加速与近似软插值用于大规模矩阵和张量补全
Accelerated and Inexact Soft-Impute for Large-Scale Matrix and Tensor Completion
AI总结：
本文提出了一种加速的软插值算法，通过近似奇异值阈值化方案降低迭代复杂度，实现低秩矩阵和张量补全的快速收敛。
摘要：
矩阵和张量补全旨在从有限观测中恢复低秩矩阵/张量，广泛应用于推荐系统和多关系数据挖掘。软插值是一种先进的矩阵补全算法，利用矩阵迭代的特殊“稀疏加低秩”结构实现高效SVD。尽管软插值是近端算法，但通常认为加速会破坏该结构而不具实用性。本文证明软插值确实可以加速而不破坏该结构。为进一步减少迭代时间复杂度，我们提出基于幂方法的近似奇异值阈值化方案。理论分析表明，所提算法仍保持加速近端算法的快速O(1/T²)收敛速度。我们进一步将所提算法扩展到张量补全，使用缩放的潜在核范数正则化器。我们证明存在类似的“稀疏加低秩”结构，导致低迭代复杂度和快速O(1/T²)收敛速度。大量实验表明，所提算法比软插值和其他最先进的矩阵和张量补全算法快得多。
英文摘要：
Matrix and tensor completion aim to recover a low-rank matrix / tensor from limited observations and have been commonly used in applications such as recommender systems and multi-relational data mining. A state-of-the-art matrix completion algorithm is Soft-Impute, which exploits the special "sparse plus low-rank" structure of the matrix iterates to allow efficient SVD in each iteration. Though Soft-Impute is a proximal algorithm, it is generally believed that acceleration destroys the special structure and is thus not useful. In this paper, we show that Soft-Impute can indeed be accelerated without comprising this structure. To further reduce the iteration time complexity, we propose an approximate singular value thresholding scheme based on the power method. Theoretical analysis shows that the proposed algorithm still enjoys the fast $O(1/T^2)$ convergence rate of accelerated proximal algorithms. We further extend the proposed algorithm to tensor completion with the scaled latent nuclear norm regularizer. We show that a similar "sparse plus low-rank" structure also exists, leading to low iteration complexity and fast $O(1/T^2)$ convergence rate. Extensive experiments demonstrate that the proposed algorithm is much faster than Soft-Impute and other state-of-the-art matrix and tensor completion algorithms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
