DelayedTensor 1.6.0
Authors: Koki Tsuyuzaki [aut, cre]
Last modified: 2023-05-01 13:34:10
Compiled: Tue May 9 00:22:17 2023
einsum
einsum
is an easy and intuitive way to write tensor operations.
It was originally introduced by
Numpy
1 https://numpy.org/doc/stable/reference/generated/numpy.einsum.html
package of Python but similar tools have been implemented in other languages
(e.g. R, Julia) inspired by Numpy
.
In this vignette, we will use CRAN einsum package first.
einsum
is named after
Einstein summation2 https://en.wikipedia.org/wiki/Einstein_notation
introduced by Albert Einstein,
which is a notational convention that implies summation over
a set of indexed terms in a formula.
Here, we consider a simple example of einsum
; matrix multiplication.
If we naively implement the matrix multiplication,
the calculation would look like the following in a for loop.
A <- matrix(runif(3*4), nrow=3, ncol=4)
B <- matrix(runif(4*5), nrow=4, ncol=5)
C <- matrix(0, nrow=3, ncol=5)
I <- nrow(A)
J <- ncol(A)
K <- ncol(B)
for(i in 1:I){
for(j in 1:J){
for(k in 1:K){
C[i,k] = C[i,k] + A[i,j] * B[j,k]
}
}
}
Therefore, any programming language can implement this. However, when analyzing tensor data, such operations tend to be more complicated and increase the possibility of causing bugs because the order of tensors is larger or more tensors are handled simultaneously. In addition, several programming languages, especially R, are known to significantly slow down the speed of computation if the code is written in for loop.
Obviously, in the case of the R language, it should be executed using the built-in matrix multiplication function (%*%) prepared by the R, as shown below.
C <- A %*% B
However, more complex operations than matrix multiplication are not always provided by programming languages as standard.
einsum
is a function that solves such a problem.
To put it simply, einsum
is a wrapper for the for loop above.
Like the Einstein summation, it omits many notations such as for,
array size (e.g. I, J, and K), brackets (e.g. {}, (), and []),
and even addition operator (+) and
extracts the array subscripts (e.g. i, j, and k)
to concisely express the tensor operation as follows.
suppressPackageStartupMessages(library("einsum"))
C <- einsum('ij,jk->ik', A, B)
DelayedTensor
CRAN einsum is easy to use because the syntax is almost
the same as that of Numpy
‘s einsum
,
except that it prohibits the implicit modes that do not use’->’.
It is extremely fast because the internal calculation
is actually performed by C++.
When the input tensor is huge, however,
it is not scalable because it assumes that the input is R’s standard array.
Using einsum
of DelayedTensor,
we can augment the CRAN einsum
’s functionality;
in DelayedTensor,
the input DelayedArray objects are divided into
multiple block tensors and the CRAN einsum
is incremently applied in the block processing.
A surprisingly large number of tensor operations can be handled
uniformly in einsum
.
In more detail, einsum
is capable of performing any tensor operation
that can be described by a combination of the following
three operations3 https://ajcr.net/Basic-guide-to-einsum/.
Some typical operations are introduced below. Here we use the arrays and DelayedArray objects below.
suppressPackageStartupMessages(library("DelayedTensor"))
suppressPackageStartupMessages(library("DelayedArray"))
arrA <- array(runif(3), dim=c(3))
arrB <- array(runif(3*3), dim=c(3,3))
arrC <- array(runif(3*4), dim=c(3,4))
arrD <- array(runif(3*3*3), dim=c(3,3,3))
arrE <- array(runif(3*4*5), dim=c(3,4,5))
darrA <- DelayedArray(arrA)
darrB <- DelayedArray(arrB)
darrC <- DelayedArray(arrC)
darrD <- DelayedArray(arrD)
darrE <- DelayedArray(arrE)
If the same subscript is written on both sides of ->,
einsum
will simply output the object without any calculation.
einsum::einsum('i->i', arrA)
## [1] 0.8313071 0.7872728 0.2325024
DelayedTensor::einsum('i->i', darrA)
## <3> DelayedArray object of type "double":
## [1] [2] [3]
## 0.8313071 0.7872728 0.2325024
einsum::einsum('ij->ij', arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.4250974 0.25918365 0.9500579 0.2463734
## [2,] 0.2328048 0.13154237 0.5790767 0.5689555
## [3,] 0.1161191 0.03348791 0.3014943 0.7246664
DelayedTensor::einsum('ij->ij', darrC)
## <3 x 4> DelayedArray object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.42509740 0.25918365 0.95005793 0.24637335
## [2,] 0.23280477 0.13154237 0.57907668 0.56895549
## [3,] 0.11611907 0.03348791 0.30149429 0.72466644
einsum::einsum('ijk->ijk', arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4729177 0.74627039 0.2784774 0.9405186
## [2,] 0.3388440 0.02732007 0.3195326 0.2941132
## [3,] 0.4247545 0.54143598 0.8038448 0.4925841
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8431826 0.5736781 0.2418506 0.385659778
## [2,] 0.4267298 0.5190271 0.3126905 0.002581954
## [3,] 0.3037314 0.2643363 0.2678443 0.134663860
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3831704 0.4971430 0.5385042 0.9307688
## [2,] 0.9526385 0.2405471 0.5427669 0.4294612
## [3,] 0.3874379 0.3126416 0.3793875 0.6806921
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7998042 0.1527263 0.8378668 0.1099577
## [2,] 0.8086172 0.7332603 0.3543911 0.9853874
## [3,] 0.5284007 0.6255601 0.4079075 0.4429649
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3645215 0.46494386 0.1181888 0.3162536
## [2,] 0.5659917 0.02129203 0.4769420 0.4511131
## [3,] 0.9734661 0.10383278 0.6496032 0.6415539
DelayedTensor::einsum('ijk->ijk', darrE)
## <3 x 4 x 5> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.47291767 0.74627039 0.27847745 0.94051856
## [2,] 0.33884400 0.02732007 0.31953265 0.29411315
## [3,] 0.42475447 0.54143598 0.80384482 0.49258410
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 0.843182555 0.573678129 0.241850572 0.385659778
## [2,] 0.426729757 0.519027079 0.312690529 0.002581954
## [3,] 0.303731362 0.264336331 0.267844269 0.134663860
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.3831704 0.4971430 0.5385042 0.9307688
## [2,] 0.9526385 0.2405471 0.5427669 0.4294612
## [3,] 0.3874379 0.3126416 0.3793875 0.6806921
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.7998042 0.1527263 0.8378668 0.1099577
## [2,] 0.8086172 0.7332603 0.3543911 0.9853874
## [3,] 0.5284007 0.6255601 0.4079075 0.4429649
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.36452148 0.46494386 0.11818878 0.31625360
## [2,] 0.56599174 0.02129203 0.47694198 0.45111315
## [3,] 0.97346612 0.10383278 0.64960321 0.64155392
We can also extract the diagonal elements as follows.
einsum::einsum('ii->i', arrB)
## [1] 0.1510672 0.2246076 0.4261232
DelayedTensor::einsum('ii->i', darrB)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.1510672 0.2246076 0.4261232
einsum::einsum('iii->i', arrD)
## [1] 0.6755720 0.6702890 0.6797056
DelayedTensor::einsum('iii->i', darrD)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.6755720 0.6702890 0.6797056
By using multiple arrays or DelayedArray objects as input and writing “,” on the right side of ->, multiplication will be performed.
Hadamard Product can also be implemented in einsum
,
multiplying by the product of each element.
einsum::einsum('i,i->i', arrA, arrA)
## [1] 0.69107148 0.61979847 0.05405736
DelayedTensor::einsum('i,i->i', darrA, darrA)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.69107148 0.61979847 0.05405736
einsum::einsum('ij,ij->ij', arrC, arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.18070780 0.06717617 0.90261008 0.06069983
## [2,] 0.05419806 0.01730340 0.33532980 0.32371035
## [3,] 0.01348364 0.00112144 0.09089881 0.52514145
DelayedTensor::einsum('ij,ij->ij', darrC, darrC)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.18070780 0.06717617 0.90261008 0.06069983
## [2,] 0.05419806 0.01730340 0.33532980 0.32371035
## [3,] 0.01348364 0.00112144 0.09089881 0.52514145
einsum::einsum('ijk,ijk->ijk', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2236511 0.5569194908 0.07754969 0.88457516
## [2,] 0.1148153 0.0007463863 0.10210111 0.08650255
## [3,] 0.1804164 0.2931529212 0.64616649 0.24263909
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.71095682 0.3291066 0.05849170 1.487335e-01
## [2,] 0.18209829 0.2693891 0.09777537 6.666486e-06
## [3,] 0.09225274 0.0698737 0.07174055 1.813436e-02
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1468196 0.24715121 0.2899868 0.8663305
## [2,] 0.9075201 0.05786292 0.2945959 0.1844369
## [3,] 0.1501081 0.09774479 0.1439349 0.4633417
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.6396868 0.02332533 0.7020208 0.01209069
## [2,] 0.6538618 0.53767066 0.1255931 0.97098832
## [3,] 0.2792073 0.39132540 0.1663885 0.19621791
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1328759 0.2161727905 0.01396859 0.1000163
## [2,] 0.3203467 0.0004533506 0.22747365 0.2035031
## [3,] 0.9476363 0.0107812457 0.42198434 0.4115914
DelayedTensor::einsum('ijk,ijk->ijk', darrE, darrE)
## <3 x 4 x 5> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.2236511205 0.5569194908 0.0775496879 0.8845751646
## [2,] 0.1148152550 0.0007463863 0.1021011134 0.0865025457
## [3,] 0.1804163567 0.2931529212 0.6461664890 0.2426390935
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 7.109568e-01 3.291066e-01 5.849170e-02 1.487335e-01
## [2,] 1.820983e-01 2.693891e-01 9.777537e-02 6.666486e-06
## [3,] 9.225274e-02 6.987370e-02 7.174055e-02 1.813436e-02
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.14681957 0.24715121 0.28998677 0.86633048
## [2,] 0.90752007 0.05786292 0.29459590 0.18443688
## [3,] 0.15010810 0.09774479 0.14393489 0.46334173
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.63968677 0.02332533 0.70202081 0.01209069
## [2,] 0.65386181 0.53767066 0.12559308 0.97098832
## [3,] 0.27920728 0.39132540 0.16638851 0.19621791
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.1328759080 0.2161727905 0.0139685876 0.1000163367
## [2,] 0.3203466534 0.0004533506 0.2274736485 0.2035030722
## [3,] 0.9476362890 0.0107812457 0.4219843361 0.4115914302
The outer product can also be implemented in einsum
,
in which the subscripts in the input array are all different,
and all of them are kept.
einsum::einsum('i,j->ij', arrA, arrA)
## [,1] [,2] [,3]
## [1,] 0.6910715 0.6544655 0.19328088
## [2,] 0.6544655 0.6197985 0.18304281
## [3,] 0.1932809 0.1830428 0.05405736
DelayedTensor::einsum('i,j->ij', darrA, darrA)
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.69107148 0.65446547 0.19328088
## [2,] 0.65446547 0.61979847 0.18304281
## [3,] 0.19328088 0.18304281 0.05405736
einsum::einsum('ij,klm->ijklm', arrC, arrE)
## , , 1, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.20103607 0.12257253 0.4492992 0.1165143
## [2,] 0.11009749 0.06220871 0.2738556 0.2690691
## [3,] 0.05491476 0.01583702 0.1425820 0.3427076
##
## , , 2, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.14404170 0.08782282 0.3219214 0.08348213
## [2,] 0.07888450 0.04457234 0.1962167 0.19278715
## [3,] 0.03934625 0.01134718 0.1021595 0.24554887
##
## , , 3, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.18056202 0.11008941 0.4035414 0.1046482
## [2,] 0.09888487 0.05587321 0.2459654 0.2416664
## [3,] 0.04932210 0.01422414 0.1280610 0.3078053
##
## , , 1, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.31723760 0.19342108 0.7090001 0.1838611
## [2,] 0.17373531 0.09816618 0.4321478 0.4245946
## [3,] 0.08665623 0.02499104 0.2249963 0.5407971
##
## , , 2, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.011613691 0.0070809157 0.025955650 0.006730937
## [2,] 0.006360243 0.0035937469 0.015820416 0.015543904
## [3,] 0.003172381 0.0009148921 0.008236845 0.019797938
##
## , , 3, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.23016303 0.14033135 0.5143955 0.1333954
## [2,] 0.12604888 0.07122177 0.3135329 0.3080530
## [3,] 0.06287104 0.01813156 0.1632399 0.3923605
##
## , , 1, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.11838004 0.072176801 0.26456971 0.06860942
## [2,] 0.06483088 0.036631583 0.16125979 0.15844127
## [3,] 0.03233654 0.009325628 0.08395936 0.20180326
##
## , , 2, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.13583250 0.08281764 0.30357453 0.07872433
## [2,] 0.07438873 0.04203208 0.18503390 0.18179986
## [3,] 0.03710384 0.01070048 0.09633727 0.23155459
##
## , , 3, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.34171234 0.20834343 0.7636991 0.1980459
## [2,] 0.18713891 0.10573965 0.4654878 0.4573519
## [3,] 0.09334172 0.02691908 0.2423546 0.5825194
##
## , , 1, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3998120 0.243767 0.8935471 0.2317187
## [2,] 0.2189572 0.123718 0.5446324 0.5351132
## [3,] 0.1092121 0.031496 0.2835610 0.6815622
##
## , , 2, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.12502674 0.076229321 0.27942453 0.07246164
## [2,] 0.06847094 0.038688341 0.17031407 0.16733729
## [3,] 0.03415215 0.009849235 0.08867344 0.21313393
##
## , , 3, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.20939622 0.12766975 0.4679834 0.1213596
## [2,] 0.11467593 0.06479568 0.2852440 0.2802584
## [3,] 0.05719841 0.01649561 0.1485113 0.3569592
##
## , , 1, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.35843471 0.21853913 0.8010723 0.2077377
## [2,] 0.19629692 0.11091423 0.4882674 0.4797333
## [3,] 0.09790958 0.02823642 0.2542147 0.6110261
##
## , , 2, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.18140171 0.11060138 0.4054180 0.1051348
## [2,] 0.09934472 0.05613304 0.2471093 0.2427902
## [3,] 0.04955146 0.01429029 0.1286566 0.3092367
##
## , , 3, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.12911541 0.07872220 0.28856239 0.07483131
## [2,] 0.07071011 0.03995354 0.17588375 0.17280963
## [3,] 0.03526900 0.01017133 0.09157327 0.22010392
##
## , , 1, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.24386908 0.14868799 0.5450275 0.1413390
## [2,] 0.13355501 0.07546298 0.3322036 0.3263973
## [3,] 0.06661497 0.01921128 0.1729607 0.4157253
##
## , , 2, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.22063706 0.13452333 0.4931058 0.1278744
## [2,] 0.12083198 0.06827405 0.3005565 0.2953033
## [3,] 0.06026894 0.01738113 0.1564837 0.3761215
##
## , , 3, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.11236869 0.068511656 0.25113483 0.06512543
## [2,] 0.06153876 0.034771428 0.15307100 0.15039561
## [3,] 0.03069449 0.008852071 0.07969589 0.19155567
##
## , , 1, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.10281005 0.06268371 0.22977205 0.05958554
## [2,] 0.05630397 0.03181360 0.14005003 0.13760221
## [3,] 0.02808346 0.00809907 0.07291657 0.17526099
##
## , , 2, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.13292393 0.08104427 0.29707412 0.07703861
## [2,] 0.07279585 0.04113205 0.18107179 0.17790699
## [3,] 0.03630933 0.01047135 0.09427441 0.22659633
##
## , , 3, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.11385990 0.069420856 0.25446757 0.06598969
## [2,] 0.06235542 0.035232870 0.15510237 0.15239147
## [3,] 0.03110183 0.008969545 0.08075352 0.19409775
##
## , , 1, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.16394297 0.09995671 0.3663991 0.09501629
## [2,] 0.08978344 0.05073060 0.2233266 0.21942325
## [3,] 0.04478246 0.01291494 0.1162742 0.27947470
##
## , , 2, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.0010975819 6.692003e-04 0.0024530059 0.0006361247
## [2,] 0.0006010912 3.396364e-04 0.0014951493 0.0014690169
## [3,] 0.0002998141 8.646424e-05 0.0007784444 0.0018710554
##
## , , 3, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.05724526 0.034902671 0.12793847 0.03317759
## [2,] 0.03135039 0.017714003 0.07798070 0.07661774
## [3,] 0.01563704 0.004509611 0.04060038 0.09758638
##
## , , 1, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.16288475 0.09931151 0.3640341 0.09440298
## [2,] 0.08920390 0.05040315 0.2218851 0.21800692
## [3,] 0.04449339 0.01283158 0.1155237 0.27767075
##
## , , 2, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4049641 0.24690832 0.9050617 0.2347047
## [2,] 0.2217788 0.12531232 0.5516507 0.5420089
## [3,] 0.1106195 0.03190187 0.2872151 0.6903451
##
## , , 3, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.16469883 0.10041756 0.3680884 0.09545436
## [2,] 0.09019738 0.05096449 0.2243562 0.22043490
## [3,] 0.04498893 0.01297448 0.1168103 0.28076322
##
## , , 1, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.21133422 0.12885135 0.4723147 0.1224828
## [2,] 0.11573727 0.06539538 0.2878839 0.2828523
## [3,] 0.05772779 0.01664828 0.1498858 0.3602629
##
## , , 2, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.10225596 0.062345885 0.22853371 0.0592644
## [2,] 0.05600052 0.031642140 0.13929524 0.1368606
## [3,] 0.02793211 0.008055421 0.07252359 0.1743164
##
## , , 3, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.13290314 0.08103160 0.29702766 0.07702657
## [2,] 0.07278446 0.04112562 0.18104347 0.17787917
## [3,] 0.03630366 0.01046971 0.09425967 0.22656089
##
## , , 1, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.22891674 0.13957148 0.5116102 0.1326731
## [2,] 0.12536635 0.07083612 0.3118352 0.3063849
## [3,] 0.06253061 0.01803338 0.1623559 0.3902359
##
## , , 2, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.23072880 0.14067630 0.5156600 0.1337233
## [2,] 0.12635872 0.07139684 0.3143036 0.3088102
## [3,] 0.06302559 0.01817613 0.1636411 0.3933249
##
## , , 3, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.16127665 0.09833104 0.3604401 0.09347097
## [2,] 0.08832322 0.04990553 0.2196945 0.21585461
## [3,] 0.04405413 0.01270489 0.1143832 0.27492940
##
## , , 1, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3956674 0.2412400 0.8842842 0.2293166
## [2,] 0.2166874 0.1224355 0.5389865 0.5295660
## [3,] 0.1080800 0.0311695 0.2806215 0.6744969
##
## , , 2, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.18256282 0.11130931 0.4080130 0.1058078
## [2,] 0.09998061 0.05649234 0.2486909 0.2443443
## [3,] 0.04986863 0.01438176 0.1294801 0.3112161
##
## , , 3, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.28936044 0.17642426 0.6466969 0.1677044
## [2,] 0.15846837 0.08953985 0.3941729 0.3872835
## [3,] 0.07904134 0.02279496 0.2052248 0.4932747
##
## , , 1, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.33999469 0.20729618 0.7598603 0.1970504
## [2,] 0.18619824 0.10520814 0.4631480 0.4550530
## [3,] 0.09287252 0.02678377 0.2411364 0.5795913
##
## , , 2, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.34374108 0.2095804 0.7682332 0.1992217
## [2,] 0.18824995 0.1063674 0.4682514 0.4600672
## [3,] 0.09389588 0.0270789 0.2437935 0.5859778
##
## , , 3, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2246218 0.13695282 0.5020113 0.1301838
## [2,] 0.1230142 0.06950708 0.3059845 0.3006365
## [3,] 0.0613574 0.01769503 0.1593098 0.3829142
##
## , , 1, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06492357 0.039584170 0.14509887 0.03762770
## [2,] 0.03555542 0.020089985 0.08844026 0.08689449
## [3,] 0.01773444 0.005114486 0.04604612 0.11067565
##
## , , 2, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.31170705 0.19004908 0.6966398 0.1806558
## [2,] 0.17070649 0.09645480 0.4246139 0.4171925
## [3,] 0.08514551 0.02455535 0.2210738 0.5313691
##
## , , 3, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.26592396 0.16213494 0.5943183 0.1541213
## [2,] 0.14563337 0.08228766 0.3622472 0.3559158
## [3,] 0.07263946 0.02094870 0.1886028 0.4533224
##
## , , 1, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.35617501 0.21716138 0.7960220 0.2064281
## [2,] 0.19505939 0.11021499 0.4851891 0.4767089
## [3,] 0.09729232 0.02805841 0.2526121 0.6071740
##
## , , 2, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.15065075 0.09185239 0.3366921 0.08731253
## [2,] 0.08250395 0.04661745 0.2052196 0.20163278
## [3,] 0.04115157 0.01186782 0.1068469 0.25681536
##
## , , 3, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.17340041 0.10572295 0.3875357 0.1004975
## [2,] 0.09496281 0.05365712 0.2362097 0.2320812
## [3,] 0.04736584 0.01365997 0.1229818 0.2955969
##
## , , 1, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04674272 0.028499229 0.10446615 0.02709064
## [2,] 0.02559867 0.014464092 0.06367392 0.06256102
## [3,] 0.01276818 0.003682252 0.03315161 0.07968263
##
## , , 2, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4188856 0.25539630 0.9361751 0.2427732
## [2,] 0.2294029 0.12962019 0.5706149 0.5606416
## [3,] 0.1144223 0.03299856 0.2970887 0.7140772
##
## , , 3, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.18830323 0.11480926 0.4208423 0.1091347
## [2,] 0.10312434 0.05826865 0.2565106 0.2520273
## [3,] 0.05143667 0.01483397 0.1335514 0.3210018
##
## , , 1, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.15495713 0.09447801 0.3463165 0.08980838
## [2,] 0.08486234 0.04795002 0.2110859 0.20739650
## [3,] 0.04232790 0.01220706 0.1099011 0.26415648
##
## , , 2, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.24060162 0.14669581 0.5377249 0.1394453
## [2,] 0.13176558 0.07445190 0.3277526 0.3220241
## [3,] 0.06572244 0.01895388 0.1706433 0.4101552
##
## , , 3, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4138179 0.25230650 0.9248492 0.2398361
## [2,] 0.2266276 0.12805204 0.5637115 0.5538589
## [3,] 0.1130380 0.03259935 0.2934945 0.7054382
##
## , , 1, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.19764643 0.12050585 0.4417236 0.1145498
## [2,] 0.10824115 0.06115982 0.2692381 0.2645324
## [3,] 0.05398885 0.01557000 0.1401779 0.3369292
##
## , , 2, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.009051187 0.0055185464 0.020228663 0.005245789
## [2,] 0.004956886 0.0028008043 0.012329719 0.012114218
## [3,] 0.002472411 0.0007130256 0.006419426 0.015429620
##
## , , 3, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04413904 0.026911758 0.09864715 0.02558163
## [2,] 0.02417277 0.013658410 0.06012714 0.05907623
## [3,] 0.01205697 0.003477143 0.03130499 0.07524413
##
## , , 1, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.05024174 0.030632599 0.11228619 0.02911857
## [2,] 0.02751491 0.015546832 0.06844037 0.06724416
## [3,] 0.01372397 0.003957895 0.03563324 0.08564744
##
## , , 2, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.20274680 0.12361556 0.4531225 0.1175058
## [2,] 0.11103437 0.06273808 0.2761860 0.2713588
## [3,] 0.05538206 0.01597179 0.1437953 0.3456238
##
## , , 3, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.27614464 0.16836653 0.6171607 0.1600449
## [2,] 0.15123073 0.08545035 0.3761701 0.3695953
## [3,] 0.07543132 0.02175385 0.1958517 0.4707456
##
## , , 1, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.13443858 0.08196776 0.30045924 0.07791646
## [2,] 0.07362535 0.04160075 0.18313508 0.17993422
## [3,] 0.03672307 0.01059067 0.09534865 0.22917837
##
## , , 2, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.19176703 0.11692115 0.4285836 0.1111423
## [2,] 0.10502129 0.05934049 0.2612291 0.2566633
## [3,] 0.05238284 0.01510684 0.1360080 0.3269066
##
## , , 3, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.27272290 0.16628029 0.6095134 0.1580618
## [2,] 0.14935681 0.08439152 0.3715089 0.3650156
## [3,] 0.07449665 0.02148430 0.1934248 0.4649126
DelayedTensor::einsum('ij,klm->ijklm', darrC, darrE)
## <3 x 4 x 3 x 4 x 5> HDF5Array object of type "double":
## ,,1,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.20103607 0.12257253 0.44929918 0.11651431
## [2,] 0.11009749 0.06220871 0.27385559 0.26906910
## [3,] 0.05491476 0.01583702 0.14258198 0.34270756
##
## ,,2,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.14404170 0.08782282 0.32192143 0.08348213
## [2,] 0.07888450 0.04457234 0.19621666 0.19278715
## [3,] 0.03934625 0.01134718 0.10215953 0.24554887
##
## ,,3,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.18056202 0.11008941 0.40354135 0.10464818
## [2,] 0.09888487 0.05587321 0.24596541 0.24166639
## [3,] 0.04932210 0.01422414 0.12806105 0.30780531
##
## ...
##
## ,,1,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.13443858 0.08196776 0.30045924 0.07791646
## [2,] 0.07362535 0.04160075 0.18313508 0.17993422
## [3,] 0.03672307 0.01059067 0.09534865 0.22917837
##
## ,,2,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.19176703 0.11692115 0.42858363 0.11114226
## [2,] 0.10502129 0.05934049 0.26122910 0.25666330
## [3,] 0.05238284 0.01510684 0.13600804 0.32690656
##
## ,,3,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.27272290 0.16628029 0.60951339 0.15806179
## [2,] 0.14935681 0.08439152 0.37150891 0.36501563
## [3,] 0.07449665 0.02148430 0.19342484 0.46491259
If there is a vanishing subscript on the left or right side of ->, the summation is done for that subscript.
einsum::einsum('i->', arrA)
## [1] 1.851082
DelayedTensor::einsum('i->', darrA)
## <1> HDF5Array object of type "double":
## [1]
## 1.851082
einsum::einsum('ij->', arrC)
## [1] 4.568859
DelayedTensor::einsum('ij->', darrC)
## <1> HDF5Array object of type "double":
## [1]
## 4.568859
einsum::einsum('ijk->', arrE)
## [1] 28.1663
DelayedTensor::einsum('ijk->', darrE)
## <1> HDF5Array object of type "double":
## [1]
## 28.1663
einsum::einsum('ij->i', arrC)
## [1] 1.880712 1.512379 1.175768
DelayedTensor::einsum('ij->i', darrC)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 1.880712 1.512379 1.175768
einsum::einsum('ij->j', arrC)
## [1] 0.7740212 0.4242139 1.8306289 1.5399953
DelayedTensor::einsum('ij->j', darrC)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 0.7740212 0.4242139 1.8306289 1.5399953
einsum::einsum('ijk->i', arrE)
## [1] 9.996404 8.803248 9.366643
DelayedTensor::einsum('ijk->i', darrE)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 9.996404 8.803248 9.366643
einsum::einsum('ijk->j', arrE)
## [1] 8.574208 5.824015 6.529798 7.238274
DelayedTensor::einsum('ijk->j', darrE)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 8.574208 5.824015 6.529798 7.238274
einsum::einsum('ijk->k', arrE)
## [1] 5.680613 4.275976 6.275159 6.786844 5.147703
DelayedTensor::einsum('ijk->k', darrE)
## <5> HDF5Array object of type "double":
## [1] [2] [3] [4] [5]
## 5.680613 4.275976 6.275159 6.786844 5.147703
These are the same as what the modeSum
function does.
einsum::einsum('ijk->ij', arrE)
## [,1] [,2] [,3] [,4]
## [1,] 2.863596 2.434762 2.014888 2.683158
## [2,] 3.092821 1.541447 2.006323 2.162657
## [3,] 2.617790 1.847807 2.508587 2.392459
DelayedTensor::einsum('ijk->ij', darrE)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 2.863596 2.434762 2.014888 2.683158
## [2,] 3.092821 1.541447 2.006323 2.162657
## [3,] 2.617790 1.847807 2.508587 2.392459
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.236516 1.5736437 1.723247 2.136822 1.9039793
## [2,] 1.315026 1.3570415 1.050332 1.511547 0.5900687
## [3,] 1.401855 0.8223854 1.460659 1.600165 1.2447340
## [4,] 1.727216 0.5229056 2.040922 1.538310 1.4089207
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.2365161 1.5736437 1.7232468 2.1368221 1.9039793
## [2,] 1.3150264 1.3570415 1.0503318 1.5115467 0.5900687
## [3,] 1.4018549 0.8223854 1.4606586 1.6001654 1.2447340
## [4,] 1.7272158 0.5229056 2.0409220 1.5383100 1.4089207
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.236516 1.5736437 1.723247 2.136822 1.9039793
## [2,] 1.315026 1.3570415 1.050332 1.511547 0.5900687
## [3,] 1.401855 0.8223854 1.460659 1.600165 1.2447340
## [4,] 1.727216 0.5229056 2.040922 1.538310 1.4089207
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.2365161 1.5736437 1.7232468 2.1368221 1.9039793
## [2,] 1.3150264 1.3570415 1.0503318 1.5115467 0.5900687
## [3,] 1.4018549 0.8223854 1.4606586 1.6001654 1.2447340
## [4,] 1.7272158 0.5229056 2.0409220 1.5383100 1.4089207
If we take the diagonal elements of a matrix
and add them together, we get trace
.
einsum::einsum('ii->', arrB)
## [1] 0.801798
DelayedTensor::einsum('ii->', darrB)
## <1> HDF5Array object of type "double":
## [1]
## 0.801798
By changing the order of the indices on the left and right side of ->, we can get a sorted array or DelayedArray.
einsum::einsum('ij->ji', arrB)
## [,1] [,2] [,3]
## [1,] 0.1510672 0.4619190 0.01416964
## [2,] 0.2511940 0.2246076 0.65651124
## [3,] 0.5405729 0.1780173 0.42612321
DelayedTensor::einsum('ij->ji', darrB)
## <3 x 3> DelayedArray object of type "double":
## [,1] [,2] [,3]
## [1,] 0.15106717 0.46191902 0.01416964
## [2,] 0.25119396 0.22460764 0.65651124
## [3,] 0.54057295 0.17801729 0.42612321
einsum::einsum('ijk->jki', arrD)
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.6755720 0.4685084 0.9096058
## [2,] 0.4874284 0.5640649 0.4744207
## [3,] 0.3713618 0.6930825 0.7874355
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.4095722 0.6075899 0.7938976
## [2,] 0.4555237 0.6702890 0.5892084
## [3,] 0.1013367 0.3161663 0.7663911
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.48375967 0.8548515 0.2909169
## [2,] 0.51505899 0.6619586 0.7830392
## [3,] 0.02876147 0.1999329 0.6797056
DelayedTensor::einsum('ijk->jki', darrD)
## <3 x 3 x 3> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3]
## [1,] 0.6755720 0.4685084 0.9096058
## [2,] 0.4874284 0.5640649 0.4744207
## [3,] 0.3713618 0.6930825 0.7874355
##
## ,,2
## [,1] [,2] [,3]
## [1,] 0.4095722 0.6075899 0.7938976
## [2,] 0.4555237 0.6702890 0.5892084
## [3,] 0.1013367 0.3161663 0.7663911
##
## ,,3
## [,1] [,2] [,3]
## [1,] 0.48375967 0.85485151 0.29091685
## [2,] 0.51505899 0.66195860 0.78303919
## [3,] 0.02876147 0.19993293 0.67970562
Some examples of combining Multiplication and Summation are shown below.
Inner Product first calculate Hadamard Product and collapses it to 0D tensor (norm).
einsum::einsum('i,i->', arrA, arrA)
## [1] 1.364927
DelayedTensor::einsum('i,i->', darrA, darrA)
## <1> HDF5Array object of type "double":
## [1]
## 1.364927
einsum::einsum('ij,ij->', arrC, arrC)
## [1] 2.572381
DelayedTensor::einsum('ij,ij->', darrC, darrC)
## <1> HDF5Array object of type "double":
## [1]
## 2.572381
einsum::einsum('ijk,ijk->', arrE, arrE)
## [1] 17.01281
DelayedTensor::einsum('ijk,ijk->', darrE, darrE)
## <1> HDF5Array object of type "double":
## [1]
## 17.01281
The inner product is an operation that eliminates all subscripts, while the outer product is an operation that leaves all subscripts intact. In the middle of the two, the operation that eliminates some subscripts while keeping others by summing them is called contracted product.
einsum::einsum('ijk,ijk->jk', arrE, arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5188827 0.9853078 1.2044477 1.5727559 1.4008589
## [2,] 0.8508188 0.6683694 0.4027589 0.9523214 0.2274074
## [3,] 0.8258173 0.2280076 0.7285176 0.9940024 0.6634266
## [4,] 1.2137168 0.1668745 1.5141091 1.1792969 0.7151108
DelayedTensor::einsum('ijk,ijk->jk', darrE, darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5188827 0.9853078 1.2044477 1.5727559 1.4008589
## [2,] 0.8508188 0.6683694 0.4027589 0.9523214 0.2274074
## [3,] 0.8258173 0.2280076 0.7285176 0.9940024 0.6634266
## [4,] 1.2137168 0.1668745 1.5141091 1.1792969 0.7151108
Matrix Multiplication is considered a contracted product.
einsum::einsum('ij,jk->ik', arrC, t(arrC))
## [,1] [,2] [,3]
## [1,] 1.2111939 0.8233902 0.5230170
## [2,] 0.8233902 0.7305416 0.6183294
## [3,] 0.5230170 0.6183294 0.6306453
DelayedTensor::einsum('ij,jk->ik', darrC, t(darrC))
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 1.2111939 0.8233902 0.5230170
## [2,] 0.8233902 0.7305416 0.6183294
## [3,] 0.5230170 0.6183294 0.6306453
Some examples of combining Multiplication and Permutation are shown below.
einsum::einsum('ij,ij->ji', arrC, arrC)
## [,1] [,2] [,3]
## [1,] 0.18070780 0.05419806 0.01348364
## [2,] 0.06717617 0.01730340 0.00112144
## [3,] 0.90261008 0.33532980 0.09089881
## [4,] 0.06069983 0.32371035 0.52514145
DelayedTensor::einsum('ij,ij->ji', darrC, darrC)
## <4 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.18070780 0.05419806 0.01348364
## [2,] 0.06717617 0.01730340 0.00112144
## [3,] 0.90261008 0.33532980 0.09089881
## [4,] 0.06069983 0.32371035 0.52514145
einsum::einsum('ijk,ijk->jki', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.22365112 0.7109568 0.1468196 0.63968677 0.13287591
## [2,] 0.55691949 0.3291066 0.2471512 0.02332533 0.21617279
## [3,] 0.07754969 0.0584917 0.2899868 0.70202081 0.01396859
## [4,] 0.88457516 0.1487335 0.8663305 0.01209069 0.10001634
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.1148152550 1.820983e-01 0.90752007 0.6538618 0.3203466534
## [2,] 0.0007463863 2.693891e-01 0.05786292 0.5376707 0.0004533506
## [3,] 0.1021011134 9.777537e-02 0.29459590 0.1255931 0.2274736485
## [4,] 0.0865025457 6.666486e-06 0.18443688 0.9709883 0.2035030722
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.1804164 0.09225274 0.15010810 0.2792073 0.94763629
## [2,] 0.2931529 0.06987370 0.09774479 0.3913254 0.01078125
## [3,] 0.6461665 0.07174055 0.14393489 0.1663885 0.42198434
## [4,] 0.2426391 0.01813436 0.46334173 0.1962179 0.41159143
DelayedTensor::einsum('ijk,ijk->jki', darrE, darrE)
## <4 x 5 x 3> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.22365112 0.71095682 0.14681957 0.63968677 0.13287591
## [2,] 0.55691949 0.32910660 0.24715121 0.02332533 0.21617279
## [3,] 0.07754969 0.05849170 0.28998677 0.70202081 0.01396859
## [4,] 0.88457516 0.14873346 0.86633048 0.01209069 0.10001634
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.148153e-01 1.820983e-01 9.075201e-01 6.538618e-01 3.203467e-01
## [2,] 7.463863e-04 2.693891e-01 5.786292e-02 5.376707e-01 4.533506e-04
## [3,] 1.021011e-01 9.777537e-02 2.945959e-01 1.255931e-01 2.274736e-01
## [4,] 8.650255e-02 6.666486e-06 1.844369e-01 9.709883e-01 2.035031e-01
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.18041636 0.09225274 0.15010810 0.27920728 0.94763629
## [2,] 0.29315292 0.06987370 0.09774479 0.39132540 0.01078125
## [3,] 0.64616649 0.07174055 0.14393489 0.16638851 0.42198434
## [4,] 0.24263909 0.01813436 0.46334173 0.19621791 0.41159143
Some examples of combining Summation and Permutation are shown below.
einsum::einsum('ijk->ki', arrE)
## [,1] [,2] [,3]
## [1,] 2.438184 0.9798099 2.2626194
## [2,] 2.044371 1.2610293 0.9705758
## [3,] 2.349586 2.1654137 1.7601591
## [4,] 1.900355 2.8816560 2.0048331
## [5,] 1.263908 1.5153389 2.3684560
DelayedTensor::einsum('ijk->ki', darrE)
## <5 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 2.4381841 0.9798099 2.2626194
## [2,] 2.0443710 1.2610293 0.9705758
## [3,] 2.3495864 2.1654137 1.7601591
## [4,] 1.9003550 2.8816560 2.0048331
## [5,] 1.2639077 1.5153389 2.3684560
Finally, we will show a more complex example, combining Multiplication, Summation, and Permutation.
einsum::einsum('i,ij,ijk,ijk,ji->jki',
arrA, arrC, arrE, arrE, t(arrC))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.03359769 0.10680255 0.02205578 0.0960960998 0.019961108
## [2,] 0.03110062 0.01837864 0.01380192 0.0013025805 0.012071958
## [3,] 0.05818911 0.04388902 0.21759046 0.5267586229 0.010481276
## [4,] 0.04463584 0.00750512 0.04371521 0.0006100985 0.005046844
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 4.899013e-03 7.769890e-03 0.0387226655 0.027899407 1.366876e-02
## [2,] 1.016764e-05 3.669751e-03 0.0007882373 0.007324415 6.175765e-06
## [3,] 2.695429e-02 2.581231e-02 0.0777721477 0.033156074 6.005214e-02
## [4,] 2.204503e-02 1.698943e-06 0.0470034370 0.247454778 5.186242e-02
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 5.656014e-04 2.892103e-04 4.705856e-04 0.0008753088 2.970819e-03
## [2,] 7.643596e-05 1.821869e-05 2.548573e-05 0.0001020332 2.811075e-06
## [3,] 1.365621e-02 1.516178e-03 3.041947e-03 0.0035164862 8.918297e-03
## [4,] 2.962542e-02 2.214144e-03 5.657247e-02 0.0239575475 5.025393e-02
DelayedTensor::einsum('i,ij,ijk,ijk,ji->jki',
darrA, darrC, darrE, darrE, t(darrC))
## <4 x 5 x 3> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0335976940 0.1068025487 0.0220557765 0.0960960998 0.0199611076
## [2,] 0.0311006247 0.0183786362 0.0138019179 0.0013025805 0.0120719582
## [3,] 0.0581891107 0.0438890220 0.2175904591 0.5267586229 0.0104812761
## [4,] 0.0446358380 0.0075051201 0.0437152076 0.0006100985 0.0050468442
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 4.899013e-03 7.769890e-03 3.872267e-02 2.789941e-02 1.366876e-02
## [2,] 1.016764e-05 3.669751e-03 7.882373e-04 7.324415e-03 6.175765e-06
## [3,] 2.695429e-02 2.581231e-02 7.777215e-02 3.315607e-02 6.005214e-02
## [4,] 2.204503e-02 1.698943e-06 4.700344e-02 2.474548e-01 5.186242e-02
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 5.656014e-04 2.892103e-04 4.705856e-04 8.753088e-04 2.970819e-03
## [2,] 7.643596e-05 1.821869e-05 2.548573e-05 1.020332e-04 2.811075e-06
## [3,] 1.365621e-02 1.516178e-03 3.041947e-03 3.516486e-03 8.918297e-03
## [4,] 2.962542e-02 2.214144e-03 5.657247e-02 2.395755e-02 5.025393e-02
einsum
By using einsum
and other DelayedTensor functions,
it is possible to implement your original tensor calculation functions.
It is intended to be applied to Delayed Arrays,
which can scale to large-scale data
since the calculation is performed internally by block processing.
For example, kronecker
can be easily implmented by eimsum
and other DelayedTensor functions4 https://stackoverflow.com/
questions/56067643/speeding-up-kronecker-products-numpy
(the kronecker
function inside DelayedTensor
has a more efficient implementation though).
darr1 <- DelayedArray(array(1:6, dim=c(2,3)))
darr2 <- DelayedArray(array(20:1, dim=c(4,5)))
mykronecker <- function(darr1, darr2){
stopifnot((length(dim(darr1)) == 2) && (length(dim(darr2)) == 2))
# Outer Product
tmpdarr <- DelayedTensor::einsum('ij,kl->ikjl', darr1, darr2)
# Reshape
DelayedTensor::unfold(tmpdarr, row_idx=c(2,1), col_idx=c(4,3))
}
identical(as.array(DelayedTensor::kronecker(darr1, darr2)),
as.array(mykronecker(darr1, darr2)))
## [1] TRUE
## R version 4.3.0 RC (2023-04-13 r84266)
## Platform: aarch64-apple-darwin20 (64-bit)
## Running under: macOS Monterey 12.6.1
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
##
## locale:
## [1] C/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## time zone: America/New_York
## tzcode source: internal
##
## attached base packages:
## [1] stats4 stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] einsum_0.1.0 DelayedRandomArray_1.8.0 HDF5Array_1.28.1
## [4] rhdf5_2.44.0 DelayedArray_0.26.2 S4Arrays_1.0.1
## [7] IRanges_2.34.0 S4Vectors_0.38.1 MatrixGenerics_1.12.0
## [10] matrixStats_0.63.0 BiocGenerics_0.46.0 Matrix_1.5-4
## [13] DelayedTensor_1.6.0 BiocStyle_2.28.0
##
## loaded via a namespace (and not attached):
## [1] jsonlite_1.8.4 compiler_4.3.0 BiocManager_1.30.20
## [4] crayon_1.5.2 rsvd_1.0.5 Rcpp_1.0.10
## [7] rhdf5filters_1.12.1 parallel_4.3.0 jquerylib_0.1.4
## [10] BiocParallel_1.34.1 yaml_2.3.7 fastmap_1.1.1
## [13] lattice_0.21-8 R6_2.5.1 ScaledMatrix_1.8.1
## [16] knitr_1.42 bookdown_0.33 bslib_0.4.2
## [19] rlang_1.1.0 cachem_1.0.7 xfun_0.38
## [22] sass_0.4.5 cli_3.6.1 Rhdf5lib_1.22.0
## [25] BiocSingular_1.16.0 digest_0.6.31 grid_4.3.0
## [28] irlba_2.3.5.1 rTensor_1.4.8 dqrng_0.3.0
## [31] evaluate_0.20 codetools_0.2-19 beachmat_2.16.0
## [34] rmarkdown_2.21 tools_4.3.0 htmltools_0.5.5