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

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