Huazhong University of Science and Technology, China.
IEEE Trans Image Process. 2010 May;19(5):1126-37. doi: 10.1109/TIP.2009.2039050. Epub 2009 Dec 22.
We present in this paper a general formulation for nonnegative data factorization, called projective nonnegative graph embedding (PNGE), which 1) explicitly decomposes the data into two nonnegative components favoring the characteristics encoded by the so-called intrinsic and penalty graphs , respectively, and 2) explicitly describes how to transform each new testing sample into its low-dimensional nonnegative representation. In the past, such a nonnegative decomposition was often obtained for the training samples only, e.g., nonnegative matrix factorization (NMF) and its variants, nonnegative graph embedding (NGE) and its refined version multiplicative nonnegative graph embedding (MNGE). Those conventional approaches for out-of-sample extension either suffer from the high computational cost or violate the basic nonnegative assumption. In this work, PNGE offers a unified solution to out-of-sample extension problem, and the nonnegative coefficient vector of each datum is assumed to be projected from its original feature representation with a universal nonnegative transformation matrix. A convergency provable multiplicative nonnegative updating rule is then derived to learn the basis matrix and transformation matrix. Extensive experiments compared with the state-of-the-art algorithms on nonnegative data factorization demonstrate the algorithmic properties in convergency, sparsity, and classification power.
我们在本文中提出了一种用于非负数据分解的通用公式，称为投影非负图嵌入（PNGE），它 1）明确地将数据分解为两个非负分量，分别有利于由所谓的内在图和惩罚图编码的特征，以及 2）明确地描述了如何将每个新的测试样本转换为其低维非负表示。在过去，这种非负分解通常仅针对训练样本获得，例如非负矩阵分解（NMF）及其变体、非负图嵌入（NGE）及其改进版乘法非负图嵌入（MNGE）。那些用于样本外扩展的常规方法要么受到高计算成本的影响，要么违反了基本的非负假设。在这项工作中，PNGE 为样本外扩展问题提供了一个统一的解决方案，并且每个数据点的非负系数向量被假设从其原始特征表示通过通用非负变换矩阵进行投影。然后推导出可证明收敛的乘法非负更新规则来学习基矩阵和变换矩阵。与非负数据分解的最新算法进行的广泛实验证明了算法在收敛性、稀疏性和分类能力方面的特性。
