Deep Rank-One Tensor Functional Factorization for Multi-Dimensional Data Recovery

Abstract Many real-world data are inherently multi-dimensional, e.g., color images, videos, and hyperspectral images. How to effectively and compactly represent these multi-dimensional data within a unified framework is an important pursuit. Previous methods focus on tensor factorizations, convolutional networks, or diffusion models for multi-dimensional data representation, which may not fully utilize inherent data structures and may lead to redundant parameters. In this work, we propose a Deep Rank-One Tensor Functional Factorization (DRO-TFF), which internally utilizes more comprehensive data priors facilitated by much fewer parameters. Concretely, our DRO-TFF consists of three organically integrated blocks: compact rank-one factorizations in the spatial domain, a deep transform to capture underlying low-dimensional structures, and smooth factors parameterized by implicit neural representations. Through a series of theoretical analysis, we show the rich data priors encoded in the DRO-TFF structure, e.g., Lipschitz smoothness and low-rankness. Extensive experiments on multi-dimensional data recovery problems, such as image and video inpainting, image denoising, and hyperspectral mixed noise removal, showcase the effectiveness of the proposed method.