In mathematics, the Khatri–Rao product or block Kronecker product of two
partitioned matrices and is defined as[1][2][3]
in which the ij-th block is the mipi × njqj 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 (Σi mipi) × (Σj njqj).
For example, if A and B both are 2 × 2 partitioned matrices e.g.:
The column-wise
Kronecker product of two matrices is a special case of the Khatri-Rao product as defined above, and 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: nj = pj = 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:
so that:
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
An 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:
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
can be written as :
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]
The Face-splitting product and the Block Face-splitting product used in the
tensor-matrix theory of
digital antenna arrays. These operations are also used in:
^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.
^Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods, 38:19, p. 3501
[1]
^
abC. 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.
^
abcdThomas 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.
^
abEilers, 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.
^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]
Rao C.R.; Rao M. Bhaskara (1998), Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, p. 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
In mathematics, the Khatri–Rao product or block Kronecker product of two
partitioned matrices and is defined as[1][2][3]
in which the ij-th block is the mipi × njqj 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 (Σi mipi) × (Σj njqj).
For example, if A and B both are 2 × 2 partitioned matrices e.g.:
The column-wise
Kronecker product of two matrices is a special case of the Khatri-Rao product as defined above, and 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: nj = pj = 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:
so that:
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
An 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:
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
can be written as :
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]
The Face-splitting product and the Block Face-splitting product used in the
tensor-matrix theory of
digital antenna arrays. These operations are also used in:
^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.
^Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods, 38:19, p. 3501
[1]
^
abC. 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.
^
abcdThomas 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.
^
abEilers, 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.
^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]
Rao C.R.; Rao M. Bhaskara (1998), Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, p. 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