IEEE Geoscience and Remote Sensing Magazine - March 2018 - 19

Examples of dimensionality reduction for a 2-D two-class
multimodal synthetic data set using LDA and LFDA are illustrated in Figure 2. Figure 2(a) depicts an original two-class
multimodal classification problem, along with the projecting directions learned using LDA and LFDA. We can clearly observe that LDA distorts the information contained in
multimodal distributions because there is a large overlap
between two classes in its projected subspace, while LFDA is
able to preserve the multimodal structure of the data in the
projected subspace.
OTHER ALGORITHMS USING
A LOCALITY-PRESERVING STRATEGY
Locality-preserving dimensionality reduction, proven to
be effective for hyperspectral data processing, has recently attracted increasing attention involving two different
types. The first type uses the locality-preserving strategy
with only spectral information. For example, the study in
[101] discussed a supervised Laplacian eigenmaps method,
which made full use of class label information to guide
the procedure of nonlinear dimensionality reduction by
adopting the large-margin concept. The authors of [102]
studied a local-scaling-cut criterion that could handle
heteroscedastic and multimodal data. Similarly, some researchers designed local discriminant analysis [56], [103]
to preserve local neighborhood information while simultaneously maximizing the class discrimination; yet others
discussed local tensor discriminant analysis [104]. Later,
Yang et al. investigated a semisupervised dual-geometric
subspace projection approach [105], where they defined a
local consistency-constrained geometric matrix to reveal
the geometric structure among the data. The research in
[106] developed sparse discriminant embedding (SDE);

GRAPh-EMBEDDING DISCRIMINANT ANALYSIS
SGDA and CGDA are recently developed dimensionalityreduction techniques [61]-[63] that belong to the graphembedding framework. An undirected weighted graph,
representing the desired statistical or geometrical properties of the data, reflects the similarities of vertex pairs. The

(i)

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By LDA
By LFDA

compared to sparsity preserving projections (SPPs), SDE
maintains the merits of both the intermanifold structure
and the sparsity property.
The other type of locality-preserving method uses spectral-spatial information. A genetic algorithm-based LFDA
(GA-LFDA) [107] used spectral-spatial information for dimensionality reduction. The
authors of [108] further employed GA-LFDA to exploit
the high correlation between
TRADITIONAL LINEAR
successive spectral bands. AddiSUBSPACE LEARNING
tionally, researchers developed
METhODS PROVIDE
an improved locally linear emSATISFACTORY
bedding [109] method based
PERFORMANCE ONLY IF
on robust spatial informaThE CLASS-CONDITIONAL
tion; they presented spatial
DISTRIBUTIONS ARE
neighbor sorting and filtering
hOMOSCEDASTIC
to ensure the robustness of
GAUSSIAN.
the spectral-spatial distance.
Zhou et al. [110] proposed a
spatial- and spectral-regularized local discriminant embedding method for dimensionality reduction, where they
calculated the discriminative projection by minimizing a
local spatial-spectral scatter and maximizing a modified
total data scatter.

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FIGURE 2. (a) A synthetic 2-D multimodal data plot and the directions of LDA and LFDA. (b) The data distribution after projection into a

one-dimensional subspace: (i) by LDA and (ii) by LFDA.
march 2018

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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2018