In mathematics, the Khatri–Rao product is defined as[1] [2]
A
∗
B
=
(
A
i
j
⊗
B
i
j
)
i
j
{\displaystyle \mathbf {A} \ast \mathbf {B} =\left(\mathbf {A} _{ij}\otimes \mathbf {B} _{ij}\right)_{ij}}
in which the ij -th block is the mi pi × nj qj Kronecker product of the corresponding blocks of A and B , assuming the number of row and column partitions of both matrices is equal. The size of the product is then (Σi mi pi j nj qj .
For example, if A and B both are 2 × 2 partitioned matrices e.g.:
A
=
[
A
11
A
12
A
21
A
22
]
=
[
1
2
3
4
5
6
7
8
9
]
,
B
=
[
B
11
B
12
B
21
B
22
]
=
[
1
4
7
2
5
8
3
6
9
]
,
{\displaystyle \mathbf {A} =\left[{\begin{array}{c | c}\mathbf {A} _{11}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right]=\left[{\begin{array}{c c | c}1&2&3\\4&5&6\\\hline 7&8&9\end{array}}\right],\quad \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right]=\left[{\begin{array}{c | c c}1&4&7\\\hline 2&5&8\\3&6&9\end{array}}\right],}
we obtain:
A
∗
B
=
[
A
11
⊗
B
11
A
12
⊗
B
12
A
21
⊗
B
21
A
22
⊗
B
22
]
=
[
1
2
12
21
4
5
24
42
14
16
45
72
21
24
54
81
]
.
{\displaystyle \mathbf {A} \ast \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {A} _{11}\otimes \mathbf {B} _{11}&\mathbf {A} _{12}\otimes \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\otimes \mathbf {B} _{21}&\mathbf {A} _{22}\otimes \mathbf {B} _{22}\end{array}}\right]=\left[{\begin{array}{c c | c c}1&2&12&21\\4&5&24&42\\\hline 14&16&45&72\\21&24&54&81\end{array}}\right].}
This is a submatrix of the Tracy–Singh product of the two matrices (each partition in this example is a partition in a corner of the Tracy–Singh product ) and also may be called the block Kronecker product.
Column-wise Kronecker product [ ] A column-wise Kronecker product of two matrices may also be called the Khatri–Rao product. This product assumes the partitions of the matrices are their columns. In this case m 1 = m p 1 = p n = q j : nj = pj = 1mp × n A and B . Using the matrices from the previous examples with the columns partitioned:
C
=
[
C
1
C
2
C
3
]
=
[
1
2
3
4
5
6
7
8
9
]
,
D
=
[
D
1
D
2
D
3
]
=
[
1
4
7
2
5
8
3
6
9
]
,
{\displaystyle \mathbf {C} =\left[{\begin{array}{c | c | c}\mathbf {C} _{1}&\mathbf {C} _{2}&\mathbf {C} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c}1&2&3\\4&5&6\\7&8&9\end{array}}\right],\quad \mathbf {D} =\left[{\begin{array}{c | c | c }\mathbf {D} _{1}&\mathbf {D} _{2}&\mathbf {D} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c }1&4&7\\2&5&8\\3&6&9\end{array}}\right],}
so that:
C
∗
D
=
[
C
1
⊗
D
1
C
2
⊗
D
2
C
3
⊗
D
3
]
=
[
1
8
21
2
10
24
3
12
27
4
20
42
8
25
48
12
30
54
7
32
63
14
40
72
21
48
81
]
.
{\displaystyle \mathbf {C} \ast \mathbf {D} =\left[{\begin{array}{c | c | c }\mathbf {C} _{1}\otimes \mathbf {D} _{1}&\mathbf {C} _{2}\otimes \mathbf {D} _{2}&\mathbf {C} _{3}\otimes \mathbf {D} _{3}\end{array}}\right]=\left[{\begin{array}{c | c | c }1&8&21\\2&10&24\\3&12&27\\4&20&42\\8&25&48\\12&30&54\\7&32&63\\14&40&72\\21&48&81\end{array}}\right].}
This column-wise version of the Khatri–Rao product is useful in linear algebra approaches to data analytical processing[3] [4] [5]
In 1996 the Column-wise Khatri–Rao product was proposed to estimate the angles of arrival (AOAs) and delays of multipath signals[6] [7] digital antenna array .
Face-splitting product [ ]
Face splitting product of matrices
The alternative concept of the matrix product, which uses row-wise splitting of matrices with a given quantity of rows, was proposed by V. Slyusar [8] [7] [9] [10] [11] [12]
This matrix operation was named the "face-splitting product" of matrices[9] [11]
C
=
[
C
1
C
2
C
3
]
=
[
1
2
3
4
5
6
7
8
9
]
,
D
=
[
D
1
D
2
D
3
]
=
[
1
4
7
2
5
8
3
6
9
]
,
{\displaystyle \mathbf {C} ={\begin{bmatrix}\mathbf {C} _{1}\\\hline \mathbf {C} _{2}\\\hline \mathbf {C} _{3}\\\end{bmatrix}}={\begin{bmatrix}1&2&3\\\hline 4&5&6\\\hline 7&8&9\end{bmatrix}},\quad \mathbf {D} ={\begin{bmatrix}\mathbf {D} _{1}\\\hline \mathbf {D} _{2}\\\hline \mathbf {D} _{3}\\\end{bmatrix}}={\begin{bmatrix}1&4&7\\\hline 2&5&8\\\hline 3&6&9\end{bmatrix}},}
the result can be obtained:[7] [9] [11]
C
∙
D
=
[
C
1
⊗
D
1
C
2
⊗
D
2
C
3
⊗
D
3
]
=
[
1
4
7
2
8
14
3
12
21
8
20
32
10
25
40
12
30
48
21
42
63
24
48
72
27
54
81
]
.
{\displaystyle \mathbf {C} \bullet \mathbf {D} ={\begin{bmatrix}\mathbf {C} _{1}\otimes \mathbf {D} _{1}\\\hline \mathbf {C} _{2}\otimes \mathbf {D} _{2}\\\hline \mathbf {C} _{3}\otimes \mathbf {D} _{3}\\\end{bmatrix}}={\begin{bmatrix}1&4&7&2&8&14&3&12&21\\\hline 8&20&32&10&25&40&12&30&48\\\hline 21&42&63&24&48&72&27&54&81\end{bmatrix}}.}
Main properties [ ] Transpose V. Slyusar , 1996[7] [9] [10]
(
A
∙
B
)
T
=
A
T
∗
B
T
{\displaystyle \left(\mathbf {A} \bullet \mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}}
Bilinearity and associativity :[7] [9] [10]
A
∙
(
B
+
C
)
=
A
∙
B
+
A
∙
C
,
(
B
+
C
)
∙
A
=
B
∙
A
+
C
∙
A
,
(
k
A
)
∙
B
=
A
∙
(
k
B
)
=
k
(
A
∙
B
)
,
(
A
∙
B
)
∙
C
=
A
∙
(
B
∙
C
)
,
{\displaystyle {\begin{aligned}\mathbf {A} \bullet (\mathbf {B} +\mathbf {C} )&=\mathbf {A} \bullet \mathbf {B} +\mathbf {A} \bullet \mathbf {C} ,\\(\mathbf {B} +\mathbf {C} )\bullet \mathbf {A} &=\mathbf {B} \bullet \mathbf {A} +\mathbf {C} \bullet \mathbf {A} ,\\(k\mathbf {A} )\bullet \mathbf {B} &=\mathbf {A} \bullet (k\mathbf {B} )=k(\mathbf {A} \bullet \mathbf {B} ),\\(\mathbf {A} \bullet \mathbf {B} )\bullet \mathbf {C} &=\mathbf {A} \bullet (\mathbf {B} \bullet \mathbf {C} ),\\\end{aligned}}}
where A , B and C are matrices, and k is a scalar ,
a
∙
B
=
B
∙
a
{\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a}
[10]
a
{\displaystyle a}
vector ,The mixed-product property (V. Slyusar , 1997[10]
(
A
∙
B
)
(
A
T
∗
B
T
)
=
(
A
A
T
)
∘
(
B
B
T
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )\left(\mathbf {A} ^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}\right)=\left(\mathbf {A} \mathbf {A} ^{\textsf {T}}\right)\circ \left(\mathbf {B} \mathbf {B} ^{\textsf {T}}\right)}
(
A
∙
B
)
(
C
∗
D
)
=
(
A
C
)
∘
(
B
D
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\circ (\mathbf {B} \mathbf {D} )}
(
A
∙
B
∙
C
∙
D
)
(
L
∗
M
∗
N
∗
P
)
=
(
A
L
)
∘
(
B
M
)
∘
(
C
N
)
∘
(
D
P
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} \bullet \mathbf {C} \bullet \mathbf {D} )(\mathbf {L} \ast \mathbf {M} \ast \mathbf {N} \ast \mathbf {P} )=(\mathbf {A} \mathbf {L} )\circ (\mathbf {B} \mathbf {M} )\circ (\mathbf {C} \mathbf {N} )\circ (\mathbf {D} \mathbf {P} )}
[13]
(
A
∗
B
)
T
(
A
∗
B
)
=
(
A
T
A
)
∘
(
B
T
B
)
{\displaystyle (\mathbf {A} \ast \mathbf {B} )^{\textsf {T}}(\mathbf {A} \ast \mathbf {B} )=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)\circ \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)}
[14]
∘
{\displaystyle \circ }
Hadamard product ,
(
A
∘
B
)
∙
(
C
∘
D
)
=
(
A
∙
C
)
∘
(
B
∙
D
)
{\displaystyle (\mathbf {A} \circ \mathbf {B} )\bullet (\mathbf {C} \circ \mathbf {D} )=(\mathbf {A} \bullet \mathbf {C} )\circ (\mathbf {B} \bullet \mathbf {D} )}
[10]
A
⊗
(
B
∙
C
)
=
(
A
⊗
B
)
∙
C
{\displaystyle \mathbf {A} \otimes (\mathbf {B} \bullet \mathbf {C} )=(\mathbf {A} \otimes \mathbf {B} )\bullet \mathbf {C} }
[7]
(
A
⊗
B
)
(
C
∗
D
)
=
(
A
C
)
∗
(
B
D
)
{\displaystyle (\mathbf {A} \otimes \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\ast (\mathbf {B} \mathbf {D} )}
[14]
(
A
⊗
B
)
∗
(
C
⊗
D
)
=
P
[
(
A
∗
C
)
⊗
(
B
∗
D
)
]
{\displaystyle (\mathbf {A} \otimes \mathbf {B} )\ast (\mathbf {C} \otimes \mathbf {D} )=\mathbf {P} [(\mathbf {A} \ast \mathbf {C} )\otimes (\mathbf {B} \ast \mathbf {D} )]}
P
{\displaystyle \mathbf {P} }
[5]
(
A
∙
B
)
(
C
⊗
D
)
=
(
A
C
)
∙
(
B
D
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=(\mathbf {A} \mathbf {C} )\bullet (\mathbf {B} \mathbf {D} )}
[11] [13]
(
A
∙
L
)
(
B
⊗
M
)
⋯
(
C
⊗
S
)
=
(
A
B
⋯
C
)
∙
(
L
M
⋯
S
)
{\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} )\bullet (\mathbf {L} \mathbf {M} \cdots \mathbf {S} )}
c
T
∙
d
T
=
c
T
⊗
d
T
{\displaystyle c^{\textsf {T}}\bullet d^{\textsf {T}}=c^{\textsf {T}}\otimes d^{\textsf {T}}}
[10]
c
∗
d
=
c
⊗
d
{\displaystyle c\ast d=c\otimes d}
c
{\displaystyle c}
d
{\displaystyle d}
vectors ,
(
A
∗
c
T
)
d
=
(
A
∗
d
T
)
c
{\displaystyle \left(\mathbf {A} \ast c^{\textsf {T}}\right)d=\left(\mathbf {A} \ast d^{\textsf {T}}\right)c}
[15]
d
T
(
c
∙
A
T
)
=
c
T
(
d
∙
A
T
)
{\displaystyle d^{\textsf {T}}\left(c\bullet \mathbf {A} ^{\textsf {T}}\right)=c^{\textsf {T}}\left(d\bullet \mathbf {A} ^{\textsf {T}}\right)}
(
A
∙
B
)
(
c
⊗
d
)
=
(
A
c
)
∘
(
B
d
)
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)}
[16]
c
{\displaystyle c}
d
{\displaystyle d}
vectors (it is a combine of properties 3 an 8),
Similarly:
(
A
∙
B
)
(
M
N
c
⊗
Q
P
d
)
=
(
A
M
N
c
)
∘
(
B
Q
P
d
)
,
{\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {M} \mathbf {N} c\otimes \mathbf {Q} \mathbf {P} d)=(\mathbf {A} \mathbf {M} \mathbf {N} c)\circ (\mathbf {B} \mathbf {Q} \mathbf {P} d),}
F
(
C
(
1
)
x
⋆
C
(
2
)
y
)
=
(
F
C
(
1
)
∙
F
C
(
2
)
)
(
x
⊗
y
)
=
F
C
(
1
)
x
∘
F
C
(
2
)
y
{\displaystyle {\mathcal {F}}\left(C^{(1)}x\star C^{(2)}y\right)=\left({\mathcal {F}}C^{(1)}\bullet {\mathcal {F}}C^{(2)}\right)(x\otimes y)={\mathcal {F}}C^{(1)}x\circ {\mathcal {F}}C^{(2)}y}
⋆
{\displaystyle \star }
convolution and
F
{\displaystyle {\mathcal {F}}}
Fourier transform matrix (this result is an evolving of count sketch properties[17]
A
∙
B
=
(
A
⊗
1
c
T
)
∘
(
1
k
T
⊗
B
)
{\displaystyle \mathbf {A} \bullet \mathbf {B} =\left(\mathbf {A} \otimes \mathbf {1_{c}} ^{\textsf {T}}\right)\circ \left(\mathbf {1_{k}} ^{\textsf {T}}\otimes \mathbf {B} \right)}
[18]
A
{\displaystyle \mathbf {A} }
r
×
c
{\displaystyle r\times c}
B
{\displaystyle \mathbf {B} }
r
×
k
{\displaystyle r\times k}
1
c
{\displaystyle \mathbf {1_{c}} }
c
{\displaystyle c}
1
k
{\displaystyle \mathbf {1_{k}} }
k
{\displaystyle k}
M
∙
M
=
(
M
⊗
1
T
)
∘
(
1
T
⊗
M
)
{\displaystyle \mathbf {M} \bullet \mathbf {M} =\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)\circ \left(\mathbf {1} ^{\textsf {T}}\otimes \mathbf {M} \right)}
[19]
M
{\displaystyle \mathbf {M} }
r
×
c
{\displaystyle r\times c}
∘
{\displaystyle \circ }
1
{\displaystyle \mathbf {1} }
c
{\displaystyle c}
M
∙
M
=
M
[
∘
]
(
M
⊗
1
T
)
{\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)}
[
∘
]
{\displaystyle [\circ ]}
penetrating face product of matrices.[11]
P
∗
N
=
(
P
⊗
1
c
)
∘
(
1
k
⊗
N
)
{\displaystyle \mathbf {P} \ast \mathbf {N} =(\mathbf {P} \otimes \mathbf {1_{c}} )\circ (\mathbf {1_{k}} \otimes \mathbf {N} )}
P
{\displaystyle \mathbf {P} }
c
×
r
{\displaystyle c\times r}
N
{\displaystyle \mathbf {N} }
k
×
r
{\displaystyle k\times r}
W
d
A
=
w
∙
A
{\displaystyle \mathbf {W_{d}} \mathbf {A} =\mathbf {w} \bullet \mathbf {A} }
[10]
vec
(
A
T
W
d
A
)
=
(
A
∙
A
)
T
w
{\displaystyle \operatorname {vec} \left(\mathbf {A} ^{\textsf {T}}\mathbf {W_{d}} \mathbf {A} \right)=\left(\mathbf {A} \bullet \mathbf {A} \right)^{\textsf {T}}\mathbf {w} }
[19]
w
{\displaystyle \mathbf {w} }
W
d
{\displaystyle \mathbf {W_{d}} }
vec
(
A
)
{\displaystyle \operatorname {vec} (\mathbf {A} )}
A
{\displaystyle \mathbf {A} }
(
A
∙
L
)
(
B
⊗
M
)
⋯
(
C
⊗
S
)
(
K
∗
T
)
=
(
A
B
.
.
.
C
K
)
∘
(
L
M
.
.
.
S
T
)
{\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {K} \ast \mathbf {T} )=(\mathbf {A} \mathbf {B} ...\mathbf {C} \mathbf {K} )\circ (\mathbf {L} \mathbf {M} ...\mathbf {S} \mathbf {T} )}
[11] [13]
(
A
∙
L
)
(
B
⊗
M
)
⋯
(
C
⊗
S
)
(
c
⊗
d
)
=
(
A
B
⋯
C
c
)
∘
(
L
M
⋯
S
d
)
,
(
A
∙
L
)
(
B
⊗
M
)
⋯
(
C
⊗
S
)
(
P
c
⊗
Q
d
)
=
(
A
B
⋯
C
P
c
)
∘
(
L
M
⋯
S
Q
d
)
{\displaystyle {\begin{aligned}(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(c\otimes d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} d),\\(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {P} c\otimes \mathbf {Q} d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} \mathbf {P} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} \mathbf {Q} d)\end{aligned}}}
c
{\displaystyle c}
d
{\displaystyle d}
vectors Examples[16] [ ]
(
[
1
0
0
1
1
0
]
∙
[
1
0
1
0
0
1
]
)
(
[
1
1
1
−
1
]
⊗
[
1
1
1
−
1
]
)
(
[
σ
1
0
0
σ
2
]
⊗
[
ρ
1
0
0
ρ
2
]
)
(
[
x
1
x
2
]
∗
[
y
1
y
2
]
)
=
(
[
1
0
0
1
1
0
]
∙
[
1
0
1
0
0
1
]
)
(
[
1
1
1
−
1
]
[
σ
1
0
0
σ
2
]
[
x
1
x
2
]
⊗
[
1
1
1
−
1
]
[
ρ
1
0
0
ρ
2
]
[
y
1
y
2
]
)
=
[
1
0
0
1
1
0
]
[
1
1
1
−
1
]
[
σ
1
0
0
σ
2
]
[
x
1
x
2
]
∘
[
1
0
1
0
0
1
]
[
1
1
1
−
1
]
[
ρ
1
0
0
ρ
2
]
[
y
1
y
2
]
.
{\displaystyle {\begin{aligned}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\right)\left({\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}\otimes {\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}\right)\left({\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\ast {\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]{}={}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\otimes \,{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]{}={}&{\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\circ \,{\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}.\end{aligned}}}
Theorem[16] [ ] If
M
=
T
(
1
)
∙
⋯
∙
T
(
c
)
{\displaystyle M=T^{(1)}\bullet \dots \bullet T^{(c)}}
T
(
1
)
,
…
,
T
(
c
)
{\displaystyle T^{(1)},\dots ,T^{(c)}}
T
{\displaystyle T}
T
1
,
…
,
T
m
∈
R
d
{\displaystyle T_{1},\dots ,T_{m}\in \mathbb {R} ^{d}}
E
[
(
T
1
x
)
2
]
=
‖
x
‖
2
2
{\displaystyle E\left[(T_{1}x)^{2}\right]=\left\|x\right\|_{2}^{2}}
E
[
(
T
1
x
)
p
]
1
p
≤
a
p
‖
x
‖
2
{\displaystyle E\left[(T_{1}x)^{p}\right]^{\frac {1}{p}}\leq {\sqrt {ap}}\|x\|_{2}}
then for any vector
x
{\displaystyle x}
|
‖
M
x
‖
2
−
‖
x
‖
2
|
<
ε
‖
x
‖
2
{\displaystyle \left|\left\|Mx\right\|_{2}-\left\|x\right\|_{2}\right|<\varepsilon \left\|x\right\|_{2}}
with probability
1
−
δ
{\displaystyle 1-\delta }
m
=
(
4
a
)
2
c
ε
−
2
log
1
/
δ
+
(
2
a
e
)
ε
−
1
(
log
1
/
δ
)
c
.
{\displaystyle m=(4a)^{2c}\varepsilon ^{-2}\log 1/\delta +(2ae)\varepsilon ^{-1}(\log 1/\delta )^{c}.}
In particular, if the entries of
T
{\displaystyle T}
±
1
{\displaystyle \pm 1}
m
=
O
(
ε
−
2
log
1
/
δ
+
ε
−
1
(
1
c
log
1
/
δ
)
c
)
{\displaystyle m=O\left(\varepsilon ^{-2}\log 1/\delta +\varepsilon ^{-1}\left({\frac {1}{c}}\log 1/\delta \right)^{c}\right)}
which matches the Johnson–Lindenstrauss lemma of
m
=
O
(
ε
−
2
log
1
/
δ
)
{\displaystyle m=O\left(\varepsilon ^{-2}\log 1/\delta \right)}
ε
{\displaystyle \varepsilon }
Block face-splitting product [ ]
Transposed block face-splitting product in the context of a multi-face radar model
[13] According to the definition of V. Slyusar [7] [11] partitioned matrices with a given quantity of rows in blocks
A
=
[
A
11
A
12
A
21
A
22
]
,
B
=
[
B
11
B
12
B
21
B
22
]
,
{\displaystyle \mathbf {A} =\left[{\begin{array}{c | c}\mathbf {A} _{11}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right],\quad \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right],}
can be written as :
A
[
∙
]
B
=
[
A
11
∙
B
11
A
12
∙
B
12
A
21
∙
B
21
A
22
∙
B
22
]
.
{\displaystyle \mathbf {A} [\bullet ]\mathbf {B} =\left[{\begin{array}{c | c}\mathbf {A} _{11}\bullet \mathbf {B} _{11}&\mathbf {A} _{12}\bullet \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\bullet \mathbf {B} _{21}&\mathbf {A} _{22}\bullet \mathbf {B} _{22}\end{array}}\right].}
The transposed block face-splitting product (or Block column-wise version of the Khatri–Rao product) of two partitioned matrices with a given quantity of columns in blocks has a view:[7] [11]
A
[
∗
]
B
=
[
A
11
∗
B
11
A
12
∗
B
12
A
21
∗
B
21
A
22
∗
B
22
]
.
{\displaystyle \mathbf {A} [\ast ]\mathbf {B} =\left[{\begin{array}{c | c}\mathbf {A} _{11}\ast \mathbf {B} _{11}&\mathbf {A} _{12}\ast \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\ast \mathbf {B} _{21}&\mathbf {A} _{22}\ast \mathbf {B} _{22}\end{array}}\right].}
Main properties [ ] Transpose :
(
A
[
∗
]
B
)
T
=
A
T
[
∙
]
B
T
{\displaystyle \left(\mathbf {A} [\ast ]\mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}[\bullet ]\mathbf {B} ^{\textsf {T}}}
[13] Applications [ ] The Face-splitting product and the Block Face-splitting product used in the tensor -matrix theory of digital antenna arrays . These operations used also in:
Artificial Intelligence and Machine learning systems to minimization of convolution and tensor sketch operations,[16] A popular Natural Language Processing models, and hypergraph models of similarity,[20]
Generalized linear array model in statistics [19] Two- and multidimensional P-spline approximation of data,[18]
Studies of genotype x environment interactions.[21] See also [ ] Notes [ ]
^ Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions" . Sankhya 30 : 167–180. Archived from the original (PDF) on 2010-10-23. Retrieved 2008-08-21 . ^ Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes , 2 : 117–124 ^ See e.g. H. D. Macedo and J.N. Oliveira. A linear algebra approach to OLAP . Formal Aspects of Computing, 27(2):283–307, 2015.
^ Lev-Ari, Hanoch (2005-01-01). "Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing" . Communications in Information & Systems . 05 (1): 123–130. doi :10.4310/CIS.2005.v5.n1.a5 ISSN 1526-7555 . ^ a b Masiero, B.; Nascimento, V. H. (2017-05-01). "Revisiting the Kronecker Array Transform" . IEEE Signal Processing Letters . 24 (5): 525–529. Bibcode :2017ISPL...24..525M . doi :10.1109/LSP.2017.2674969 . ISSN 1070-9908 . S2CID 14166014 . ^ Vanderveen, M. C., Ng, B. C., Papadias, C. B., & Paulraj, A. (n.d.). Joint angle and delay estimation (JADE) for signals in multipath environments . Conference Record of The Thirtieth Asilomar Conference on Signals, Systems and Computers. – DOI:10.1109/acssc.1996.599145
^ a b c d e f g h Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications" (PDF) . Radioelectronics and Communications Systems . 41 (3): 50–53. ^ Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods , 38:19, p. 3501 [1]
^ a b c d e Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products" (PDF) . Proc. ICATT-97, Kyiv : 108–109. ^ a b c d e f g h Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars" (PDF) . Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv. : 73–74. ^ a b c d e f g h Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties" (PDF) . Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999 . 35 (3): 379–384. doi :10.1007/BF02733426 . S2CID 119661450 . ^ Slyusar, V. I. (2003). "Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels" (PDF) . Radioelectronics and Communications Systems . 46 (10): 9–17. ^ a b c d e Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1
^ a b C. Radhakrishna Rao . Estimation of Heteroscedastic Variances in Linear Models.//Journal of the American Statistical Association, Vol. 65, No. 329 (Mar., 1970), pp. 161–172^ Kasiviswanathan, Shiva Prasad, et al. «The price of privately releasing contingency tables and the spectra of random matrices with correlated rows.» Proceedings of the forty-second ACM symposium on Theory of computing. 2010.
^ a b c d Thomas D. Ahle, Jakob Bæk Tejs Knudsen. Almost Optimal Tensor Sketch. Published 2019. Mathematics, Computer Science, ArXiv
^ Ninh, Pham; Pagh, Rasmus (2013). Fast and scalable polynomial kernels via explicit feature maps . SIGKDD international conference on Knowledge discovery and data mining. Association for Computing Machinery. doi :10.1145/2487575.2487591 . ^ a b Eilers, Paul H.C.; Marx, Brian D. (2003). "Multivariate calibration with temperature interaction using two-dimensional penalized signal regression". Chemometrics and Intelligent Laboratory Systems . 66 (2): 159–174. doi :10.1016/S0169-7439(03)00029-7 . ^ a b c Currie, I. D.; Durban, M.; Eilers, P. H. C. (2006). "Generalized linear array models with applications to multidimensional smoothing". Journal of the Royal Statistical Society 68 (2): 259–280. doi :10.1111/j.1467-9868.2006.00543.x . ^ Bryan Bischof. Higher order co-occurrence tensors for hypergraphs via face-splitting. Published 15 February 2020, Mathematics, Computer Science, ArXiv
^ Johannes W. R. Martini, Jose Crossa, Fernando H. Toledo, Jaime Cuevas. On Hadamard and Kronecker products in covariance structures for genotype x environment interaction.//Plant Genome. 2020;13:e20033. Page 5. [2]
References [ ]
![\mathbf{A} =
\left[
\begin{array} {c | c}
\mathbf{A}_{11} & \mathbf{A}_{12} \\
\hline
\mathbf{A}_{21} & \mathbf{A}_{22}
\end{array}
\right]
=
\left[
\begin{array} {c c | c}
1 & 2 & 3 \\
4 & 5 & 6 \\
\hline
7 & 8 & 9
\end{array}
\right]
,\quad
\mathbf{B} =
\left[
\begin{array} {c | c}
\mathbf{B}_{11} & \mathbf{B}_{12} \\
\hline
\mathbf{B}_{21} & \mathbf{B}_{22}
\end{array}
\right]
=
\left[
\begin{array} {c | c c}
1 & 4 & 7 \\
\hline
2 & 5 & 8 \\
3 & 6 & 9
\end{array}
\right]
,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53cde028551100b310943f25b2028a764436687)
![\mathbf{A} \ast \mathbf{B} =
\left[
\begin{array} {c | c}
\mathbf{A}_{11} \otimes \mathbf{B}_{11} & \mathbf{A}_{12} \otimes \mathbf{B}_{12} \\
\hline
\mathbf{A}_{21} \otimes \mathbf{B}_{21} & \mathbf{A}_{22} \otimes \mathbf{B}_{22}
\end{array}
\right]
=
\left[
\begin{array} {c c | c c}
1 & 2 & 12 & 21 \\
4 & 5 & 24 & 42 \\
\hline
14 & 16 & 45 & 72 \\
21 & 24 & 54 & 81
\end{array}
\right].](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a086acf8812fc6ff3dec00cf40423b150b9942)
![\mathbf{C} =
\left[
\begin{array} { c | c | c}
\mathbf{C}_1 & \mathbf{C}_2 & \mathbf{C}_3
\end{array}
\right]
=
\left[
\begin{array} {c | c | c}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}
\right]
,\quad
\mathbf{D} =
\left[
\begin{array} { c | c | c }
\mathbf{D}_1 & \mathbf{D}_2 & \mathbf{D}_3
\end{array}
\right]
=
\left[
\begin{array} { c | c | c }
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9
\end{array}
\right]
,](https://wikimedia.org/api/rest_v1/media/math/render/svg/311fb96a2459096ea05d8f0461e67a8b49f5ee43)
![\mathbf{C} \ast \mathbf{D}
=
\left[
\begin{array} { c | c | c }
\mathbf{C}_1 \otimes \mathbf{D}_1 & \mathbf{C}_2 \otimes \mathbf{D}_2 & \mathbf{C}_3 \otimes \mathbf{D}_3
\end{array}
\right]
=
\left[
\begin{array} { c | c | c }
1 & 8 & 21 \\
2 & 10 & 24 \\
3 & 12 & 27 \\
4 & 20 & 42 \\
8 & 25 & 48 \\
12 & 30 & 54 \\
7 & 32 & 63 \\
14 & 40 & 72 \\
21 & 48 & 81
\end{array}
\right].](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e951f306d0dd52a9a56a35d767f2117db8a5ee6)
![{\displaystyle (\mathbf {A} \otimes \mathbf {B} )\ast (\mathbf {C} \otimes \mathbf {D} )=\mathbf {P} [(\mathbf {A} \ast \mathbf {C} )\otimes (\mathbf {B} \ast \mathbf {D} )]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0be3bf1fc6226631d8af11b6bb868812171113)
![{\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ef70c0e73175eb22cda75ecb898d7779fb0441)
![{\displaystyle [\circ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae862b2ddef9fd1f7eb59ccec0f5f8ce6d30ae1)
![{\displaystyle {\begin{aligned}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\right)\left({\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}\otimes {\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}\right)\left({\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\ast {\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]{}={}&\left({\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}\bullet {\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}\right)\left({\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\otimes \,{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}\right)\\[5pt]{}={}&{\begin{bmatrix}1&0\\0&1\\1&0\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\,\circ \,{\begin{bmatrix}1&0\\1&0\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}\rho _{1}&0\\0&\rho _{2}\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/077e2f2bb61c0f8393874c7f0ceb5f1bef78355c)
![{\displaystyle E\left[(T_{1}x)^{2}\right]=\left\|x\right\|_{2}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45663822e2a8c1b8b9b87a1a530b573089063f38)
![{\displaystyle E\left[(T_{1}x)^{p}\right]^{\frac {1}{p}}\leq {\sqrt {ap}}\|x\|_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/443c2d06bae7487f43814c4526a92bb3282cf0e8)
![{\displaystyle \mathbf {A} =\left[{\begin{array}{c | c}\mathbf {A} _{11}&\mathbf {A} _{12}\\\hline \mathbf {A} _{21}&\mathbf {A} _{22}\end{array}}\right],\quad \mathbf {B} =\left[{\begin{array}{c | c}\mathbf {B} _{11}&\mathbf {B} _{12}\\\hline \mathbf {B} _{21}&\mathbf {B} _{22}\end{array}}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86b027efdf1b4ea7866bdc28bc7ac1b3809137bf)
![{\displaystyle \mathbf {A} [\bullet ]\mathbf {B} =\left[{\begin{array}{c | c}\mathbf {A} _{11}\bullet \mathbf {B} _{11}&\mathbf {A} _{12}\bullet \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\bullet \mathbf {B} _{21}&\mathbf {A} _{22}\bullet \mathbf {B} _{22}\end{array}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5d6918b358808c418ec16679a5f86b9d93d4aa)
![{\displaystyle \mathbf {A} [\ast ]\mathbf {B} =\left[{\begin{array}{c | c}\mathbf {A} _{11}\ast \mathbf {B} _{11}&\mathbf {A} _{12}\ast \mathbf {B} _{12}\\\hline \mathbf {A} _{21}\ast \mathbf {B} _{21}&\mathbf {A} _{22}\ast \mathbf {B} _{22}\end{array}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1177613d963926bd03de425c42177de8661e5f55)
![{\displaystyle \left(\mathbf {A} [\ast ]\mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}[\bullet ]\mathbf {B} ^{\textsf {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907ec4c862586658a29c538a04d6f973e129a19c)
