Dictionary learning and component analysis models are fundamental for learning compact representations that are relevant to a given task (feature extraction, dimensionality reduction, denoising, etc.). The model complexity is encoded by means of specific structure, such as sparsity, low-rankness, or nonnegativity. Unfortunately, approaches like K-SVD - that learn dictionaries for sparse coding via Singular Value Decomposition (SVD) - are hard to scale to high-volume and high-dimensional visual data, and fragile in the presence of outliers. Conversely, robust component analysis methods such as the Robust Principal Component Analysis (RPCA) are able to recover low-complexity (e.g., low-rank) representations from data corrupted with noise of unknown magnitude and support, but do not provide a dictionary that respects the structure of the data (e.g., images), and also involve expensive computations. In this paper, we propose a novel Kronecker-decomposable component analysis model, coined as Robust Kronecker Component Analysis (RKCA), that combines ideas from sparse dictionary learning and robust component analysis. RKCA has several appealing properties, including robustness to gross corruption; it can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization, and analyze its optimality and low-rankness properties. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising and completion, by performing a thorough comparison with the current state of the art.

中文翻译：

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鲁棒克罗内克分量分析


字典学习和成分分析模型是学习与给定任务（特征提取、降维、去噪等）相关的紧凑表示的基础。模型复杂性通过特定结构进行编码，例如稀疏性、低秩性或非负性。不幸的是，像 K-SVD 这样的方法（通过奇异值分解（SVD）学习稀疏编码字典）很难扩展到大容量和高维视觉数据，并且在存在异常值的情况下很脆弱。相反，诸如鲁棒主成分分析（RPCA）之类的鲁棒成分分析方法能够从被未知大小和支持的噪声损坏的数据中恢复低复杂性（例如，低秩）表示，但不提供尊重的字典数据的结构（例如图像），并且还涉及昂贵的计算。在本文中，我们提出了一种新颖的克罗内克可分解成分分析模型，称为鲁棒克罗内克成分分析（RKCA），它结合了稀疏字典学习和鲁棒成分分析的思想。 RKCA 有几个吸引人的特性，包括对严重腐败的稳健性；它可用于低秩建模，并利用可分离性来解决更小的问题。我们通过绘制受限形式的张量分解的链接来设计一种有效的学习算法，并分析其最优性和低秩特性。通过与当前技术水平进行彻底比较，所提出的方法的有效性在实际应用中得到了证明，即背景扣除和图像去噪和完成。