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On the Performance of Manhattan Nonnegative Matrix Factorization

机译：曼哈顿非负矩阵分解的性能

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摘要

Extracting low-rank and sparse structures from matrices has been extensively studied in machine learning, compressed sensing, and conventional signal processing, and has been widely applied to recommendation systems, image reconstruction, visual analytics, and brain signal processing. Manhattan nonnegative matrix factorization (MahNMF) is an extension of the conventional NMF, which models the heavy-tailed Laplacian noise by minimizing the Manhattan distance between a nonnegative matrix X and the product of two nonnegative low-rank factor matrices. Fast algorithms have been developed to restore the low-rank and sparse structures of X in the MahNMF. In this paper, we study the statistical performance of the MahNMF in the frame of the statistical learning theory. We decompose the expected reconstruction error of the MahNMF into the estimation error and the approximation error. The estimation error is bounded by the generalization error bounds of the MahNMF, while the approximation error is analyzed using the asymptotic results of the minimum distortion of vector quantization. The generalization error bound is valuable for determining the size of the training sample needed to guarantee a desirable upper bound for the defect between the expected and empirical reconstruction errors. Statistical performance analysis shows how the reduced dimensionality affects the estimation and approximation errors. Our framework can also be used for analyzing the performance of the NMF.
机译：从矩阵中提取低秩和稀疏结构已在机器学习，压缩感测和常规信号处理中进行了广泛研究，并且已广泛应用于推荐系统，图像重建，视觉分析和脑信号处理。曼哈顿非负矩阵分解（MahNMF）是常规NMF的扩展，该模型通过最小化非负矩阵X与两个非负低秩因子矩阵的乘积之间的曼哈顿距离来建模重尾拉普拉斯噪声。已经开发了快速算法来还原MahNMF中X的低秩和稀疏结构。在本文中，我们在统计学习理论的框架内研究MahNMF的统计性能。我们将MahNMF的预期重构误差分解为估计误差和近似误差。估计误差由MahNMF的泛化误差范围界定，而逼近误差则使用矢量量化最小失真的渐近结果进行分析。泛化误差边界对于确定训练样本的大小非常有价值，该样本需要保证期望的和经验的重建误差之间的缺陷的理想上限。统计性能分析显示降维如何影响估计和近似误差。我们的框架还可以用于分析NMF的性能。

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