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首页> 外文期刊>Neural Networks and Learning Systems, IEEE Transactions on >Local Coordinates Alignment With Global Preservation for Dimensionality Reduction
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Local Coordinates Alignment With Global Preservation for Dimensionality Reduction

机译：局部坐标与全局保留对齐以减少维数

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摘要

Dimensionality reduction is vital in many fields, and alignment-based methods for nonlinear dimensionality reduction have become popular recently because they can map the high-dimensional data into a low-dimensional subspace with the property of local isometry. However, the relationships between patches in original high-dimensional space cannot be ensured to be fully preserved during the alignment process. In this paper, we propose a novel method for nonlinear dimensionality reduction called local coordinates alignment with global preservation. We first introduce a reasonable definition of topology-preserving landmarks (TPLs), which not only contribute to preserving the global structure of datasets and constructing a collection of overlapping linear patches, but they also ensure that the right landmark is allocated to the new test point. Then, an existing method for dimensionality reduction that has good performance in preserving the global structure is used to derive the low-dimensional coordinates of TPLs. Local coordinates of each patch are derived using tangent space of the manifold at the corresponding landmark, and then these local coordinates are aligned into a global coordinate space with the set of landmarks in low-dimensional space as reference points. The proposed alignment method, called landmarks-based alignment, can produce a closed-form solution without any constraints, while most previous alignment-based methods impose the unit covariance constraint, which will result in the deficiency of global metrics and undesired rescaling of the manifold. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed algorithm.
机译：降维在许多领域都至关重要，近来基于对齐的非线性降维方法已经流行，因为它们可以将高维数据映射到具有局部等距特性的低维子空间中。但是，在对齐过程中无法确保完全保留原始高维空间中的小块之间的关系。在本文中，我们提出了一种新的非线性降维方法，称为局部坐标对齐和全局保留。我们首先介绍一个合理的拓扑保留界标（TPL）定义，它不仅有助于保存数据集的全局结构并构造重叠的线性补丁集合，而且还可以确保将正确的界标分配给新的测试点。然后，使用在保留全局结构方面具有良好性能的现有降维方法来导出TPL的低维坐标。使用斑块的切线空间在相应地标处导出每个面片的局部坐标，然后将这些局部坐标对齐为全局坐标空间，并以低维空间中的一组地标作为参考点。所提出的对齐方法称为基于地标的对齐，可以产生没有任何约束的封闭形式的解决方案，而大多数以前的基于对齐的方法都施加了单位协方差约束，这将导致全局度量的不足和流形的不期望的重新缩放。在合成和真实数据集上的实验都证明了该算法的有效性。

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