Uncertainty quantification is crucial for the deployment of image restoration models in safety-critical domains, like autonomous driving and biological imaging.

In ill-posed imaging inverse problems, there can exist many hypotheses that fit both the observed measurements and prior knowledge of the true image.

Reasoning is the fundamental capability of large language models (LLMs). Due to the rapid progress of LLMs, there are two main issues of current benchmarks: i) these benchmarks can be crushed in a short time (less than 1 year), and ii) these benchmarks may be easily hacked. To handle these issues, we propose the ever-scalingness for building the benchmarks which are uncrushable, unhackable, auto-verifiable and general. This paper presents Nondeterministic Polynomial-time Problem Challenge (NPPC), an ever-scaling reasoning benchmark for LLMs. Specifically, the NPPC has three main modules: i) npgym, which provides a unified interface of 25 well-known NP-complete problems and can generate any number of instances with any levels of complexities, ii) npsolver: which provides a unified interface to evaluate the problem instances with both online and offline models via APIs and local deployments, respectively, and iii) npeval: which provides the comprehensive and ready-to-use tools to analyze the performances of LLMs over different problems, the number of tokens, the aha moments, the reasoning errors and the solution errors. Extensive experiments over widely-used LLMs demonstrate: i) NPPC can successfully decrease the performances of advanced LLMs' performances to below 10%, demonstrating that NPPC is uncrushable, ii) DeepSeek-R1, Claude-3.7-Sonnet, and o1/o3-mini are the most powerful LLMs, where DeepSeek-R1 outperforms Claude-3.7-Sonnet and o1/o3-mini in most NP-complete problems considered, and iii) the numbers of tokens, aha moments in the advanced LLMs, e.g., Claude-3.7-Sonnet and DeepSeek-R1, are observed first to increase and then decrease when the problem instances become more and more difficult. We believe that NPPC is the first ever-scaling reasoning benchmark, serving as the uncrushable and unhackable testbed for LLMs toward artificial general intelligence (AGI).

In ill-posed imaging inverse problems, there can exist many hypotheses that fit both the observed measurements and prior knowledge of the true image. Rather than returning just one hypothesis of that image, posterior samplers aim to explore the full solution space by generating many probable hypotheses, which can later be used to quantify uncertainty or construct recoveries that appropriately navigate the perception/distortion trade-off. In this work, we propose a fast and accurate posterior-sampling conditional generative adversarial network (cGAN) that, through a novel form of regularization, aims for correctness in the posterior mean as well as the trace and K principal components of the posterior covariance matrix. Numerical experiments demonstrate that our method outperforms contemporary cGANs and diffusion models in imaging inverse problems like denoising, large-scale inpainting, and accelerated MRI recovery.

Uncertainty quantification is crucial for the deployment of image restoration models in safety-critical domains, like autonomous driving and biological imaging. To date, methods for uncertainty visualization have mainly focused on per-pixel estimates. Yet, a heatmap of per-pixel variances is typically of little practical use, as it does not capture the strong correlations between pixels. A more natural measure of uncertainty corresponds to the variances along the principal components (PCs) of the posterior distribution. Theoretically, the PCs can be computed by applying PCA on samples generated from a conditional generative model for the input image. However, this requires generating a very large number of samples at test time, which is painfully slow with the current state-of-the-art (diffusion) models. In this work, we present a method for predicting the PCs of the posterior distribution for any input image, in a single forward pass of a neural network. Our method can either wrap around a pre-trained model that was trained to minimize the mean square error (MSE), or can be trained from scratch to output both a predicted image and the posterior PCs. We showcase our method on multiple inverse problems in imaging, including denoising, inpainting, super-resolution, and biological image-to-image translation. Our method reliably conveys instance-adaptive uncertainty directions, achieving uncertainty quantification comparable with posterior samplers while being orders of magnitude faster. Code and examples are available at https://eliasnehme.github.io/NPPC/