Linear local tangent space alignment and application to face recognition

Effect of Dimension Reduction on Prediction Performance of Multivariate Nonlinear Time Series

“…Neighborhood Preserving Embedding (NPE) is based on LLE (He et al, 2005). Linear Local Tangent Space Alignment (LLTSA) constructs the linear transformation that minimizes the objective function in LTSA (Zhang et al, 2007).…”

Adaptive Neighborhood Graph for LTSA Learning Algorithm without FreeParameter

“…To preserve the local structure of each neighborhood X i , like original LTSA [7] and LLTSA [8], the local linear approximation for the data points in X i by using tangent space should be given as …”

“…The global learning algorithms include Isomap [1,2], C-Isomap [3] and L-Isomap [3]. The local learning algorithms include LLE [4,5], Lapacian Eigenmap(LE) [6], LTSA [7], LLTSA [8], NPE [9], SNE [10], LPP [11], RML [12], etc.. Basically, almost all of nonlinear dimensionality reduction algorithms usually concerns a foundational concept of neighborhood, because it is of central importance not only in studies of bijective map between high and low dimensional space, due to every point in low dimension embedding space has a neighborhood homeomorphic to an open set of high dimensional real space from viewpoint of topology, but also in the analysis of algorithm's robustness related to the problem of topological stability [13,14]. Indeed, all learning algorithm mentioned above, except SNE, are closely related to the information about the representation of local neighborhood structure, i.e., the choice of nearest neighbors that may be used naturally to results in a corresponding neighborhood in each data points.…”

“…Indeed, all learning algorithm mentioned above, except SNE, are closely related to the information about the representation of local neighborhood structure, i.e., the choice of nearest neighbors that may be used naturally to results in a corresponding neighborhood in each data points. The first step involved in the procedures of manifold learning algorithms is always to extract and construct an efficient representation about neighborhood in high-dimensional input space that can be viewed as an infrastructure corresponding to data space, and then describing certain geometric or algebraic properties upon this local structure, for example, LTSA [7] and LLTSA [8] algorithms require information about the local tangent space based on every neighborhood in each data point.…”

“…Solutions presented by original LTSA can also be interpreted as an invariant subspaces spanned by true low-dimensional representations of the observations [17], which aims at the performance analysis related to the worst-case upper bound on the angle between the estimated linear invariant subspace and the true linear invariant subspace. For the problem of face recognition, two variant versions of LTSA-based algorithms are presented recently, one is the linear local tangent space alignment (LLTSA) [8], and the other is the orthogonal discriminant linear local tangent space alignment (ODLLTSA) [18].…”