IEEE Circuits and Systems Magazine - Q2 2023 - 11
low-rank SVD and reshaping if the matrix is too thin.
As illustrated in Fig. 3, a matrix of size 810×
can be
decomposed into five matrices G1 through G5 by recursively
applying reshaping and low-rank SVD. Then
these five matrices can be folded into five TT-cores of
sizes 12 1××r , rr rr rr12 23 3422 2
respectively.
×× ×× ××,,
and r4 51
×× ,
C. Canonical Polyadic Decomposition
In 1927, the idea of expressing a tensor as the sum of a finite
number of rank-one tensors was proposed by Hitchcock
[42], [43]. In 1970, the concept was generalized as
CANDECOMP (canonical decomposition) by Carroll and
Chang [44] and as PARAFAC (parallel factors) by Harshman
et al. [45].
The Canonical Polyadic (CP) decomposition factorFor
example, a third-order tensor X∈R ××
approximated as:
R
X ≈
∑
r=1
where denotes the outer product of vectors. Parameter
R is a positive integer and abr
cr ∈R for rR
∈∈ JRR,
I
r
K
=...
of tensor X can be computed as:
xij k
R
,, ≈∑ab c
r=1
ir jr kr,
where iIjJkK=... =... =...11 1
Bb bb
=...
,, ,, ,, ,, . The factor matrices
are defined as the combination of the vectors
from the rank-one components: Aa aa
=...
izes a tensor into a sum of component rank-one tensors. [] []
12 R and Cc cc
=...
12 R .
[]
12 R ,
and
1, , . As shown in Fig. 4, each element
ab cr ,
rr
IJ K
can be
Figure 3. Tensor Train Decomposition of a matrix of size 8 × 10 by recursively applying reshaping and low-rank
SVD. After decomposition and reshaping, the dimensions of the five TT-cores are 1 × 2 × r1, r1 × 2 × r2, r2 × 2 ×
r3, r3 × 2 × r4, and r4 × 5 × 1, respectively.
SECOND QUARTER 2023
IEEE CIRCUITS AND SYSTEMS MAGAZINE
11
IEEE Circuits and Systems Magazine - Q2 2023
Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2023
Contents
IEEE Circuits and Systems Magazine - Q2 2023 - Cover1
IEEE Circuits and Systems Magazine - Q2 2023 - Cover2
IEEE Circuits and Systems Magazine - Q2 2023 - Contents
IEEE Circuits and Systems Magazine - Q2 2023 - 2
IEEE Circuits and Systems Magazine - Q2 2023 - 3
IEEE Circuits and Systems Magazine - Q2 2023 - 4
IEEE Circuits and Systems Magazine - Q2 2023 - 5
IEEE Circuits and Systems Magazine - Q2 2023 - 6
IEEE Circuits and Systems Magazine - Q2 2023 - 7
IEEE Circuits and Systems Magazine - Q2 2023 - 8
IEEE Circuits and Systems Magazine - Q2 2023 - 9
IEEE Circuits and Systems Magazine - Q2 2023 - 10
IEEE Circuits and Systems Magazine - Q2 2023 - 11
IEEE Circuits and Systems Magazine - Q2 2023 - 12
IEEE Circuits and Systems Magazine - Q2 2023 - 13
IEEE Circuits and Systems Magazine - Q2 2023 - 14
IEEE Circuits and Systems Magazine - Q2 2023 - 15
IEEE Circuits and Systems Magazine - Q2 2023 - 16
IEEE Circuits and Systems Magazine - Q2 2023 - 17
IEEE Circuits and Systems Magazine - Q2 2023 - 18
IEEE Circuits and Systems Magazine - Q2 2023 - 19
IEEE Circuits and Systems Magazine - Q2 2023 - 20
IEEE Circuits and Systems Magazine - Q2 2023 - 21
IEEE Circuits and Systems Magazine - Q2 2023 - 22
IEEE Circuits and Systems Magazine - Q2 2023 - 23
IEEE Circuits and Systems Magazine - Q2 2023 - 24
IEEE Circuits and Systems Magazine - Q2 2023 - 25
IEEE Circuits and Systems Magazine - Q2 2023 - 26
IEEE Circuits and Systems Magazine - Q2 2023 - 27
IEEE Circuits and Systems Magazine - Q2 2023 - 28
IEEE Circuits and Systems Magazine - Q2 2023 - 29
IEEE Circuits and Systems Magazine - Q2 2023 - 30
IEEE Circuits and Systems Magazine - Q2 2023 - 31
IEEE Circuits and Systems Magazine - Q2 2023 - 32
IEEE Circuits and Systems Magazine - Q2 2023 - 33
IEEE Circuits and Systems Magazine - Q2 2023 - 34
IEEE Circuits and Systems Magazine - Q2 2023 - 35
IEEE Circuits and Systems Magazine - Q2 2023 - 36
IEEE Circuits and Systems Magazine - Q2 2023 - 37
IEEE Circuits and Systems Magazine - Q2 2023 - 38
IEEE Circuits and Systems Magazine - Q2 2023 - 39
IEEE Circuits and Systems Magazine - Q2 2023 - 40
IEEE Circuits and Systems Magazine - Q2 2023 - 41
IEEE Circuits and Systems Magazine - Q2 2023 - 42
IEEE Circuits and Systems Magazine - Q2 2023 - 43
IEEE Circuits and Systems Magazine - Q2 2023 - 44
IEEE Circuits and Systems Magazine - Q2 2023 - 45
IEEE Circuits and Systems Magazine - Q2 2023 - 46
IEEE Circuits and Systems Magazine - Q2 2023 - 47
IEEE Circuits and Systems Magazine - Q2 2023 - 48
IEEE Circuits and Systems Magazine - Q2 2023 - 49
IEEE Circuits and Systems Magazine - Q2 2023 - 50
IEEE Circuits and Systems Magazine - Q2 2023 - 51
IEEE Circuits and Systems Magazine - Q2 2023 - 52
IEEE Circuits and Systems Magazine - Q2 2023 - 53
IEEE Circuits and Systems Magazine - Q2 2023 - 54
IEEE Circuits and Systems Magazine - Q2 2023 - 55
IEEE Circuits and Systems Magazine - Q2 2023 - 56
IEEE Circuits and Systems Magazine - Q2 2023 - 57
IEEE Circuits and Systems Magazine - Q2 2023 - 58
IEEE Circuits and Systems Magazine - Q2 2023 - 59
IEEE Circuits and Systems Magazine - Q2 2023 - 60
IEEE Circuits and Systems Magazine - Q2 2023 - 61
IEEE Circuits and Systems Magazine - Q2 2023 - 62
IEEE Circuits and Systems Magazine - Q2 2023 - 63
IEEE Circuits and Systems Magazine - Q2 2023 - 64
IEEE Circuits and Systems Magazine - Q2 2023 - 65
IEEE Circuits and Systems Magazine - Q2 2023 - 66
IEEE Circuits and Systems Magazine - Q2 2023 - 67
IEEE Circuits and Systems Magazine - Q2 2023 - 68
IEEE Circuits and Systems Magazine - Q2 2023 - 69
IEEE Circuits and Systems Magazine - Q2 2023 - 70
IEEE Circuits and Systems Magazine - Q2 2023 - 71
IEEE Circuits and Systems Magazine - Q2 2023 - 72
IEEE Circuits and Systems Magazine - Q2 2023 - Cover3
IEEE Circuits and Systems Magazine - Q2 2023 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com