IEEE Geoscience and Remote Sensing Magazine - December 2017 - 45

to extract spatial (local neighborhood) information that is
very useful for classification. It was shown that OTVCA is
highly robust to noise, because it exploits a penalized leastsquares minimization framework.
SUPERVISED DIMENSIONALITY REDUCTION
Supervised methods rely on the existence of labeled samples to infer class separability. Several widely used supervised DR methods for HSIs are linear discriminant analysis (LDA) [37], nonparametric weighted feature extraction
(NWFE) [38], band selection based on Jeffries-Matsushita
(J-M) distance [39], and mutual information [40]. Many
extensions of these methods have been proposed in past
decades, including modified Fisher's LDA [41], regularized
LDA [42], modified NWFE using spatial and spectral information [43], kernel NWFE [44], extended J-M to multiclass
cases [40], J-M distance for spatially invariant features [45],
minimal-redundancy/maximal-relevance based on mutual
information [46], and normalized mutual information [47].
Recent supervised DR methods for hyperspectral data
exploit the local neighborhood properties of data. Li et al.
[48] employed local Fisher's LDA [49] to reduce the dimensionality of the data while preserving the corresponding
multimodal structure. In [50], local neighborhood information was exploited in both the spectral and spatial domains to find a discriminative projection for DR of hyperspectral data. Cao et al. [51] proposed a supervised band
selection, by introducing the local spatial smoothness of
the HSI into the wrapper method. Dong et al. [52] presented
an ensemble-discriminative local-metric-learning method
for DR, where local spatial information was incorporated
into distance metric learning to learn a subspace, keeping
the samples from the same class closer while pushing those
from different classes farther away.
Sparse graph embedding (SGE) explores the sparsity
structure of the data for hyperspectral DR. Ly et al. [53]
proposed block sparse-graph-based discriminant analysis,
which learns a block sparse graph for a supervised DR. Xue
et al. [54] proposed a spatially and spectrally regularized local discriminant embedding method for DR, where spatial
information was integrated into the sparse graph learning
process. In [55], a discriminative sparse multimodal learning was developed for multiple-feature selection. However,
the sparse coding used in SGE is helpful for learning under
conditions where the coding is local [56], which means locality is more important than sparsity. Unfortunately, the
converse is not true: sparsity does not always guarantee locality [56]. He et al. [57] proposed a weighted sparse graph to
overcome the drawback of sparse coding in SGE, where both
the locality and sparsity of the training pixels are integrated.
Other trends in supervised DR methods exploit various
algorithms and learning techniques from soft computing,
artificial intelligence, and machine learning. Genetic algorithms (GAs) [58], particle swarm optimization (PSO) [59],
and the combination of GAs and PSO are used to optimize
feature selection [60], [61]. Deep learning techniques, e.g.,
DECEMBER 2017

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

stacked autoencoders [62] and convolutional neural networks (CNNs) [63], are used for spectral-spatial feature extraction for HSI classification [64], [65].
SEMISUPERVISED DIMENSIONALITY REDUCTION
In real-world applications, labeled data are usually very limited, and labeling a large amount of data may sometimes
require considerable human resources or expertise. On the
other hand, unlabeled data are available in large quantities at
very low cost. For this reason, semisupervised methods [66]-
[68], which aim at improved
classification by utilizing both
unlabeled and limited labeled
SOME SEMISUPERVISED
data, have gained popularity
FEATURE EXTRACTION
in the machine-learning comMETHODS ADD A
munity. Some of the representative semisupervised learning
REGULARIZATION TERM
methods include cotraining
TO PRESERVE CERTAIN
[66], transductive support vecPOTENTIAL PROPERTIES
tor machines (SVMs) [67], and
OF THE DATA.
graph-based semisupervised
learning methods [68].
Some semisupervised feature extraction methods add a regularization term to preserve certain potential properties of the data. For example,
semisupervised discriminant analysis (SDA) [69] adds a
regularizer into the objective function of LDA. The resulting
method makes use of a limited number of labeled samples
to maximize class discrimination and employs both labeled
and unlabeled samples to preserve the local properties of
the data. The approach of [70] proposed a general semisupervised DR framework based on pairwise constraints and
employed regularization with sparse representation (SR). A
semisupervised pairwise band selection method [71] was
proposed for HSIs, in which an individual band selection
process was performed only on each pair of classes. Other
semisupervised feature extraction methods combine supervised methods with unsupervised ones using a tradeoff
parameter, such as semisupervised local Fisher's (SELF) discriminant analysis [72].
It may not be easy, however, to specify the optimal parameter values in these and similar semisupervised techniques, as mentioned in [70] and [72]. Liao et al. [73] proposed a semisupervised local discriminant (SELD) analysis
to overcome this problem by combining unsupervised methods (LLFE [23]-[25]) and a supervised method (LDA [37]) in
a novel framework without any free parameters. They found
an optimal projection matrix that preserves the local neighborhood information inferred from unlabeled samples,
while simultaneously maximizing the class discrimination
of the data inferred from the labeled samples. The approach
of [74] improved SELD [73] by better modeling the differences and similarities between samples. Specifically, this
method built a semisupervised graph where labeled samples
were connected according to their label information and unlabeled samples by their nearest-neighborhood information.
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