Ill-Posed Inverse Problem

Maximum A Posteriori (MAP) estimation is a cornerstone framework for blind inverse problems, where an image and a forward operator are jointly estimated as the maximizers of a posterior distribution. In applications such as blind deblurring, this principle is used to recover sharp images from degraded observations. In this paper, we analyze the recovery guarantees of MAP-based methods by adopting a \emph{Morse--Bott framework}. We model the image potential as a Morse--Bott function, where natural images are modeled as residing locally on a critical submanifold. This means that while the potential is locally flat along the natural'' directions of the image manifold, it is strictly convex in the directions normal to it. We demonstrate that this Morse--Bott hypothesis aligns with the structural properties of state-of-the-art learned priors, a finding we validate through an experimental analysis of the potential landscape and its Hessian spectrum. Our theoretical results show that, in a neighborhood of the ground-truth image and operator, the posterior admits local minimizers that are stable both with respect to initialization (gradient descents converge to the same minimizer) and to small perturbations of the data (solutions vary smoothly with the observations). This local stability potentially provides a theoretical justification for the empirical success of well designed gradient-based optimization in these settings. However, we also demonstrate that this local stability is a \textbf{local} property: the blurry trap'', well-known for sparse priors in blind deconvolution, persists even with state-of-the-art learned priors. Our findings demonstrate that the failure of MAP in blind deconvolution is not a limitation of prior quality, but an intrinsic characteristic of the landscape. We conclude that successful recovery depends on strategic initialization around favorable local minima.

Diffusion models represent the state-of-the-art for solving inverse problems such as image restoration tasks. Diffusion-based inverse solvers incorporate a likelihood term to guide prior sampling, generating data consistent with the posterior distribution. However, due to the intractability of the likelihood, most methods rely on isotropic Gaussian approximations, which can push estimates off the data manifold and produce inconsistent, poor reconstructions. We propose Equivariance Regularized (EquiReg) diffusion, a general plug-in framework that improves posterior sampling by penalizing trajectories that deviate from the data manifold. EquiReg formalizes manifold-preferential equivariant functions that exhibit low equivariance error for on-manifold samples and high error for off-manifold ones, thereby guiding sampling toward symmetry-preserving regions of the solution space. We highlight that such functions naturally emerge when training non-equivariant models with augmentation or on data with symmetries. EquiReg's largest gains are under reduced sampling and measurement consistency steps, where many methods suffer severe quality degradation. By regularizing trajectories toward the manifold, EquiReg implicitly accelerates convergence and enables high-quality reconstructions. EquiReg consistently improves performance in linear and nonlinear image restoration tasks and solving partial differential equations. Our code is available at https://github.com/Anima-Lab/EquiReg

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO$^\star$ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

We introduce Superstate Quantum Mechanics (SQM), a theory that considers states in Hilbert space subject to multiple quadratic constraints, with energy'' also expressed as a quadratic function of these states. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization with energy expressed as a quadratic form involving the Hamiltonian. When SQM represents states as unitary operators, the stationary problem becomes a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. Any stationary SQM problem is equivalent to a new algebraic problem that we address in this paper. The non-stationary SQM problem considers the evolution of the system itself, involving the same energy'' operator as in the stationary case. Two possible options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. As an immediately available practical application of the theory, we consider using a quantum channel as a classical computational model; this type of computation can be performed on a classical computer.

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to $64$, where the recovered prior is evaluated in a single forward pass.