Decomposition in Transition II: Adaptive Tensor Factorization - Realtime Data Mining - page 182

N d Notation 9.1 Multi-indexes are denoted by bold small letters, i.e. , ( i 1 , ... , i d ) ∈

is denoted by i . Furthermore, the multi-index set n 1 ... n d , induced by

n ¼ ( n 1 , ... , n d ) ,isdenotedby n . Given multi-indexes m ¼ m 1 ; ...; ð m d 1 Þ , n ¼

ð n 1 ; ...; n d 2 Þ, their concatenation is denoted by

ð m ; n Þ :¼ m 1 , ð ... m d 1 , n 1 , ... , n d 2 Þ:

We state the following fundamental operations with tensors:

1. Addition/subtraction

m m m ,

R ! R

R

b j j ∈ m aðÞ b j

i ∈ m :

aðÞ i ∈ m ; :¼ a i b j

2. Contraction
Let p ¼ ( p , q ), n ¼ (1, ... , m p 1 , m p +1 , ... , m q 1 , m q +1 , ... , m d )

m n

! R

R

0 1

X

a i ;ðÞ i ;ð ∈ n ;ðÞ

@ A

aðÞ:¼ a i ;ðÞ :

k ∈ p

i ∈ n

3. Inner product

m m

R ! R ,

R

:¼ X
i ∈ m

b j j ∈ m b j aðÞ i ∈ m ; aðÞ; a i b i :

4. Outer product (tensor product)

m n ð m ; n Þ ,

R ! R

R

b j j ∈ n aðÞ b j :¼ a i b j

i ;ð ∈ m

aðÞ i ∈ m ;

Þ : ð ; n

5. Contracted product (multilinear product)
Let m N d 1 N d 2 N δ :

∈ n ∈ p ∈

, ,

ð m, n Þ , ð m ; p Þ R n ;ðÞ ! R

δ : R

0 1

X

j ;ð ∈ n ;ðÞ

a i ;ðÞ δ

:¼

@ A i ;ð ∈ m ; n

a i, ðÞ i ;ð ∈ m ; p

Þ ;

b j ;ðÞ b j ;ðÞ a i ;ð b j ;ðÞ :

ð

k ∈ p

ð Þ