenge in inverse problems is that there are many undesired images that fit to the observed measurements, thus requiring image priors to constrain the space of possible solutions to more plausible reconstructions. However, in many applications it is difficult or potentially impossible to obtain ground-truth images to learn an image prior. Thus, inaccurate priors are often used, which inevitably result in biased solutions. Rather than solving an inverse problem using priors that encode the explicit structure of any one image, we propose to solve a set of inverse problems jointly by incorporating prior constraints on the collective structure of the underlying images.The key assumption of our work is that the ground-truth images we aim to reconstruct share common, low-dimensional structure. We show that such a set of inverse problems can be solved simultaneously by learning a shared image generator with a low-dimensional latent space. The parameters of the generator and latent embedding are learned by maximizing a proxy for the Evidence Lower Bound (ELBO). Once learned, the generator and latent embeddings can be combined to provide reconstructions for each inverse problem. The framework we propose can handle general forward model corruptions, and we show that measurements derived from only a few ground-truth images (O(10)) are sufficient for image reconstruction without explicit priors.

Optimal Convex and Nonconvex Regularizers for a Data Source

In optimization-based approaches to inverse problems and to statistical estimation, it is common to augment the objective with a regularizer to address challenges associated with ill-posedness. The choice of a suitable regularizer is typically driven by prior domain information and computational considerations. Convex regularizers are attractive as they are endowed with certificates of optimality as well as the toolkit of convex analysis, but exhibit a computational scaling that makes them ill-suited beyond moderate-sized problem instances. On the other hand, nonconvex regularizers can often be deployed at scale, but do not enjoy the certification properties associated with convex regularizers. In this paper, we seek a systematic understanding of the power and the limitations of convex regularization by investigating the following questions: Given a distribution, what are the optimal regularizers, both convex and nonconvex, for data drawn from the distribution? What properties of a data source govern whether it is amenable to convex regularization? We address these questions for the class of continuous and positively homogenous regularizers for which convex and nonconvex regularizers correspond, respectively, to convex bodies and star bodies. By leveraging dual Brunn-Minkowski theory, we show that a radial function derived from a data distribution is the key quantity for identifying optimal regularizers and for assessing the amenability of a data source to convex regularization. Using tools such as $\Gamma$-convergence, we show that our results are robust in the sense that the optimal regularizers for a sample drawn from a distribution converge to their population counterparts as the sample size grows large. Finally, we give generalization guarantees that recover previous results for polyhedral regularizers (i.e., dictionary learning) and lead to new ones for semidefinite regularizers.

Alternating Phase Langevin Sampling with Implicit Denoiser Priors for Phase Retrieval

Phase retrieval is the nonlinear inverse problem of recovering a true signal from its Fourier magnitude measurements. It arises in many applications such as astronomical imaging, X-Ray crystallography, microscopy, and more. The problem is highly ill-posed due to the phase-induced ambiguities and the large number of possible images that can fit to the given measurements. Thus, there's a rich history of enforcing structural priors to improve solutions including sparsity priors and deep-learning-based generative models. However, such priors are often limited in their representational capacity or generalizability to slightly different distributions. Recent advancements in using denoisers as regularizers for non-convex optimization algorithms have shown promising performance and generalization. We present a way of leveraging the prior implicitly learned by a denoiser to solve phase retrieval problems by incorporating it in a classical alternating minimization framework. Compared to performant denoising-based algorithms for phase retrieval, we showcase competitive performance with Fourier measurements on in-distribution images and notable improvement on out-of-distribution images.

Optimal Sample Complexity of Gradient Descent for Amplitude Flow via Non-Lipschitz Matrix Concentration

We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of gradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentration inequality when the measurement vectors are Gaussian as soon as $m = \Omega(n)$ with high probability. We then show that satisfaction of this inequality is sufficient for gradient descent with proper initialization to converge linearly to the true solution up to the global sign ambiguity. As a consequence, this guarantees local convergence for Gaussian measurements at optimal sample complexity. The concentration methods in the present work have previously been used to establish recovery guarantees for a variety of inverse problems under generative neural network priors. This paper demonstrates the applicability of these techniques to more traditional inverse problems and serves as a pedagogical introduction to those results.

Compressive Phase Retrieval: Optimal Sample Complexity with Deep Generative Priors

Advances in compressive sensing provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data-driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm in this challenging nonlinear inverse problem. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non-convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.

Low Shot Learning with Untrained Neural Networks for Imaging Inverse Problems

Employing deep neural networks as natural image priors to solve inverse problems either requires large amounts of data to sufficiently train expressive generative models or can succeed with no data via untrained neural networks. However, very few works have considered how to interpolate between these no- to high-data regimes. In particular, how can one use the availability of a small amount of data (even $5-25$ examples) to one's advantage in solving these inverse problems and can a system's performance increase as the amount of data increases as well? In this work, we consider solving linear inverse problems when given a small number of examples of images that are drawn from the same distribution as the image of interest. Comparing to untrained neural networks that use no data, we show how one can pre-train a neural network with a few given examples to improve reconstruction results in compressed sensing and semantic image recovery problems such as colorization. Our approach leads to improved reconstruction as the amount of available data increases and is on par with fully trained generative models, while requiring less than $1 \%$ of the data needed to train a generative model.