Abstract Low-rank matrix factorization (LRMF) has received much popularity owing to its successful applications in both computer vision and data mining. By assuming noise to come from a Gaussian, Laplace or mixture of Gaussian distributions, significant efforts have been made on optimizing the (weighted) L1 or L2-norm loss between an observed matrix and its bilinear factorization. However, the type of noise distribution is generally unknown in real applications and inappropriate assumptions will inevitably deteriorate the behavior of LRMF. On the other hand, real data are often corrupted by skew rather than symmetric noise. To tackle this problem, this paper presents a novel LRMF model called AQ-LRMF by modeling noise with a mixture of asymmetric Laplace distributions. An efficient algorithm based on the expectation-maximization (EM) algorithm is also offered to estimate the parameters involved in AQ-LRMF. The AQ-LRMF model possesses the advantage that it can approximate noise well no matter whether the real noise is symmetric or skew. The core idea of AQ-LRMF lies in solving a weighted L1 problem with weights being learned from data. The experiments conducted on synthetic and real data sets show that AQ-LRMF outperforms several state-of-the-art techniques. Furthermore, AQ-LRMF also has the superiority over the other algorithms in terms of capturing local structural information contained in real images.

中文翻译：

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自适应分位数低秩矩阵分解

摘要 低秩矩阵分解（LRMF）因其在计算机视觉和数据挖掘中的成功应用而广受欢迎。通过假设噪声来自高斯分布、拉普拉斯分布或高斯分布的混合，已经在优化观察矩阵与其双线性分解之间的（加权）L1 或 L2 范数损失方面做出了重大努力。然而，噪声分布的类型在实际应用中通常是未知的，不适当的假设将不可避免地恶化 LRMF 的行为。另一方面，真实数据经常被偏斜而不是对称噪声破坏。为了解决这个问题，本文提出了一种称为 AQ-LRMF 的新型 LRMF 模型，该模型通过使用非对称拉普拉斯分布的混合对噪声进行建模。还提供了一种基于期望最大化（EM）算法的有效算法来估计 AQ-LRMF 中涉及的参数。AQ-LRMF 模型的优点是无论真实噪声是对称的还是偏斜的，它都能很好地近似噪声。AQ-LRMF 的核心思想在于通过从数据中学习权重来解决加权 L1 问题。在合成和真实数据集上进行的实验表明，AQ-LRMF 优于几种最先进的技术。此外，在捕获真实图像中包含的局部结构信息方面，AQ-LRMF 也优于其他算法。AQ-LRMF 的核心思想在于通过从数据中学习权重来解决加权 L1 问题。在合成和真实数据集上进行的实验表明，AQ-LRMF 优于几种最先进的技术。此外，在捕获真实图像中包含的局部结构信息方面，AQ-LRMF 也优于其他算法。AQ-LRMF 的核心思想在于通过从数据中学习权重来解决加权 L1 问题。在合成和真实数据集上进行的实验表明，AQ-LRMF 优于几种最先进的技术。此外，在捕获真实图像中包含的局部结构信息方面，AQ-LRMF 也优于其他算法。