Nie Feiping, Wang Zheng, Wang Rong, Li Xuelong
IEEE Trans Cybern. 2020 Aug;50(8):3682-3695. doi: 10.1109/TCYB.2019.2910751. Epub 2019 Apr 26.
Due to the multimodality of non-Gaussian data, traditional globality-preserved dimensionality reduction (DR) methods, such as linear discriminant analysis (LDA) and principal component analysis (PCA) are difficult to deal with. In this paper, we present a novel local DR framework via auto-optimized graph embedding to extract the intrinsic submanifold structure of multimodal data. Specifically, the proposed model seeks to learn an embedding space which can preserve the local neighborhood structure by constructing a k -nearest neighbors ( k NNs) graph on data points. Different than previous works, our model employs the l -norm constraint and binary constraint on the similarity matrix to impose that there only be a k nonzero value in each row of the similarity matrix, which can ensure the k -connectivity in graph. More important, as the high-dimensional data probably contains some noises and redundant features, calculating the similarity matrix in the original space by using a kernel function is inaccurate. As a result, a mechanism of an auto-optimized graph is derived in the proposed model. Concretely, we learn the embedding space and similarity matrix simultaneously. In other words, the selection of neighbors is automatically executed in the optimal subspace rather than in the original space when the algorithm reaches convergence, which can alleviate the affect of noises and improve the robustness of the proposed model. In addition, four supervised and semisupervised local DR methods are derived by the proposed framework which can extract the discriminative features while preserving the submanifold structure of data. Last but not least, since two variables need to be optimized simultaneously in the proposed methods, and the constraints on the similarity matrix are difficult to satisfy, which is an NP-hard problem. Consequently, an efficient iterative optimization algorithm is introduced to solve the proposed problems. Extensive experiments conducted on synthetic data and several real-world datasets have demonstrated the advantages of the proposed methods in robustness and recognition accuracy.
由于非高斯数据的多模态性，传统的全局保持降维（DR）方法，如线性判别分析（LDA）和主成分分析（PCA）难以处理。在本文中，我们提出了一种通过自动优化图嵌入的新颖局部DR框架，以提取多模态数据的内在子流形结构。具体而言，所提出的模型旨在学习一个嵌入空间，该空间可以通过在数据点上构建k近邻（k-NNs）图来保留局部邻域结构。与先前的工作不同，我们的模型对相似性矩阵采用l范数约束和二进制约束，以确保相似性矩阵的每行中只有k个非零值，这可以保证图中的k连通性。更重要的是，由于高维数据可能包含一些噪声和冗余特征，使用核函数在原始空间中计算相似性矩阵是不准确的。因此，在所提出的模型中推导了一种自动优化图的机制。具体来说，我们同时学习嵌入空间和相似性矩阵。换句话说，当算法收敛时，邻居的选择是在最优子空间中自动执行的，而不是在原始空间中，这可以减轻噪声的影响并提高所提出模型的鲁棒性。此外，所提出的框架还推导了四种监督和半监督局部DR方法，这些方法可以在保留数据子流形结构的同时提取判别特征。最后但同样重要的是，由于在所提出的方法中需要同时优化两个变量，并且相似性矩阵上的约束难以满足，这是一个NP难问题。因此，引入了一种高效的迭代优化算法来解决所提出的问题。在合成数据和几个真实世界数据集上进行的大量实验证明了所提出方法在鲁棒性和识别准确性方面的优势。
