Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: 1) the local geometry of samples is well preserved for low dimensional data representation, 2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, 3) the projection matrix of MEN improves the parsimony in computation, 4) the elastic net penalty reduces the over-fitting problem, and 5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.Comment: 33 pages, 12 figure

Evaluating Graph Signal Processing for Neuroimaging Through Classification and Dimensionality Reduction

Graph Signal Processing (GSP) is a promising framework to analyze multi-dimensional neuroimaging datasets, while taking into account both the spatial and functional dependencies between brain signals. In the present work, we apply dimensionality reduction techniques based on graph representations of the brain to decode brain activity from real and simulated fMRI datasets. We introduce seven graphs obtained from a) geometric structure and/or b) functional connectivity between brain areas at rest, and compare them when performing dimension reduction for classification. We show that mixed graphs using both a) and b) offer the best performance. We also show that graph sampling methods perform better than classical dimension reduction including Principal Component Analysis (PCA) and Independent Component Analysis (ICA).Comment: 5 pages, GlobalSIP 201

Dimensionality Reduction for Classification: Comparison of Techniques and Dimension Choice

We investigate the effects of dimensionality reduction using different techniques and different dimensions on six two-class data sets with numerical attributes as pre-processing for two classification algorithms. Besides reducing the dimensionality with the use of principal components and linear discriminants, we also introduce four new techniques. After this dimensionality reduction two algorithms are applied. The first algorithm takes advantage of the reduced dimensionality itself while the second one directly exploits the dimensional ranking. We observe that neither a single superior dimensionality reduction technique nor a straightforward way to select the optimal dimension can be identified. On the other hand we show that a good choice of technique and dimension can have a major impact on the classification power, generating classifiers that can rival industry standards. We conclude that dimensionality reduction should not only be used for visualisation or as pre-processing on very high dimensional data, but also as a general preprocessing technique on numerical data to raise the classification power. The difficult choice of both the dimensionality reduction technique and the reduced dimension however, should be directly based on the effects on the classification power