We propose a new operator-sketching paradigm for designing efficient iterative data-driven reconstruction (IDR) schemes, e.g. Plug-and-Play algorithms and deep unrolling networks. These IDR schemes are currently the state-of-the-art solutions for imaging inverse problems. However, for high-dimensional imaging tasks, especially X-ray CT and MRI imaging, these IDR schemes typically become inefficient both in terms of computation, due to the need of computing multiple times the high-dimensional forward and adjoint operators. In this work, we explore and propose a universal dimensionality reduction framework for accelerating IDR schemes in solving imaging inverse problems, based on leveraging the sketching techniques from stochastic optimization. Using this framework, we derive a number of accelerated IDR schemes, such as the plug-and-play multi-stage sketched gradient (PnP-MS2G) and sketching-based primal-dual (LSPD and Sk-LSPD) deep unrolling networks. Meanwhile, for fully accelerating PnP schemes when the denoisers are computationally expensive, we provide novel stochastic lazy denoising schemes (Lazy-PnP and Lazy-PnP-EQ), leveraging the ProxSkip scheme in optimization and equivariant image denoisers, which can massively accelerate the PnP algorithms with improved practicality. We provide theoretical analysis for recovery guarantees of instances of the proposed framework. Our numerical experiments on natural image processing and tomographic image reconstruction demonstrate the remarkable effectiveness of our sketched IDR schemes.

翻译：我们提出了一种新的算子草图范式，用于设计高效的迭代数据驱动重建（IDR）方案，例如即插即用算法和深度展开网络。这些IDR方案是目前成像逆问题的最先进解决方案。然而，对于高维成像任务，特别是X射线CT和MRI成像，由于需要多次计算高维前向算子和伴随算子，这些IDR方案通常在计算效率方面变得低下。在本工作中，我们基于随机优化中的草图技术，探索并提出了一种通用的降维框架，用于加速求解成像逆问题的IDR方案。利用该框架，我们推导出多种加速的IDR方案，例如即插即用多阶段草图梯度（PnP-MS2G）和基于草图的原始对偶（LSPD和Sk-LSPD）深度展开网络。同时，为了在去噪器计算成本高昂时全面加速PnP方案，我们利用优化中的ProxSkip方案和等变图像去噪器，提出了新颖的随机惰性去噪方案（Lazy-PnP和Lazy-PnP-EQ），这能大幅加速PnP算法并提升其实用性。我们为所提出框架的实例提供了恢复保证的理论分析。在自然图像处理和断层图像重建上的数值实验证明了我们草图IDR方案的显著有效性。