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.:
we obtain:
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).
Column-wise Kronecker product
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 = qj = 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
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:
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:
^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.
^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" (PDF). Communications in Information & Systems. 05 (1): 123–130. doi:10.4310/CIS.2005.v5.n1.a5. ISSN 1526-7555.
^ a bMasiero, 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 hSlyusar, V. I. (December 27, 1996). "End matrix products in radar applications" (PDF). Izvestiya VUZ: Radioelektronika. 41 (3): 71–75.
^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 eSlyusar, 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 hSlyusar, 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 i jSlyusar, 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 eVadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1
^ a bC. 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 dThomas 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 bEilers, 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 cCurrie, 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. S2CID 10261944.
^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. Archived from the original on 2010-10-23. Retrieved 2008-08-21.
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