A unified semi-supervised dimensionality reduction framework for manifold learning

Abstract

We present a general framework of semi-supervised dimensionality reduction for manifold learning which naturally generalizes existing supervised and unsupervised learning frameworks which apply the spectral decomposition. Algorithms derived under our framework are able to employ both labeled and unlabeled examples and are able to handle complex problems where data form separate clusters of manifolds. Our framework offers simple views, explains relationships among existing frameworks and provides further extensions which can improve existing algorithms. Furthermore, a new semi-supervised kernelization framework called "KPCA trick" is proposed to handle non-linear problems.

Hyperspectral Dimensionality Reduction by Tensor Sparse and Low-Rank Graph-Based Discriminant Analysis

“…To obtain the lth class graph weight matrix W (l) , the alternating direction method of multipliers (ADMM) [35] is adopted to solve problem (12). Two auxiliary variables Z (l) and J (l) are first introduced to make the objective function separable arg min…”

“…The global similarity matrix W will be obtained depending on Equation (13) when each sub-similarity matrix corresponding to each class is calculated from problem (12). Until now, a tensor sparse and low-rank graph G = {X, W} is completely constructed with vertex set X and similarity matrix W. How to obtain a set of projection matrices {U n ∈ R B n ×I n , B n ≤ I n , n = 1, 2, .…”

Local subspace smoothness alignment for constrained local model fitting

“…However, there are still many difficulties when applying these manifold methods in the real-world applications. Besides the challenges concerning the generalization, such as dealing with sparse, non-uniform data or incorporating discriminant information into the model [35,36,37,38,39], finding the number of inherent dimensionality is another important problem that needs to be addressed, which is unfortunately still open to now. Although some ad hoc methods can be used sometimes, if the dimensionality of data is considerably high, it is quite time-consuming to estimate the dimensionality of manifold.…”

A novel semi-supervised learning for face recognition

“…For semisupervised dimensionality reduction, graph-based semi-supervised manifold learning techniques are successful and effective in many applications such as face recognition, action recognition, and image retrieval [19][20][21]. These semi-supervised approaches can be unified within the graph-based semi-supervised framework [22,23].…”