IEEE Systems, Man and Cybernetics Magazine - January 2021 - 32

opportunities with focus in areas such as anomaly detecand feature selection; in these cases, mining large
tion, coclustering with multiway data, and applications in
amounts of information to find one or more small subsets
unsupervised machine learning.
satisfying specific criteria is key.
In other large data applications,
Conclusion
incremental tensor analysis (ITA)
We provided an overview of the
For both image sets,
was developed to deal with tensors
PCA variants known as 2DPCA,
with essentially unlimited tempoCSA was shown
B2DPCA, and CSA. These variants
ral dimension [28], and in [29], tento outperform
operate directly on grayscale imagsor train (TT) decomposition can
es in matrix form. Some diverse
provide an efficient representation
conventional PCA in
areas of application have been
of high order tensors for which
terms of classification
noted, and some advantages and
required memory scales linearly
disadvantages of these variants
with tensor order. Such methods
accuracy over a
briefly described in relation to PCA.
can reduce dimensionality and disrange of numbers of
The performance of CSA in image
till large amounts of data into a
principal components.
classification has been demonstratsimpler more meaningful represened in comparison to PCA using two
tation, which reveals important
well-known image data sets: the
hidden interrelationships and corMNIST handwritten digits set and
relations within these data. MethFashion-MNIST set. For both image sets, CSA was shown to
ods providing these characteristics have obvious benefits
outperform conventional PCA in terms of classification
in the area of big data: as Tien emphasizes in [30], simply
accuracy over a range of numbers of principal components
having huge amounts of information may be worthless
in each demonstration. Image reconstruction with CSA was
without analysis or processing that can " ...yield critical
also discussed and demonstrated in comparison to reconinformation. " As applications of multiway big data grow,
struction with PCA; the CSA-based reconstruction provided
new extensions of existing tools for analysis and processmore efficient and better quality reconstruction. We pointed
ing analogous to ITA and TT may be beneficial. Although
out that the PCA variants described herein can be put into
we highlight the theme of large amounts of data, tensor
the broader context of tensor-based methods that can operPCA variants and related methods should provide research
ate directly on data in the form of not just second-order tensors or matrices but also on tensors of order three and
higher. Finally, we mentioned some areas of potentially
fruitful research utilizing some related tensor-based methods, including streaming tensor data and coclustering with
multiway data.

(f)

(b)

(g)

(c)

(h)

(d)

(i)

(e)

(j)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 4. Image reconstruction for the MNIST digits

(rows 1 and 2) and Fashion-MNIST (rows 3 and 4):
(a) and (f) are the original image, (b) the top 50
eigenvectors (PCA), (c) the top 36 eigenvectors
(PCA), (d) the top 20 eigenvectors (PCA), (e) the top
12 eigenvectors (PCA), and the CSA feature matrix:
(g) 25 # 25,  (h)  18 # 18,  (i)  10 # 10,   and (j)  6 # 6.  
32	

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Janu ar y 2021

About the Authors
Richard A. Nelson (ran11f@my.fsu.edu) earned his B.S.
and M.S. degrees in electrical engineering from Florida State
University, Tallahassee. Currently, he is a Ph.D. candidate in

Eigenvalue

(a)

14
12
10
8
6
4
2
0

PCA
CSA: Left (U ) Subspace
CSA: Right (V ) Subspace

10

20
30
Eigenvalue Index

40

50

Figure 5. The eigenvalues of the CSA left subspace,

CSA right subspace, and first 50 eigenvalues for PCA
for the MNIST digits reconstruction example. The plots
are combined to save space.



IEEE Systems, Man and Cybernetics Magazine - January 2021

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