General Dimensionality Reduction Framework Research Articles

A General Framework for Dimensionality Reduction of K-Means Clustering

Dimensionality reduction plays an important role in many machine learning and pattern recognition applications. Linear discriminant analysis (LDA) is the most popular supervised dimensionality reduction technique which searches for the projection matrix that makes the data points of different classes to be far from each other while requiring data points of the same class to be close to each other. In this paper, trace ratio LDA is combined with K-means clustering into a unified framework, in which K-means clustering is employed to generate class labels for unlabeled data and LDA is used to investigate low-dimensional representation of data. Therefore, by combining the subspace clustering with dimensionality reduction together, the optimal subspace can be obtained. Differing from other existing dimensionality reduction methods, our novel framework is suitable for different scenarios: supervised, semi-supervised, and unsupervised dimensionality reduction cases. Experimental results on benchmark datasets validate the effectiveness and superiority of our algorithm compared with other relevant techniques.

Graph embedding and extensions: a general framework for dimensionality reduction.

Over the past few decades, a large family of algorithms - supervised or unsupervised; stemming from statistics or geometry theory - has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called Marginal Fisher Analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional Linear Discriminant Analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions.

A General Exponential Framework for Dimensionality Reduction.

As a general framework, Laplacian embedding, based on a pairwise similarity matrix, infers low dimensional representations from high dimensional data. However, it generally suffers from three issues: 1) algorithmic performance is sensitive to the size of neighbors; 2) the algorithm encounters the well known small sample size (SSS) problem; and 3) the algorithm de-emphasizes small distance pairs. To address these issues, here we propose exponential embedding using matrix exponential and provide a general framework for dimensionality reduction. In the framework, the matrix exponential can be roughly interpreted by the random walk over the feature similarity matrix, and thus is more robust. The positive definite property of matrix exponential deals with the SSS problem. The behavior of the decay function of exponential embedding is more significant in emphasizing small distance pairs. Under this framework, we apply matrix exponential to extend many popular Laplacian embedding algorithms, e.g., locality preserving projections, unsupervised discriminant projections, and marginal fisher analysis. Experiments conducted on the synthesized data, UCI, and the Georgia Tech face database show that the proposed new framework can well address the issues mentioned above.

Graph Embedding and Extensions: A General Framework for Dimensionality Reduction

Graph Embedding and Extensions: A General Framework for Dimensionality Reduction