Locality adaptive preserving projections for linear dimensionality reduction

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. (C) 2020 Elsevier Ltd. All rights reserved.