This paper proposes a unified framework for inferring large-scale time-varying correlation networks via data-driven time-varying thresholds that can control uncertainty simultaneously. The framework allows the dimension of time series vectors to be fixed or diverging at a high polynomial rate of the sample size. It also allows the time series to exhibit changing temporal characteristics beyond stationarity without specific structural assumptions. Motivated by the practical issue that the confidence band of non-parametric estimators of correlations can exceed their natural domain [−1, 1] (see, for example [1])), we propose a simple uniform variance reduction technique. When applied to the construction of a correlation network, the new device yields more accurate thresholds, which enhance the probability of recovering the time-varying network structures. We broaden the applicability of our method by developing difference-based estimators of cross-correlations that are robust to structure breaks in the time-varying mean functions, and by allowing both a fixed and a diverging number of lags in the correlation functions. We prove the asymptotic validity of the proposed method, especially in achieving accurate family-wise error control when disclosing flexible time-varying network structures. The effectiveness of our method in finite samples is demonstrated through simulation studies and data analysis.

Diffusion probabilistic models (DPMs) have demonstrated a very promising ability in high-resolution image synthesis. However, sampling from a pre-trained DPM is time-consuming due to the multiple evaluations of the denoising network, making it more and more important to accelerate the sampling of DPMs. Despite recent progress in designing fast samplers, existing methods still cannot generate satisfying images in many applications where fewer steps (e.g., $<$10) are favored. In this paper, we develop a unified corrector (UniC) that can be applied after any existing DPM sampler to increase the order of accuracy without extra model evaluations, and derive a unified predictor (UniP) that supports arbitrary order as a byproduct. Combining UniP and UniC, we propose a unified predictor-corrector framework called UniPC for the fast sampling of DPMs, which has a unified analytical form for any order and can significantly improve the sampling quality over previous methods, especially in extremely few steps. We evaluate our methods through extensive experiments including both unconditional and conditional sampling using pixel-space and latent-space DPMs. Our UniPC can achieve 3.87 FID on CIFAR10 (unconditional) and 7.51 FID on ImageNet 256$\times$256 (conditional) with only 10 function evaluations. Code is available at https://github.com/wl-zhao/UniPC.