Produs Khatri–Rao

Produsul cu divizarea feței

Principalele proprietăți

1. Transpusa (V. Sliusar, 1996^[6][7][8]):

   ${\displaystyle \left(\mathbf {A} \bullet \mathbf {B} \right)^{\textsf {T}}={\textbf {A}}^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}}$,

2. Biliniaritate și asociativitate:^[6][7][8]

   ${\displaystyle {\begin{aligned}\mathbf {A} \bullet (\mathbf {B} +\mathbf {C} )&=\mathbf {A} \bullet \mathbf {B} +\mathbf {A} \bullet \mathbf {C} ,\\(\mathbf {B} +\mathbf {C} )\bullet \mathbf {A} &=\mathbf {B} \bullet \mathbf {A} +\mathbf {C} \bullet \mathbf {A} ,\\(k\mathbf {A} )\bullet \mathbf {B} &=\mathbf {A} \bullet (k\mathbf {B} )=k(\mathbf {A} \bullet \mathbf {B} ),\\(\mathbf {A} \bullet \mathbf {B} )\bullet \mathbf {C} &=\mathbf {A} \bullet (\mathbf {B} \bullet \mathbf {C} ),\\\end{aligned}}}$

   unde A, B și C sunt matrici, iar k este un scalar,

   ${\displaystyle a\bullet \mathbf {B} =\mathbf {B} \bullet a}$,^[8]

   unde ${\displaystyle a}$ este un vector,
3. Proprietatea produsului mixt (V. Sliusar, 1997^[8]):

   ${\displaystyle (\mathbf {A} \bullet \mathbf {B} )\left(\mathbf {A} ^{\textsf {T}}\ast \mathbf {B} ^{\textsf {T}}\right)=\left(\mathbf {A} \mathbf {A} ^{\textsf {T}}\right)\circ \left(\mathbf {B} \mathbf {B} ^{\textsf {T}}\right)}$,

   ${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\circ (\mathbf {B} \mathbf {D} )}$,^[9]

   ${\displaystyle (\mathbf {A} \bullet \mathbf {B} \bullet \mathbf {C} \bullet \mathbf {D} )(\mathbf {L} \ast \mathbf {M} \ast \mathbf {N} \ast \mathbf {P} )=(\mathbf {A} \mathbf {L} )\circ (\mathbf {B} \mathbf {M} )\circ (\mathbf {C} \mathbf {N} )\circ (\mathbf {D} \mathbf {P} )}$^[11]

   ${\displaystyle (\mathbf {A} \ast \mathbf {B} )^{\textsf {T}}(\mathbf {A} \ast \mathbf {B} )=\left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)\circ \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)}$,^[12]

   unde ${\displaystyle \circ }$ indică produsul Hadamard,
4. ${\displaystyle (\mathbf {A} \circ \mathbf {B} )\bullet (\mathbf {C} \circ \mathbf {D} )=(\mathbf {A} \bullet \mathbf {C} )\circ (\mathbf {B} \bullet \mathbf {D} )}$,^[8]
5. ${\displaystyle \mathbf {A} \otimes (\mathbf {B} \bullet \mathbf {C} )=(\mathbf {A} \otimes \mathbf {B} )\bullet \mathbf {C} }$,^[6]
6. ${\displaystyle (\mathbf {A} \otimes \mathbf {B} )(\mathbf {C} \ast \mathbf {D} )=(\mathbf {A} \mathbf {C} )\ast (\mathbf {B} \mathbf {D} )}$,^[12]
7. ${\displaystyle (\mathbf {A} \otimes \mathbf {B} )\ast (\mathbf {C} \otimes \mathbf {D} )=\mathbf {P} [(\mathbf {A} \ast \mathbf {C} )\otimes (\mathbf {B} \ast \mathbf {D} )]}$, unde ${\displaystyle \mathbf {P} }$ este matricea de permutări.^[13]
8.  

   ${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=(\mathbf {A} \mathbf {C} )\bullet (\mathbf {B} \mathbf {D} )}$,^[9][11]

   Similar:

   ${\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} )\bullet (\mathbf {L} \mathbf {M} \cdots \mathbf {S} )}$,

9.  
   ${\displaystyle c^{\textsf {T}}\bullet d^{\textsf {T}}=c^{\textsf {T}}\otimes d^{\textsf {T}}}$,^[8]
   ${\displaystyle c\ast d=c\otimes d}$,
   unde ${\displaystyle c}$ și ${\displaystyle d}$ sunt vectori,
10. ${\displaystyle \left(\mathbf {A} \ast c^{\textsf {T}}\right)d=\left(\mathbf {A} \ast d^{\textsf {T}}\right)c}$,^[14] ${\displaystyle d^{\textsf {T}}\left(c\bullet \mathbf {A} ^{\textsf {T}}\right)=c^{\textsf {T}}\left(d\bullet \mathbf {A} ^{\textsf {T}}\right)}$,
11.  

    ${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(c\otimes d)=(\mathbf {A} c)\circ (\mathbf {B} d)}$,^[15]

    unde ${\displaystyle c}$ și ${\displaystyle d}$ sunt vectori (este o combinare a proprietăților 3 și 8). Similar:

    ${\displaystyle (\mathbf {A} \bullet \mathbf {B} )(\mathbf {M} \mathbf {N} c\otimes \mathbf {Q} \mathbf {P} d)=(\mathbf {A} \mathbf {M} \mathbf {N} c)\circ (\mathbf {B} \mathbf {Q} \mathbf {P} d),}$

12.  
    ${\displaystyle {\mathcal {F}}\left(C^{(1)}x\star C^{(2)}y\right)=\left({\mathcal {F}}C^{(1)}\bullet {\mathcal {F}}C^{(2)}\right)(x\otimes y)={\mathcal {F}}C^{(1)}x\circ {\mathcal {F}}C^{(2)}y}$,
    unde ${\displaystyle \star }$ este convoluția vectorilor, iar ${\displaystyle {\mathcal {F}}}$ este matricea Fourier^⁠(d),
13.  

    ${\displaystyle \mathbf {A} \bullet \mathbf {B} =\left(\mathbf {A} \otimes \mathbf {1_{c}} ^{\textsf {T}}\right)\circ \left(\mathbf {1_{k}} ^{\textsf {T}}\otimes \mathbf {B} \right)}$,^[16]

    unde ${\displaystyle \mathbf {A} }$ este matricea ${\displaystyle r\times c}$, ${\displaystyle \mathbf {B} }$ este matricea${\displaystyle r\times k}$, ${\displaystyle \mathbf {1_{c}} }$ este vectorul cu toate elementele de lungime 1 ${\displaystyle c}$, iar ${\displaystyle \mathbf {1_{k}} }$ este vectorul cu toate elementele de lungime 1 ${\displaystyle k}$ sau

    ${\displaystyle \mathbf {M} \bullet \mathbf {M} =\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)\circ \left(\mathbf {1} ^{\textsf {T}}\otimes \mathbf {M} \right)}$,^[17]

    unde ${\displaystyle \mathbf {M} }$ este matricea ${\displaystyle r\times c}$, ${\displaystyle \circ }$ indică îmmulțirea pe elemente, iar ${\displaystyle \mathbf {1} }$ este vectorul cu toate elementele de lungime 1 ${\displaystyle c}$.

    ${\displaystyle \mathbf {M} \bullet \mathbf {M} =\mathbf {M} [\circ ]\left(\mathbf {M} \otimes \mathbf {1} ^{\textsf {T}}\right)}$,

    unde ${\displaystyle [\circ ]}$ indică produsul cu penetrarea feței al matricelor.^[9] Similar:

    ${\displaystyle \mathbf {P} \ast \mathbf {N} =(\mathbf {P} \otimes \mathbf {1_{c}} )\circ (\mathbf {1_{k}} \otimes \mathbf {N} )}$,

    unde ${\displaystyle \mathbf {P} }$ este matricea ${\displaystyle c\times r}$, iar ${\displaystyle \mathbf {N} }$ este matricea ${\displaystyle k\times r}$,
14.  

    ${\displaystyle \mathbf {W_{d}} \mathbf {A} =\mathbf {w} \bullet \mathbf {A} }$,^[8]

    ${\displaystyle vec((\mathbf {w} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {B} )=(\mathbf {B} ^{\textsf {T}}\ast \mathbf {A} )\mathbf {w} }$^[9]= ${\displaystyle vec(\mathbf {A} (\mathbf {w} \bullet \mathbf {B} ))}$,

    ${\displaystyle \operatorname {vec} \left(\mathbf {A} ^{\textsf {T}}\mathbf {W_{d}} \mathbf {A} \right)=\left(\mathbf {A} \bullet \mathbf {A} \right)^{\textsf {T}}\mathbf {w} }$,^[17]

    unde ${\displaystyle \mathbf {w} }$ este vectorul format din elementele de pe diagonala ${\displaystyle \mathbf {W_{d}} }$, ${\displaystyle \operatorname {vec} (\mathbf {A} )}$ este stivuirea coloanelor matricei ${\displaystyle \mathbf {A} }$ una peste alta pentru a forma un vector.
15.  

    ${\displaystyle (\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {K} \ast \mathbf {T} )=(\mathbf {A} \mathbf {B} ...\mathbf {C} \mathbf {K} )\circ (\mathbf {L} \mathbf {M} ...\mathbf {S} \mathbf {T} )}$.^[9][11]

    Similar:

    ${\displaystyle {\begin{aligned}(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(c\otimes d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} d),\\(\mathbf {A} \bullet \mathbf {L} )(\mathbf {B} \otimes \mathbf {M} )\cdots (\mathbf {C} \otimes \mathbf {S} )(\mathbf {P} c\otimes \mathbf {Q} d)&=(\mathbf {A} \mathbf {B} \cdots \mathbf {C} \mathbf {P} c)\circ (\mathbf {L} \mathbf {M} \cdots \mathbf {S} \mathbf {Q} d)\end{aligned}}}$,

    unde ${\displaystyle c}$ și ${\displaystyle d}$ sunt vectori.

Exemple^[15]

Aplicații

Note

1. ^ en Khatri, C. G.; Rao, C. R. (1968). „Solutions to some functional equations and their applications to characterization of probability distributions”. Sankhya. 30: 167–180. Arhivat din original (PDF) la 23 octombrie 2010. Accesat în 21 august 2008. 
2. ^ en 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. ^ en 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. ^ en 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. ^ en Anna Esteve, Eva Boj & Josep Fortiana (2009): "Interaction Terms in Distance-Based Regression," Communications in Statistics – Theory and Methods, 38:19, p. 3501 [1]
6. ^ ^{a b c d e} en Slyusar, V. I. (27 decembrie 1996). „End products in matrices in radar applications” (PDF). Radioelectronics and Communications Systems. 41 (3): 50–53. 
7. ^ ^{a b c d e} en Slyusar, V. I. (20 mai 1997). „Analytical model of the digital antenna array on a basis of face-splitting matrix products” (PDF). Proc. ICATT-97, Kyiv: 108–109. 
8. ^ ^{a b c d e f g h} en Slyusar, V. I. (15 septembrie 1997). „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. 
9. ^ ^{a b c d e f g h} en Slyusar, V. I. (13 martie 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. 
10. ^ en 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. 
11. ^ ^{a b c} en Vadym Slyusar. New Matrix Operations for DSP (Lecture). April 1999. – DOI: 10.13140/RG.2.2.31620.76164/1
12. ^ ^{a b} en C. R. Rao. Estimation of Heteroscedastic Variances in Linear Models.//Journal of the American Statistical Association, Vol. 65, No. 329 (Mar., 1970), pp. 161–172
13. ^ en Masiero, B.; Nascimento, V. H. (1 mai 2017). „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. 
14. ^ en 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.
15. ^ ^{a b c} en Thomas D. Ahle, Jakob Bæk Tejs Knudsen. Almost Optimal Tensor Sketch. Published 2019. Mathematics, Computer Science, ArXiv
16. ^ ^{a b} en 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. 
17. ^ ^{a b c} en 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. 
18. ^ en Bryan Bischof. Higher order co-occurrence tensors for hypergraphs via face-splitting. Published 15 February 2020, Mathematics, Computer Science, ArXiv
19. ^ en 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]

Bibliografie

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