Scalable Operator Learning via Nyström Approximation With Denoising Applications

In this paper, we study Nyström subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nyström subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical Hölder-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.

A Unified Framework for Forward and Inverse Problems in Subsurface Imaging using Latent Space Translations

In subsurface imaging, learning the mapping from velocity maps to seismic waveforms (forward problem) and waveforms to velocity (inverse problem) is important for several applications. While traditional techniques for solving forward and inverse problems are computationally prohibitive, there is a growing interest in leveraging recent advances in deep learning to learn the mapping between velocity maps and seismic waveform images directly from data. Despite the variety of architectures explored in previous works, several open questions still remain unanswered such as the effect of latent space sizes, the importance of manifold learning, the complexity of translation models, and the value of jointly solving forward and inverse problems. We propose a unified framework to systematically characterize prior research in this area termed the Generalized Forward-Inverse (GFI) framework, building on the assumption of manifolds and latent space translations. We show that GFI encompasses previous works in deep learning for subsurface imaging, which can be viewed as specific instantiations of GFI. We also propose two new model architectures within the framework of GFI: Latent U-Net and Invertible X-Net, leveraging the power of U-Nets for domain translation and the ability of IU-Nets to simultaneously learn forward and inverse translations, respectively. We show that our proposed models achieve state-of-the-art (SOTA) performance for forward and inverse problems on a wide range of synthetic datasets, and also investigate their zero-shot effectiveness on two real-world-like datasets.