Original Locally Linear Embedding Research Articles

Symmetric positive definite manifold learning and its application in fault diagnosis

Locally linear embedding (LLE) is an effective tool to extract the significant features from a dataset. However, most of the relevant existing algorithms assume that the original dataset resides on a Euclidean space, unfortunately nearly all the original data space is non-Euclidean. In addition, the original LLE does not use the discriminant information of the dataset, which will degrade its performance in feature extraction. To address these problems raised in the conventional LLE, we first employ the original dataset to construct a symmetric positive definite manifold, and then estimate the tangent space of this manifold. Furthermore, the local and global discriminant information are integrated into the LLE, and the improved LLE is operated in the tangent space to extract the important features. We introduce Iris dataset to analyze the capability of the proposed method to extract features. Finally, several experiments are performed on five machinery datasets, and experimental results indicate that our proposed method can extract the excellent low-dimensional representations of the original dataset. Compared with the state-of-the-art methods, the proposed algorithm shows a strong capability for fault diagnosis.

Local quasi-linear embedding based on kronecker product expansion of vectors

Locally Linear Embedding (LLE) is honored as the first algorithm of manifold learning. Generally speaking, the relation between a data and its nearest neighbors is nonlinear and LLE only extracts its linear part. Therefore, local nonlinear embedding is an important direction of improvement to LLE. However, any attempt in this direction may lead to a significant increase in computational complexity. In this paper, a novel algorithm called local quasi-linear embedding (LQLE) is proposed. In our LQLE, each high-dimensional data vector is first expanded by using Kronecker product. The expanded vector contains not only the components of the original vector, but also the polynomials of its components. Then, each expanded vector of high dimensional data is linearly approximated with the expanded vectors of its nearest neighbors. In this way, the proposed LQLE achieves a certain degree of local nonlinearity and learns the data dimensionality reduction results under the principle of keeping local nonlinearity unchanged. More importantly, LQLE does not increase computation complexity by only replacing the data vectors with their Kronecker product expansions in the original LLE program. Experimental results between our proposed methods and four comparison algorithms on various datasets demonstrate the well performance of the proposed methods.

An Improved Locally Linear Embedding Method for Feature Extraction

In this work, a feature extraction approach based on improved Locally Linear Embedding(LLE) is proposed. In the algorithm, tangent space distance is introduced to LLE, which overcomes the shortcoming of original LLE method based on Euclidean distance. It can satisfy the requirement of locally linear much better and so can express the I/O mapping quality better than classical method. Simulation results are given to demonstrate the effectiveness of the improved LLE method.

Locally linear discriminant embedding: An efficient method for face recognition

In this paper an efficient feature extraction method named as locally linear discriminant embedding (LLDE) is proposed for face recognition. It is well known that a point can be linearly reconstructed by its neighbors and the reconstruction weights are under the sum-to-one constraint in the classical locally linear embedding (LLE). So the constrained weights obey an important symmetry: for any particular data point, they are invariant to rotations, rescalings and translations. The latter two are introduced to the proposed method to strengthen the classification ability of the original LLE. The data with different class labels are translated by the corresponding vectors and those belonging to the same class are translated by the same vector. In order to cluster the data with the same label closer, they are also rescaled to some extent. So after translation and rescaling, the discriminability of the data will be improved significantly. The proposed method is compared with some related feature extraction methods such as maximum margin criterion (MMC), as well as other supervised manifold learning-based approaches, for example ensemble unified LLE and linear discriminant analysis (En-ULLELDA), locally linear discriminant analysis (LLDA). Experimental results on Yale and CMU PIE face databases convince us that the proposed method provides a better representation of the class information and obtains much higher recognition accuracies.

Incremental locally linear embedding

The locally linear embedding (LLE) algorithm belongs to a group of manifold learning methods that not only merely reduce data dimensionality, but also attempt to discover a true low dimensional structure of the data. In this paper, we propose an incremental version of LLE and experimentally demonstrate its advantages in terms of topology preservation. Also compared to the original (batch) LLE, the incremental LLE needs to solve a much smaller optimization problem.