Dimensionality reduction techniques aim to transform the high-dimensional data into a meaningful reduced representation and have been consistently playing a fundamental role in the study of intrinsic dimensionality estimation and the design of an intelligent expert system towards real-world applications. From the perspective of manifold learning, locality preserving projections is a classical and commonly used dimensionality reduction method and it essentially learns the low-dimensional embedding under the constraint of preserving the local geometry of data. However, since it determines the neighborhood relationships in the original feature space that probably contains noisy and irrelevant features, the derived similarity between the neighbors are unreliable and the corresponding local data manifold tends to be error-prone, which inevitably leads to degraded performance for subsequent data analyses. Hence, how to accurately identify the true neighbor relationships for each sample remains crucial to the robustness improvement. In this work, we propose a novel approach, termed locality adaptive preserving projections (LAPP), to adaptively determine the neighbors and their relationships in the optimal subspace rather than in the original space. Specifically, due to the absence of prior knowledge of local properties of the underlying manifold, LAPP adopts a coarse-to-fine strategy to iteratively update the projected low-dimensional subspace and optimize the identification of the local structure of the data. Moreover, an iterative algorithm with fast convergence is utilized to solve the transformation matrix for explicit out-of-sample extension. Besides, LAPP is easy to implement and its key idea can be potentially extended to other methods for neighbor-finding and similarity measurement. To evaluate the performance of LAPP, we conduct comparative experiments on numerous synthetic and real-world datasets. Experimental results show that seeking the local structure in the original feature space misleads the selection of neighbors and the calculation of similarity and that the proposed method helps alleviate the negative effect of noisy and irrelevant features, which demonstrates its effectiveness. Besides, this study has the potential to enlighten relevant studies to consider the problem of optimizing the neighborhood relationships.

降维技术旨在将高维数据转换为有意义的简化表示形式，并且在固有维数估计研究和面向实际应用的智能专家系统的设计中一直发挥着重要作用。从流形学习的角度来看，局部保留投影是一种经典且常用的降维方法，它本质上是在保留数据局部几何的约束下学习低维嵌入。但是，由于它确定了可能包含嘈杂和不相关特征的原始特征空间中的邻域关系，因此邻居之间得出的相似度不可靠，相应的本地数据流形往往容易出错，这不可避免地导致后续数据分析的性能下降。因此，如何准确识别每个样本的真实邻居关系仍然是鲁棒性提高的关键。在这项工作中，我们提出了一种新颖的方法，称为局部性自适应保留投影（LAPP），用于在最佳子空间而不是原始空间中自适应确定邻居及其关系。具体而言，由于缺乏对底层歧管局部属性的先验知识，LAPP采用了从粗到精的策略来迭代更新投影的低维子空间并优化对数据局部结构的识别。此外，利用具有快速收敛性的迭代算法来解决变换矩阵的显式样本外扩展。除了，LAPP易于实施，其关键思想可以潜在地扩展到其他用于邻居发现和相似性测量的方法。为了评估LAPP的性能，我们在许多合成和真实数据集上进行了对比实验。实验结果表明，在原始特征空间中寻找局部结构会误导邻居的选择和相似度的计算，并且所提出的方法有助于减轻噪声和无关特征的负面影响，从而证明了其有效性。此外，该研究有可能启发相关研究来考虑优化邻里关系的问题。我们在众多合成和真实数据集上进行了对比实验。实验结果表明，在原始特征空间中寻找局部结构会误导邻居的选择和相似度的计算，并且所提出的方法有助于减轻噪声和无关特征的负面影响，从而证明了其有效性。此外，该研究有可能启发相关研究来考虑优化邻里关系的问题。我们在众多合成和真实数据集上进行了对比实验。实验结果表明，在原始特征空间中寻找局部结构会误导邻居的选择和相似度的计算，并且所提出的方法有助于减轻噪声和无关特征的负面影响，从而证明了其有效性。此外，该研究有可能启发相关研究来考虑优化邻里关系的问题。证明了它的有效性。此外，该研究有可能启发相关研究来考虑优化邻里关系的问题。证明了它的有效性。此外，该研究有可能启发相关研究来考虑优化邻里关系的问题。