We study the problem of recovering an incomplete $m\times n$ matrix of rank $r$ with columns arriving online over time. This is known as the problem of life-long matrix completion, and is widely applied to recommendation system, computer vision, system identification, etc. The challenge is to design provable algorithms tolerant to a large amount of noises, with small sample complexity. In this work, we give algorithms achieving strong guarantee under two realistic noise models. In bounded deterministic noise, an adversary can add any bounded yet unstructured noise to each column. For this problem, we present an algorithm that returns a matrix of a small error, with sample complexity almost as small as the best prior results in the noiseless case. For sparse random noise, where the corrupted columns are sparse and drawn randomly, we give an algorithm that exactly recovers an $\mu_0$-incoherent matrix by probability at least $1-\delta$ with sample complexity as small as $O\left(\mu_0rn\log (r/\delta)\right)$. This result advances the state-of-the-art work and matches the lower bound in a worst case. We also study the scenario where the hidden matrix lies on a mixture of subspaces and show that the sample complexity can be even smaller. Our proposed algorithms perform well experimentally in both synthetic and real-world datasets.

本文提供了两种不同的音量模型下的算法，能够克服实践中存在的鸡兔同笼且带有噪音的矩阵完形填空问题，并给出了小样本复杂度。实验表明，提出的算法在合成和现实世界中的数据集下均表现出色。

自适应采样的噪声容忍终身矩阵补全