We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized da-ta points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approxi-mation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data pointswith respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can bequite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimension-al Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.

Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment

Abstract As basic research, it has also received increasing attention from people that the “curse of dimensionality” will lead to increase the cost of data storage and computing; it also influences the efficiency and accuracy of dealing with problems. Feature dimensionality reduction as a key link in the process of pattern recognition has become one hot and difficulty spot in the field of pattern recognition, machine learning and data mining. It is one of the most challenging research fields, which has been favored by most of the scholars’ attention. How to implement “low loss” in the process of feature dimension reduction, keep the nature of the original data, find out the best mapping and get the optimal low dimensional data are the keys aims of the research. In this paper, two-dimensionality reduction methods, feature selection and feature extraction, are introduced; the current mainstream dimensionality reduction algorithms are analyzed, including the method for small sample and method based on deep learning. For each algorithm, examples of their application are given and the advantages and disadvantages of these methods are evaluated. 

/pdf/feature-dimensionality-reduction-a-review-vtdczklf.pdf

Feature dimensionality reduction: a review

Distance and similarity-search metrics for use with soil vis-NIR spectra

Label propagation is one of the state-of-the-art methods for semi-supervised learning, which estimates labels by propagating label information through a graph. Label propagation assumes that data points (nodes) connected in a graph should have similar labels. Consequently, the label estimation heavily depends on edge weights in a graph which represent similarity of each node pair. We propose a method for a graph to capture the manifold structure of input features using edge weights parameterized by a similarity function. In this approach, edge weights represent both similarity and local reconstruction weight simultaneously, both being reasonable for label propagation. For further justification, we provide analytical considerations including an interpretation as a cross-validation of a propagation model in the feature space, and an error analysis based on a low dimensional manifold model. Experimental results demonstrated the effectiveness of our approach both in synthetic and real datasets.

/pdf/manifold-based-similarity-adaptation-for-label-propagation-j4ds104k5x.pdf

Manifold-based Similarity Adaptation for Label Propagation

As faults in the solar water heaters are structurally complicated and highly correlated, an approach of fault diagnosis on the basis of support vector machine and D-S evidence theory has been propo...

https://journals.sagepub.com/doi/pdf/10.1177/0144598718816604

SVM-DS fusion based soft fault detection and diagnosis in solar water heaters:

As a classic method of nonlinear dimensional reduction, locally linear embedding (LLE) is more and more attractive to researchers due to its ability to deal with large amounts of high dimensional data and its non-iterative way of finding the embeddings. However, several problems in the LLE algorithm still remain open, such as its sensitivity to noise, inevitable ill-conditioned eigenproblems, the lack of how to deal with the novel data, etc. The existing extensions are comprehensively reviewed and discussed classifying into different categories in this paper. Their strategies, advantages/disadvantages and performances are elaborated. By generalizing different tactics in various extensions related to different stages of LLE and evaluating their performances, several promising directions for future research have been suggested.

Jing Chen

Papers

Locally linear embedding: a survey