Low-rank tensor estimation offers a powerful approach to addressing high-dimensional data challenges and can substantially improve solutions to ill-posed inverse problems, such as image reconstruction under noisy or undersampled conditions. Meanwhile, tensor decomposition has gained prominence in federated learning (FL) due to its effectiveness in exploiting latent space structure and its capacity to enhance communication efficiency. In this paper, we present a federated image reconstruction method that applies Tucker decomposition, incorporating joint factorization and randomized sketching to manage large-scale, multimodal data. Our approach avoids reconstructing full-size tensors and supports heterogeneous ranks, allowing clients to select personalized decomposition ranks based on prior knowledge or communication capacity. Numerical results demonstrate that our method achieves superior reconstruction quality and communication compression compared to existing approaches, thereby highlighting its potential for multimodal inverse problems in the FL setting.

翻译：低秩张量估计为应对高维数据挑战提供了有力方法，并能显著改善病态逆问题（如噪声或欠采样条件下的图像重建）的求解。同时，张量分解因其在挖掘隐空间结构方面的有效性及提升通信效率的能力，在联邦学习（FL）中日益受到重视。本文提出一种基于Tucker分解的联邦图像重建方法，该方法结合联合因子化与随机草图技术以处理大规模多模态数据。我们的方法避免了重建全尺寸张量，并支持异构秩设定，允许客户端根据先验知识或通信能力选择个性化的分解秩。数值实验表明，相较于现有方法，本方法在重建质量与通信压缩方面均表现出优越性，从而凸显了其在联邦学习环境下处理多模态逆问题的潜力。