A plethora of dimensionality reduction (DR) techniques, stemming from statistics, machine learning, or graph theory, has been developed. Their ultimate objective is to eliminate data redundancy without a significant information loss. A general formulation, known as graph embedding, offers a unified framework for describing several well-known DR algorithms. In this paper, the inclusion of several DR algorithms within this unified framework is demonstrated. The enforcement of orthogonality to the projection matrix within this framework is proven to be of vital importance, since Orthogonal Neighborhood Preserving Projections, Orthogonal Locality Preserving Projections, Orthogonal Isometric Projection, Orthogonal Linear Discriminant Analysis, and Orthogonal Local Tangent Space Alignment algorithms outperform their non-orthogonal counterparts, e.g., Neighborhood Preserving Embedding, Locality Preserving Projections, Isometric Mapping, Linear Discriminant Analysis, and Local Tangent Space Alignment. One may simultaneously also impose the $\ell _{2,1}$ norm regularization on the projection matrix, seeking for row-sparsity. This leads to the known Joint Feature Selection and Subspace Learning (JFSSL) framework. All DR algorithms are employed within JFSSL. It is proven that the use of orthogonal mapping algorithms within JFSSL against their non-orthogonal counterparts does not improve the recognition rate of NN classifier.

中文翻译：

---

通过正交映射进行子空间学习和特征选择

已经开发了大量源自统计学、机器学习或图论的降维 (DR) 技术。他们的最终目标是在不丢失大量信息的情况下消除数据冗余。称为图嵌入的通用公式为描述几种著名的 DR 算法提供了统一的框架。在本文中，演示了在这个统一框架中包含几种 DR 算法。在这个框架内对投影矩阵的正交性被证明是至关重要的，因为正交邻域保持投影、正交局部保持投影、正交等距投影、正交线性判别分析和正交局部切线空间对齐算法优于它们的非正交对应物，例如，邻域保留嵌入、局部保留投影、等距映射、线性判别分析和局部切线空间对齐。也可以同时对投影矩阵施加 $\ell _{2,1}$ 范数正则化，以寻求行稀疏性。这导致了已知的联合特征选择和子空间学习 (JFSSL) 框架。JFSSL 中采用了所有 DR 算法。事实证明，在 JFSSL 中使用正交映射算法针对其非正交对应物不会提高 NN 分类器的识别率。这导致了已知的联合特征选择和子空间学习 (JFSSL) 框架。JFSSL 中采用了所有 DR 算法。事实证明，在 JFSSL 中使用正交映射算法针对其非正交对应物不会提高 NN 分类器的识别率。这导致了已知的联合特征选择和子空间学习 (JFSSL) 框架。JFSSL 中采用了所有 DR 算法。事实证明，在 JFSSL 中使用正交映射算法针对其非正交对应物不会提高 NN 分类器的识别率。