Decomposition in Transition II: Adaptive Tensor Factorization - Realtime Data Mining - page 197

n 3

n 2

u 2 u 2 u 2

+ … + + +

n 2 n 2 n 2

n 3 n 3 n 3

=

A

n 1

u 3 u 3 u 3

n 1 n

n 1

1

u 1 u 1 u 1

Fig. 9.5 Illustration of the CP-decomposition for a 3-mode tensor A. The dashed lines indicate the
boundaries of a rank-k factorization

Here, the symbol “ ” represents the outer vector product, i.e., each element of a

∘

¼ v i i∈ n k , which is called a CP rank-1-
tensor , is the product of its corresponding vector elements

tensor A ¼ v 1 v d with vectors v k

∘ ... ∘

a i 1 , ... , i d ¼ v i 1 v i 2 ... v i d :

Thus, we may write ( 9.7 ) component-wise:

a i 1 , ...i d ¼ X

t

u i 1 , j u i d , j :

j¼ 1

Therefore, ( 9.7 ) tells us that each CP-tensor may be represented by a sum of
rank-1 tensors (Fig. 9.5 ).
For d ¼ 2, the HOSVD, i.e., the “classical” SVD, coincides with a
CP-decomposition, since the “core tensor” S k in ( 8.14 ) is a diagonal matrix. Though
it is not the identity matrix, we may multiply the diagonal values into the matrices
of singular vectors, for example, into V k .
Example 9.7 The rank-2 approximation from Example 8.4 may be written as a sum
of two rank-1 tensors:

0
@

1
A

,

10 : 3

0 : 23 2 : 76 9 : 65 5 : 17

A k ¼XY ¼ 0 : 2 0 : 7

0 : 69 6 : 61 : 65 0 : 14

:

1 0

:

0 7

0
@

1
A

0
@

1
A

1
0

3
0

0 :

¼ : 2 ð Þ

| {z }
u ðÞ

0 : 23 2 : 76 9 : 65 5 : 17 þ : 7 ð Þ

| {z }
u ðÞ

0 : 69 6 : 61 : 65 0 : 14

0 : 1 0 : 7

T T

| {z }
u 1

| {z }
u 1

¼u 1 ∘ u 1

þu 2 ∘ u 2 :

■