Pretrained Transformers demonstrate remarkable in-context learning (ICL) capabilities, enabling them to adapt to new tasks from demonstrations without parameter updates. However, theoretical studies often rely on simplified architectures (e.g., omitting MLPs), plain data models (e.g., linear regression with isotropic inputs), and single-source training--limiting their relevance to realistic settings. In this work, we study ICL in pretrained Transformers with nonlinear MLP heads on nonlinear tasks drawn from multiple data sources with heterogeneous input, task, and noise distributions. We analyze a model where the MLP comprises two layers, with the first layer trained via a single gradient step and the second layer fully optimized. Under high-dimensional asymptotics, we prove that such models are equivalent in ICL error to structured polynomial predictors, leveraging results from the theory of Gaussian universality and orthogonal polynomials. This equivalence reveals that nonlinear MLPs meaningfully enhance ICL performance--particularly on nonlinear tasks--compared to linear baselines.

Diffusion models have recently demonstrated notable success in solving inverse problems. However, current diffusion model-based solutions typically require a large number of function evaluations (NFEs) to generate high-quality images conditioned on measurements, as they incorporate only limited information at each step. To accelerate the diffusion-based inverse problem-solving process, we introduce \textbf{M}easurements \textbf{O}ptimization (MO), a more efficient plug-and-play module for integrating measurement information at each step of the inverse problem-solving process. This method is comprehensively evaluated across eight diverse linear and nonlinear tasks on the FFHQ and ImageNet datasets. By using MO, we establish state-of-the-art (SOTA) performance across multiple tasks, with key advantages: (1) it operates with no more than 100 NFEs, with phase retrieval on ImageNet being the sole exception; (2) it achieves SOTA or near-SOTA results even at low NFE counts; and (3) it can be seamlessly integrated into existing diffusion model-based solutions for inverse problems, such as DPS \cite{chung2022diffusion} and Red-diff \cite{mardani2023variational}. For example, DPS-MO attains a peak signal-to-noise ratio (PSNR) of 28.71 dB on the FFHQ 256 dataset for high dynamic range imaging, setting a new SOTA benchmark with only 100 NFEs, whereas current methods require between 1000 and 4000 NFEs for comparable performance.

Diffusion models have shown strong performances in solving inverse problems through posterior sampling while they suffer from errors during earlier steps. To mitigate this issue, several Decoupled Posterior Sampling methods have been recently proposed. However, the reverse process in these methods ignores measurement information, leading to errors that impede effective optimization in subsequent steps. To solve this problem, we propose Guided Decoupled Posterior Sampling (GDPS) by integrating a data consistency constraint in the reverse process. The constraint performs a smoother transition within the optimization process, facilitating a more effective convergence toward the target distribution. Furthermore, we extend our method to latent diffusion models and Tweedie's formula, demonstrating its scalability. We evaluate GDPS on the FFHQ and ImageNet datasets across various linear and nonlinear tasks under both standard and challenging conditions. Experimental results demonstrate that GDPS achieves state-of-the-art performance, improving accuracy over existing methods.