MAP Estimation

Pretrained denoisers provide a powerful way to incorporate image priors into restoration algorithms. Plug-and-Play and RED approaches exploit fixed-noise-level denoisers within first-order optimization schemes, with convergence guarantees, but often struggle to achieve high-quality reconstruction on severely ill-posed inverse problems. In contrast, recent state-of-the-art approaches leverage denoisers derived from flow- or diffusion-based generative models and evaluate them along a sequence of decreasing noise levels. While these methods achieve strong empirical performance, their convergence theory remains limited. In this paper, we bridge this gap by specifically designing an algorithm that combines denoisers at decreasing noise levels with a schedule tailored to ensure convergence. From a Bayesian perspective, we prove that our method converges to a $\textit{Maximum a Posteriori}$ (MAP) estimate, under suitable assumptions. Subsequently, we apply our method to various ill-posed inverse problems and show that it surpasses convergent methods while competing with state-of-the-art empirical ones.

Inference-time alignment of pretrained text-to-image models is typically performed along a single control axis, such as classifier-free guidance, attention editing, or reward-based latent perturbations. This limitation prevents modeling joint dependencies between conditioning and latent variables and hinders transfer across generative transports. We propose PG-MAP, a training-free framework that formulates inference-time alignment as a trajectory-level Gibbs-MAP / proximal energy optimization over the conditioning $c$ and latent state $z_t$ via a forward-consistency coupling, optionally guided by a frozen preference reward. This joint formulation enables coordinated updates across modalities while remaining compatible with both diffusion and flow-matching models through transport-specific adaptations. Across diffusion backbones (SD~1.5, SDXL), PG-MAP consistently improves alignment metrics such as PickScore and Aesthetic, and can be effectively combined with tuned classifier-free guidance to achieve the strongest overall performance. On flow-matching models (SD3.5-medium), the framework reduces to a latent-only variant, achieving $\mathbf{91.9\%}$ PickScore and $75.7\%$ HPS win rates against a static baseline, with controlled experiments ruling out noise-related artifacts. Human evaluations further confirm consistent preference over strong baselines, including tuned CFG and compute-matched universal guidance. Finally, an oracle-routing analysis shows that the relative importance of conditioning and latent optimization depends on prompt types, surfacing further headroom that a per-prompt selector could exploit.

We recast predictive coding as continuous-time proximal gradient descent applied to a regularized maximum-a-posteriori (MAP) objective. We study first a single-level problem and then a multi-level hierarchy. For the single-level problem, we show that proximal gradient descent is precisely a leaky firing-rate network: the membrane leak, the effective recurrent matrix, the local synaptic drive, and the static nonlinearity all follow from one optimization principle, and the resulting circuit is the one proposed by Rao and Ballard. The prior selects the nonlinearity through its proximal operator, and the likelihood precision sets the gain on the observation. For the hierarchy, we show that a classical variable-splitting relaxation of the deep MAP problem yields hierarchical predictive coding as the interconnection of local and distributed solvers. In probabilistic modeling terms, this relaxation replaces the directed generative chain by an undirected Markov random field whose node potentials are the level-wise priors. Each level then applies its own activation function, namely the proximal operator of its prior.

Over the past decades, numerous Image Quality Assessment (IQA) models have emerged, aiming to predict the perceptual quality of images. However, individual models are often biased toward certain types of image content or distortions, depending on the design principle and process. An intuitive idea is to harness the strengths and mitigate the weaknesses of each IQA model, by fusing the scores of multiple models into a stronger one. Here we make one of the first attempts to seek an optimal solution for the idea and propose a general framework for unsupervised IQA score fusion using deep Maximum a Posteriori (MAP) estimation. The proposed model conducts fine-grained uncertainty estimation at the score level to increase the accuracy and reduce the uncertainty in fused predictions. Comprehensive experiments demonstrate the superiority of the proposed model over individual IQA models and other fusion methods. It also exhibits an interesting capability of rejecting ``bad" models in the fusion process.

Realizing relative localization by leveraging inter-robot local measurements is a challenging problem, especially in the presence of measurement noise. Motivated by this challenge, in this paper we propose a novel and systematic 3-D relative localization framework based on inter-robot interior angle and self-displacement measurements. Initially, we propose a linear relative localization theory comprising a distributed linear relative localization algorithm and sufficient conditions for localizability. According to this theory, robots can determine their neighbors' relative positions and orientations in a purely linear manner. Subsequently, in order to deal with measurement noise, we present an advanced Maximum a Posterior (MAP) estimator by addressing three primary challenges existing in the MAP estimator. Firstly, it is common to formulate the MAP problem as an optimization problem, whose inherent non-convexity can result in local optima. To address this issue, we reformulate the linear computation process of the linear relative localization algorithm as a Weighted Total Least Squares (WTLS) optimization problem on manifolds. The optimal solution of the WTLS problem is more accurate, which can then be used as initial values when solving the optimization problem associated with the MAP problem, thereby reducing the risk of falling into local optima. The second challenge is the lack of knowledge of the prior probability density of the robots' relative positions and orientations at the initial time, which is required as an input for the MAP estimator. To deal with it, we combine the WTLS with a Neural Density Estimator (NDE). Thirdly, to prevent the increasing size of the relative positions and orientations to be estimated as the robots continuously move when solving the MAP problem, a marginalization mechanism is designed, which ensures that the computational cost remains constant.

The calibration of extrinsic parameters and clock offsets between sensors for high-accuracy performance in underwater SLAM systems remains insufficiently explored. Existing methods for Doppler Velocity Log (DVL) calibration are either constrained to specific sensor configurations or rely on oversimplified assumptions, and none jointly estimate translational extrinsics and time offsets. We propose a Unified Iterative Calibration (UIC) framework for general DVL sensor setups, formulated as a Maximum A Posteriori (MAP) estimation with a Gaussian Process (GP) motion prior for high-fidelity motion interpolation. UIC alternates between efficient GP-based motion state updates and gradient-based calibration variable updates, supported by a provably statistically consistent sequential initialization scheme. The proposed UIC can be applied to IMU, cameras and other modalities as co-sensors. We release an open-source DVL-camera calibration toolbox. Beyond underwater applications, several aspects of UIC-such as the integration of GP priors for MAP-based calibration and the design of provably reliable initialization procedures-are broadly applicable to other multi-sensor calibration problems. Finally, simulations and real-world tests validate our approach.

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 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 **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.

Maximum a posteriori inference in a finite Markov random field is generally NP-hard. We study an instance-wise optimizer that minimizes the exact multilinear energy under product marginals. The canonical parameterization adds a GNN output to free per-node logits. This residual bypass is a global change of coordinates, so the lifting preserves the infimum, critical points, local extrema (modulo the GNN-parameter fiber), and Hessian inertia of the free-logit problem. Joint gradient flow then induces the preconditioner $I+JJ^{\top}$: the GNN cannot enlarge the feasible set, but it can add dissipation $\|J^{\top}\nabla\ell\|_{2}^{2}$. For a two-layer branch this extra coupling is four-hop local. We also show that every finite critical point of a nonconstant multilinear objective is a saddle, and that a unique MAP gap $Γ$ makes conditional-expectation rounding exact once the relaxed gap falls below $Γ$; residual gradient flow and joint gradient descent then obey explicit $O(1/t)$ and $O(1/k)$ rates in a small-gap region, with an improved bound under residual alignment. Experiments on synthetic MRFs, UAI 2022 benchmarks, and Physical Cell Identity (PCI) problems evaluate this residual geometry at scale.