Recently, learned image compression (LIC) has garnered increasing interest with its rapidly improving performance surpassing conventional codecs. A key ingredient of LIC is a hyperprior-based entropy model, where the underlying joint probability of the latent image features is modeled as a product of Gaussian distributions from each latent element. Since latents from the actual images are not spatially independent, autoregressive (AR) context based entropy models were proposed to handle the discrepancy between the assumed distribution and the actual distribution. Though the AR-based models have proven effective, the computational complexity is significantly increased due to the inherent sequential nature of the algorithm. In this paper, we present a novel alternative to the AR-based approach that can provide a significantly better trade-off between performance and complexity. To minimize the discrepancy, we introduce a correlation loss that forces the latents to be spatially decorrelated and better fitted to the independent probability model. Our correlation loss is proved to act as a general plug-in for the hyperprior (HP) based learned image compression methods. The performance gain from our correlation loss is'free' in terms of computation complexity for both inference time and decoding time. To our knowledge, our method gives the best trade-off between the complexity and performance: combined with the Checkerboard-CM, it attains 90% and when combined with ChARM-CM, it attains 98% of the AR-based BD-Rate gains yet is around 50 times and 30 times faster than AR-based methods respectively.

Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.

Gaussian processes are flexible function approximators, with inductive biases controlledbyacovariancekernel.

In the digital world, visual media has proven to be a mixed blessing.

Most recent methods for learning-based, lossy image compression adopt an approach based on transform coding [1]. In this approach, image compression is achieved by first mapping pixel data into a quantized latent representation and then losslessly compressing the latents.

This paper presents a comprehensive analysis of hyperparameter estimation within the empirical Bayes framework (EBF) for sparse learning. By studying the influence of hyperpriors on the solution of EBF, we establish a theoretical connection between the choice of the hyperprior and the sparsity as well as the local optimality of the resulting solutions. We show that some strictly increasing hyperpriors, such as half-Laplace and half-generalized Gaussian with the power in $(0,1)$, effectively promote sparsity and improve solution stability with respect to measurement noise. Based on this analysis, we adopt a proximal alternating linearized minimization (PALM) algorithm with convergence guaranties for both convex and concave hyperpriors. Extensive numerical tests on two-dimensional image deblurring problems demonstrate that introducing appropriate hyperpriors significantly promotes the sparsity of the solution and enhances restoration accuracy. Furthermore, we illustrate the influence of the noise level and the ill-posedness of inverse problems to EBF solutions.

Complex inference tasks, such as those encountered in Pulsar Timing Array (PTA) data analysis, rely on Bayesian frameworks. The high-dimensional parameter space and the strong interdependencies among astrophysical, pulsar noise, and nuisance parameters introduce significant challenges for efficient learning and robust inference. These challenges are emblematic of broader issues in decision science, where model over-parameterization and prior sensitivity can compromise both computational tractability and the reliability of the results. We address these issues in the framework of hierarchical Bayesian modeling by introducing a reparameterization strategy. Our approach employs Normalizing Flows (NFs) to decorrelate the parameters governing hierarchical priors from those of astrophysical interest. The use of NF-based mappings provides both the flexibility to realize the reparametrization and the tractability to preserve proper probability densities. We further adopt i-nessai, a flow-guided nested sampler, to accelerate exploration of complex posteriors. This unified use of NFs improves statistical robustness and computational efficiency, providing a principled methodology for addressing hierarchical Bayesian inference in PTA analysis.

Pulsar Timing Arrays provide a powerful framework to measure low-frequency gravitational waves, but accuracy and robustness of the results are challenged by complex noise processes that must be accurately modeled. Standard PTA analyses assign fixed uniform noise priors to each pulsar, an approach that can introduce systematic biases when combining the array. To overcome this limitation, we adopt a hierarchical Bayesian modeling strategy in which noise priors are parametrized by higher-level hyperparameters. We further address the challenge posed by the correlations between hyperparameters and physical noise parameters, focusing on those describing red noise and dispersion measure variations. To decorrelate these quantities, we introduce an orthogonal reparametrization of the hierarchical model implemented with Normalizing Flows. We also employ i-nessai, a flow-guided nested sampler, to efficiently explore the resulting higher-dimensional parameter space. We apply our method to a minimal 3-pulsar case study, performing a simultaneous inference of noise and SGWB parameters. Despite the limited dataset, the results consistently show that the hierarchical treatment constrains the noise parameters more tightly and partially alleviates the red-noise-SGWB degeneracy, while the orthogonal reparametrization further enhances parameter independence without affecting the correlations intrinsic to the power-law modeling of the physical processes involved.