On the Performance of Manhattan Nonnegative Matrix Factorization
On the Performance of Manhattan Nonnegative Matrix Factorization
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DOI：
10.1109/tnnls.2015.2458986
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发表时间：
2016-09
影响因子：
10.4
通讯作者：
Tongliang Liu;D. Tao
中科院分区：
文献类型：
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作者：
Tongliang Liu;D. Tao
从矩阵中提取低秩和稀疏结构在机器学习、压缩感知和传统信号处理中得到了广泛的研究，并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）是传统NMF的扩展，它通过最小化非负矩阵X与两个非负低秩因子矩阵乘积之间的曼哈顿距离来模拟重尾拉普拉斯噪声。已经开发了快速算法来恢复MahNMF中X的低秩和稀疏结构。本文在统计学习理论的框架下研究了MahNMF的统计性能。我们将MahNMF的预期重建误差分解为估计误差和逼近误差。估计误差的MahNMF的推广误差界的范围内，而近似误差分析使用的矢量量化的最小失真的渐近结果。泛化误差界对于确定训练样本的大小是有价值的，所述训练样本的大小需要保证期望重构误差和经验重构误差之间的缺陷的期望上界。统计性能分析显示了降维如何影响估计和近似误差。我们的框架也可以用于分析的NMF的性能。
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.