IEEE Geoscience and Remote Sensing Magazine - December 2020 - 72
[47] and its linearized LPP [48], LLE [46], sparse GDA (SGDA)
[65], and collaborative GDA (CGDA) [66]. Figure 8 visualizes the affinity matrices given by five different strategies in
a four-class case selected from the Houston 2013 data set.
LDA-LIKE AFFINITY MATRIX
In essence, LDA is vested in a special case of the GGE framework with D (LDA) = I, whose affinity matrix can be represented as
(LDA)
W ij
=(
1/N k, if x i and x j ! C k;
(20)
0,
otherwise,
where N k is the number of samples belonging to kth class.
LPP- OR LE-BASED AFFINITY MATRIX
One is to be constructed in kernel space with a higher dimension via similarity measurement, i.e., extensively using (15).
LLE-BASED AFFINITY MATRIX
Different from the handcrafted graph, LLE reconstructs
each given sample with its k-nearest neighbors by exploiting linear regression techniques [67], [68]. As a result, the
reconstruction coefficients (A) can be obtained by solving the optimization problem of (10). With the known A,
it is straightforward to derive the needful affinity matrix,
denoted as W (LLE),
(LLE)
W ij
=(
A ij + A ijT - A ij A ijT,
0,
if x j ! z k(x i);
(21)
otherwise,
thereby inducing the Laplacian matrix as L(LLE) = D (LLE) - W (LLE) =
(I - A)T (I - A).
SGDA- AND CGDA-GUIDED AFFINITY MATRIX
Similar to LLE, the affinity matrix can be estimated using
data-driven representation learning, i.e., sparse and collaborative representations [69]-[71]. Accordingly, the two
learning strategies can be equivalent to respectively solving
the constrained l 1- norm optimization problem,
min W
W
1
s.t.
Xm W - Xm
2
F
# e, (22)
and the l 2- norm optimization problem,
min W
W
2
F
s. t.
Xm W - Xm
2
F
# e. (23)
These affinity matrices can be unified to the GGE framework of (12).
In addition to SGDA and CGDA (the two baselines),
Huang et al. [72] learned a set of sparse coefficients on
manifolds and then preserved the sparse manifold structure in the embedded space. In [73], Xue et al. extended the
existing SGDA to the spatial-spectral graph embedding
to address issues of spatial variability and spectral multimodality. With the embedding of the intrinsic geometric
structure of the data, a Laplacian regularizer CGDA [74]
was developed to further improve the graph's confidence.
Li et al. [75] simultaneously integrated sparsity and low
rankness into the graph to capture a more robust structure
of the data locally and globally. Furthermore, Pan et al.
[76] improved the work by Li et al. [75] by unfolding the
HSI data with the form of a tensor.
KDA
In reality, the HSI usually exhibits a highly nonlinear data
distribution, which may result in difficulties in effectively identifying the materials. The solution to this issue
makes use of a so-called kernel trick [77] that can map
the data of the input space into a new Hilbert space with
a higher feature dimension. In the kernel-induced space,
the complex nonlinearity of the HSI can be well analyzed in a linearized system. Comparatively, the input to
KDA is an inner product of original data pairs, defined as
k(x i, x j), which can be given by (15). By introducing the
kernel Gram matrix K with K i, j = U(x i)T U (x j) = k(x i, x j),
most of the previous LDA-based methods can be simp----
ly e x tended to t he cor responding ke r ne l i z e d versions; i.e., kernelized LDA (KLDA) and kernelized LFDA
-(KLFDA) can calculate their projections P by solving a
GED p
- roblem of
KLKP = m (KBK + cI)P. (24)
Note that B = I in KLDA, whereas L = L w and B = L b are
computed by D w - W w and D b - W b in the kernel space,
respectively, for KLFDA. Furthermore, for kernelized SGDA
1
0.5
0
(a)
(b)
(c)
(d)
(e)
FIGURE 8. A four-class showcase for affinity matrices ( W ) with respect to five different approaches, where the connectivity (or edge) of W is
computed within each class: (a) LDA, (b) LPP, (c) LLE, (d) SGDA, and (e) CGDA.
72
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
DECEMBER 2020
IEEE Geoscience and Remote Sensing Magazine - December 2020
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