Khatri–Rao product

Short description: Type of product of matrices

In mathematics, the Khatri–Rao product of matrices is defined as^[1]^[2]^[3]

[math]\displaystyle{ \mathbf{A} \ast \mathbf{B} = \left(\mathbf{A}_{ij} \otimes \mathbf{B}_{ij}\right)_{ij} }[/math]

in which the ij-th block is the m_ip_i × n_jq_j sized 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 (Σ_i m_ip_i) × (Σ_j n_jq_j).

For example, if A and B both are 2 × 2 partitioned matrices e.g.:

[math]\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] , }[/math]

we obtain:

[math]\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]. }[/math]

This is a submatrix of the Tracy–Singh product ^[4] 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 m₁ = m, p₁ = p, n = q and for each j: n_j = p_j = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:

[math]\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] , }[/math]

so that:

[math]\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]. }[/math]

This column-wise version of the Khatri–Rao product is useful in linear algebra approaches to data analytical processing^[5] and in optimizing the solution of inverse problems dealing with a diagonal matrix.^[6]^[7]

In 1996 the Column-wise Khatri–Rao product was proposed to estimate the angles of arrival (AOAs) and delays of multipath signals^[8] and four coordinates of signals sources^[9] at a 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^[10] in 1996.^[9]^[11]^[12]^[13]^[14]

This matrix operation was named the "face-splitting product" of matrices^[11]^[13] or the "transposed Khatri–Rao product". This type of operation is based on row-by-row Kronecker products of two matrices. Using the matrices from the previous examples with the rows partitioned:

[math]\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} , }[/math]

the result can be obtained:^[9]^[11]^[13]

[math]\displaystyle{ \mathbf{C} \bull \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}. }[/math]

Main properties

Transpose (V. Slyusar, 1996^[9]^[11]^[12]):
[math]\displaystyle{ \left(\mathbf{A} \bull \mathbf{B}\right)^\textsf{T} = \textbf{A}^\textsf{T} \ast \mathbf{B}^\textsf{T} }[/math],
Bilinearity and associativity:^[9]^[11]^[12]

[math]\displaystyle{ \begin{align} \mathbf{A} \bull (\mathbf{B} + \mathbf{C}) &= \mathbf{A} \bull \mathbf{B} + \mathbf{A} \bull \mathbf{C}, \\ (\mathbf{B} + \mathbf{C}) \bull \mathbf{A} &= \mathbf{B} \bull \mathbf{A} + \mathbf{C} \bull \mathbf{A}, \\ (k\mathbf{A}) \bull \mathbf{B} &= \mathbf{A} \bull (k\mathbf{B}) = k(\mathbf{A} \bull \mathbf{B}), \\ (\mathbf{A} \bull \mathbf{B}) \bull \mathbf{C} &= \mathbf{A} \bull (\mathbf{B} \bull \mathbf{C}), \\ \end{align} }[/math]

where A, B and C are matrices, and k is a scalar,

[math]\displaystyle{ a \bull \mathbf{B} = \mathbf{B} \bull a }[/math],^[12]
where [math]\displaystyle{ a }[/math] is a vector,
The mixed-product property (V. Slyusar, 1997^[12]):
[math]\displaystyle{ (\mathbf{A} \bull \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) }[/math],

[math]\displaystyle{ (\mathbf{A} \bull \mathbf{B})(\mathbf{C} \ast \mathbf{D}) = (\mathbf{A}\mathbf{C}) \circ (\mathbf{B} \mathbf{D}) }[/math],^[13]

[math]\displaystyle{ (\mathbf{A} \bull \mathbf{B} \bull \mathbf{C} \bull \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}) }[/math]^[15]

[math]\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) }[/math],^[16]
where [math]\displaystyle{ \circ }[/math] denotes the Hadamard product,
[math]\displaystyle{ (\mathbf{A} \circ \mathbf{B}) \bull (\mathbf{C} \circ \mathbf{D}) = (\mathbf{A} \bull \mathbf{C}) \circ (\mathbf{B} \bull \mathbf{D}) }[/math],^[12]
[math]\displaystyle{ \mathbf{A} \otimes (\mathbf{B} \bull \mathbf{C}) = (\mathbf{A} \otimes \mathbf{B}) \bull \mathbf{C} }[/math],^[9]
[math]\displaystyle{ (\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \ast \mathbf{D}) = (\mathbf{A}\mathbf{C}) \ast (\mathbf{B} \mathbf{D}) }[/math],^[16]
[math]\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})] }[/math], where [math]\displaystyle{ \mathbf{P} }[/math] is a permutation matrix.^[7]
[math]\displaystyle{ (\mathbf{A} \bull \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{A}\mathbf{C}) \bull (\mathbf{B} \mathbf{D}) }[/math],^[13]^[15]
Similarly:
[math]\displaystyle{ (\mathbf{A} \bull \mathbf{L})(\mathbf{B} \otimes \mathbf{M}) \cdots (\mathbf{C} \otimes \mathbf{S}) = (\mathbf{A}\mathbf{B}\cdots\mathbf{C}) \bull (\mathbf{L}\mathbf{M}\cdots\mathbf{S}) }[/math],
[math]\displaystyle{ c^\textsf{T} \bull d^\textsf{T} = c^\textsf{T} \otimes d^\textsf{T} }[/math],^[12]

[math]\displaystyle{ c \ast d = c \otimes d }[/math],
where [math]\displaystyle{ c }[/math] and [math]\displaystyle{ d }[/math] are vectors,
[math]\displaystyle{ \left(\mathbf{A} \ast c^\textsf{T}\right)d = \left(\mathbf{A} \ast d^\textsf{T}\right)c }[/math],^[17] [math]\displaystyle{ d^\textsf{T}\left(c \bull \mathbf{A}^\textsf{T}\right) = c^\textsf{T}\left(d \bull \mathbf{A}^\textsf{T}\right) }[/math],
[math]\displaystyle{ (\mathbf{A} \bull \mathbf{B})(c \otimes d) = (\mathbf{A}c) \circ (\mathbf{B}d) }[/math],^[18]
where [math]\displaystyle{ c }[/math] and [math]\displaystyle{ d }[/math] are vectors (it is a combine of properties 3 an 8), Similarly:
[math]\displaystyle{ (\mathbf{A} \bull \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), }[/math]
[math]\displaystyle{ \mathcal F\left(C^{(1)}x \star C^{(2)}y\right) = \left(\mathcal F C^{(1)} \bull \mathcal F C^{(2)}\right)(x \otimes y) = \mathcal F C^{(1)}x \circ \mathcal F C^{(2)}y }[/math],
where [math]\displaystyle{ \star }[/math] is vector convolution and [math]\displaystyle{ \mathcal F }[/math] is the Fourier transform matrix (this result is an evolving of count sketch properties^[19]),
[math]\displaystyle{ \mathbf{A} \bull \mathbf{B} = \left(\mathbf {A} \otimes \mathbf {1_c}^\textsf{T}\right) \circ \left(\mathbf {1_k}^\textsf{T} \otimes \mathbf {B}\right) }[/math],^[20]
where [math]\displaystyle{ \mathbf {A} }[/math] is [math]\displaystyle{ r \times c }[/math] matrix, [math]\displaystyle{ \mathbf{B} }[/math] is [math]\displaystyle{ r \times k }[/math] matrix, [math]\displaystyle{ \mathbf{1_c} }[/math] is a vector of 1's of length [math]\displaystyle{ c }[/math], and [math]\displaystyle{ \mathbf{1_k} }[/math] is a vector of 1's of length [math]\displaystyle{ k }[/math] or
[math]\displaystyle{ \mathbf{M} \bull \mathbf{M} = \left(\mathbf{M} \otimes \mathbf{1}^\textsf{T}\right) \circ \left(\mathbf{1}^\textsf{T} \otimes \mathbf{M}\right) }[/math],^[21]
where [math]\displaystyle{ \mathbf{M} }[/math] is [math]\displaystyle{ r \times c }[/math] matrix, [math]\displaystyle{ \circ }[/math] means element by element multiplication and [math]\displaystyle{ \mathbf{1} }[/math] is a vector of 1's of length [math]\displaystyle{ c }[/math].
[math]\displaystyle{ \mathbf{M} \bull \mathbf{M} = \mathbf{M} [\circ] \left(\mathbf{M} \otimes \mathbf{1}^\textsf{T}\right) }[/math],
where [math]\displaystyle{ [\circ] }[/math] denotes the penetrating face product of matrices.^[13] Similarly:
[math]\displaystyle{ \mathbf{P} \ast \mathbf{N} = (\mathbf{P} \otimes \mathbf{1_c}) \circ (\mathbf {1_k} \otimes \mathbf {N}) }[/math], where [math]\displaystyle{ \mathbf {P} }[/math] is [math]\displaystyle{ c \times r }[/math] matrix, [math]\displaystyle{ \mathbf {N} }[/math] is [math]\displaystyle{ k \times r }[/math] matrix,.
[math]\displaystyle{ \mathbf{W_d}\mathbf{A} = \mathbf{w} \bull \mathbf{A} }[/math],^[12]

[math]\displaystyle{ vec((\mathbf{w}^\textsf{T} \ast \mathbf{A})\mathbf{B})= (\mathbf{B}^\textsf{T} \ast \mathbf{A}) \mathbf{w} }[/math]^[13]= [math]\displaystyle{ vec(\mathbf{A}(\mathbf{w} \bull \mathbf{B})) }[/math],

[math]\displaystyle{ \operatorname{vec}\left(\mathbf{A}^\textsf{T} \mathbf{W_d} \mathbf{A}\right) = \left(\mathbf{A} \bull \mathbf{A}\right)^\textsf{T} \mathbf{w} }[/math],^[21]
where [math]\displaystyle{ \mathbf{w} }[/math] is the vector consisting of the diagonal elements of [math]\displaystyle{ \mathbf{W_d} }[/math], [math]\displaystyle{ \operatorname{vec}(\mathbf{A}) }[/math] means stack the columns of a matrix [math]\displaystyle{ \mathbf{A} }[/math] on top of each other to give a vector.
[math]\displaystyle{ (\mathbf{A} \bull \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}) }[/math].^[13]^[15]
Similarly:
[math]\displaystyle{ \begin{align} (\mathbf{A} \bull \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} \bull \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{align} }[/math],
where [math]\displaystyle{ c }[/math] and [math]\displaystyle{ d }[/math] are vectors

Examples^[18]

[math]\displaystyle{ \begin{align} &\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{align} }[/math]

Theorem^[18]

If [math]\displaystyle{ M = T^{(1)} \bullet \dots \bullet T^{(c)} }[/math], where [math]\displaystyle{ T^{(1)}, \dots, T^{(c)} }[/math] are independent components a random matrix [math]\displaystyle{ T }[/math] with independent identically distributed rows [math]\displaystyle{ T_1, \dots, T_m\in \mathbb R^d }[/math], such that

[math]\displaystyle{ E\left[(T_1x)^2\right] = \left\|x\right\|_2^2 }[/math] and [math]\displaystyle{ E\left[(T_1 x)^p\right]^\frac{1}{p} \le \sqrt{ap}\|x\|_2 }[/math],

then for any vector [math]\displaystyle{ x }[/math]

[math]\displaystyle{ \left| \left\|Mx\right\|_2 - \left\|x\right\|_2 \right| \lt \varepsilon \left\|x\right\|_2 }[/math]

with probability [math]\displaystyle{ 1 - \delta }[/math] if the quantity of rows

[math]\displaystyle{ m = (4a)^{2c} \varepsilon^{-2} \log 1/\delta + (2ae)\varepsilon^{-1}(\log 1/\delta)^c. }[/math]

In particular, if the entries of [math]\displaystyle{ T }[/math] are [math]\displaystyle{ \pm 1 }[/math] can get

[math]\displaystyle{ m = O\left(\varepsilon^{-2}\log1/\delta + \varepsilon^{-1}\left(\frac{1}{c}\log1/\delta\right)^c\right) }[/math]

which matches the Johnson–Lindenstrauss lemma of [math]\displaystyle{ m = O\left(\varepsilon^{-2}\log1/\delta\right) }[/math] when [math]\displaystyle{ \varepsilon }[/math] is small.

Block face-splitting product

Transposed block face-splitting product in the context of a multi-face radar model^[15]

According to the definition of V. Slyusar^[9]^[13] the block face-splitting product of two partitioned matrices with a given quantity of rows in blocks

[math]\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] , }[/math]

can be written as :

[math]\displaystyle{ \mathbf{A} [\bull] \mathbf{B} = \left[ \begin{array} {c | c} \mathbf{A}_{11} \bull \mathbf{B}_{11} & \mathbf{A}_{12} \bull \mathbf{B}_{12} \\ \hline \mathbf{A}_{21} \bull \mathbf{B}_{21} & \mathbf{A}_{22} \bull \mathbf{B}_{22} \end{array} \right] . }[/math]

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:^[9]^[13]

[math]\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]. }[/math]

Main properties

Transpose:
[math]\displaystyle{ \left(\mathbf{A} [\ast] \mathbf{B} \right)^\textsf{T} = \textbf{A}^\textsf{T} [\bull] \mathbf{B}^\textsf{T} }[/math]^[15]

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,^[18]
A popular Natural Language Processing models, and hypergraph models of similarity,^[22]
Generalized linear array model in statistics^[21]
Two- and multidimensional P-spline approximation of data,^[20]
Studies of genotype x environment interactions.^[23]

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. http://library.isical.ac.in:8080/xmlui/bitstream/handle/10263/614/68.E02.pdf. Retrieved 2008-08-21.
↑ Liu, Shuangzhe (1999). "Matrix Results on the Khatri–Rao and Tracy–Singh Products". Linear Algebra and Its Applications 289 (1–3): 267–277. doi:10.1016/S0024-3795(98)10209-4.
↑ Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes 2: 117–124
↑ Liu, Shuangzhe; Trenkler, Götz (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". International Journal of Information and Systems Sciences 4 (1): 160–177.
↑ 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" (in EN). Communications in Information & Systems 05 (1): 123–130. doi:10.4310/CIS.2005.v5.n1.a5. ISSN 1526-7555. http://www.ims.cuhk.edu.hk/~cis/2005.1/05.pdf.
↑ ^7.0 ^7.1 Masiero, B.; Nascimento, V. H. (2017-05-01). "Revisiting the Kronecker Array Transform". IEEE Signal Processing Letters 24 (5): 525–529. doi:10.1109/LSP.2017.2674969. ISSN 1070-9908. Bibcode: 2017ISPL...24..525M. https://zenodo.org/record/896497.
↑ 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
↑ ^9.0 ^9.1 ^9.2 ^9.3 ^9.4 ^9.5 ^9.6 ^9.7 Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications.". Radioelectronics and Communications Systems 41 (3): 50–53. http://slyusar.kiev.ua/en/IZV_1998_3.pdf.
↑ Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods, 38:19, p. 3501 [1]
↑ ^11.0 ^11.1 ^11.2 ^11.3 ^11.4 Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products.". Proc. ICATT-97, Kyiv: 108–109. http://slyusar.kiev.ua/ICATT97.pdf.
↑ ^12.0 ^12.1 ^12.2 ^12.3 ^12.4 ^12.5 ^12.6 ^12.7 Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars". Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74. http://slyusar.kiev.ua/DIPED_1997.pdf.
↑ ^13.0 ^13.1 ^13.2 ^13.3 ^13.4 ^13.5 ^13.6 ^13.7 ^13.8 ^13.9 Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties". Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999. 35 (3): 379–384. doi:10.1007/BF02733426. http://slyusar.kiev.ua/FACE.pdf.
↑ Slyusar, V. I. (2003). "Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels". Radioelectronics and Communications Systems 46 (10): 9–17. http://slyusar.kiev.ua/en/IZV_2003_10.pdf.
↑ ^15.0 ^15.1 ^15.2 ^15.3 ^15.4 Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1
↑ ^16.0 ^16.1 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.
↑ ^18.0 ^18.1 ^18.2 ^18.3 Thomas D. Ahle, Jakob Bæk Tejs Knudsen. Almost Optimal Tensor Sketch. Published 2019. Mathematics, Computer Science, ArXiv
↑ Ninh, Pham (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.
↑ ^20.0 ^20.1 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.
↑ ^21.0 ^21.1 ^21.2 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

Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions". Sankhya 30: 167–180. http://sankhya.isical.ac.in/search/30a2/30a2019.html. Retrieved 2008-08-21.
Rao C.R.; Rao M. Bhaskara (1998), Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, pp. 216
Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes 2: 117–124
Liu Shuangzhe; Trenkler Götz (2008), "Hadamard, Khatri-Rao, Kronecker and other matrix products", International Journal of Information and Systems Sciences 4: 160–177

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Original source: https://en.wikipedia.org/wiki/Khatri–Rao product. Read more

[1] Khatri C. G., C. R. Rao (1968). "Solutions to some functional equations and their applications to characterization of probability distributions". Sankhya 30: 167–180. http://library.isical.ac.in:8080/xmlui/bitstream/handle/10263/614/68.E02.pdf. Retrieved 2008-08-21.

[2] Liu, Shuangzhe (1999). "Matrix Results on the Khatri–Rao and Tracy–Singh Products". Linear Algebra and Its Applications 289 (1–3): 267–277. doi:10.1016/S0024-3795(98)10209-4.

[3] Zhang X; Yang Z; Cao C. (2002), "Inequalities involving Khatri–Rao products of positive semi-definite matrices", Applied Mathematics E-notes 2: 117–124

[4] Liu, Shuangzhe; Trenkler, Götz (2008). "Hadamard, Khatri-Rao, Kronecker and other matrix products". International Journal of Information and Systems Sciences 4 (1): 160–177.

[5] 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.

[6] Lev-Ari, Hanoch (2005-01-01). "Efficient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing" (in EN). Communications in Information & Systems 05 (1): 123–130. doi:10.4310/CIS.2005.v5.n1.a5. ISSN 1526-7555. http://www.ims.cuhk.edu.hk/~cis/2005.1/05.pdf.

[Masiero-7] 7.0 ^7.1 Masiero, B.; Nascimento, V. H. (2017-05-01). "Revisiting the Kronecker Array Transform". IEEE Signal Processing Letters 24 (5): 525–529. doi:10.1109/LSP.2017.2674969. ISSN 1070-9908. Bibcode: 2017ISPL...24..525M. https://zenodo.org/record/896497.

[8] 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

[slyusar-9] 9.0 ^9.1 ^9.2 ^9.3 ^9.4 ^9.5 ^9.6 ^9.7 Slyusar, V. I. (December 27, 1996). "End products in matrices in radar applications.". Radioelectronics and Communications Systems 41 (3): 50–53. http://slyusar.kiev.ua/en/IZV_1998_3.pdf.

[Fortiana-10] Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods, 38:19, p. 3501 [1]

[slyusar1-11] 11.0 ^11.1 ^11.2 ^11.3 ^11.4 Slyusar, V. I. (1997-05-20). "Analytical model of the digital antenna array on a basis of face-splitting matrix products.". Proc. ICATT-97, Kyiv: 108–109. http://slyusar.kiev.ua/ICATT97.pdf.

[DIPED-12] 12.0 ^12.1 ^12.2 ^12.3 ^12.4 ^12.5 ^12.6 ^12.7 Slyusar, V. I. (1997-09-15). "New operations of matrices product for applications of radars". Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.: 73–74. http://slyusar.kiev.ua/DIPED_1997.pdf.

[slyusar2-13] 13.0 ^13.1 ^13.2 ^13.3 ^13.4 ^13.5 ^13.6 ^13.7 ^13.8 ^13.9 Slyusar, V. I. (March 13, 1998). "A Family of Face Products of Matrices and its Properties". Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz. 1999. 35 (3): 379–384. doi:10.1007/BF02733426. http://slyusar.kiev.ua/FACE.pdf.

[14] Slyusar, V. I. (2003). "Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels". Radioelectronics and Communications Systems 46 (10): 9–17. http://slyusar.kiev.ua/en/IZV_2003_10.pdf.

[lecture-15] 15.0 ^15.1 ^15.2 ^15.3 ^15.4 Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1

[Rao-16] 16.0 ^16.1 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

[k2010-17] 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.

[tensorsketch-18] 18.0 ^18.1 ^18.2 ^18.3 Thomas D. Ahle, Jakob Bæk Tejs Knudsen. Almost Optimal Tensor Sketch. Published 2019. Mathematics, Computer Science, ArXiv

[ninh-19] Ninh, Pham (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.

[spline-20] 20.0 ^20.1 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.

[GLAM-21] 21.0 ^21.1 ^21.2 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.

[22] Bryan Bischof. Higher order co-occurrence tensors for hypergraphs via face-splitting. Published 15 February 2020, Mathematics, Computer Science, ArXiv

[23] 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]

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Column-wise Kronecker product

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