DelayedTensor 1.9.0
Authors: Koki Tsuyuzaki [aut, cre]
Last modified: 2023-10-20 14:48:21.036969
Compiled: Tue Oct 24 19:03:35 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] 0.2629702 0.3360933 0.6081032
DelayedTensor::einsum('i->i', darrA)
## <3> DelayedArray object of type "double":
## [1] [2] [3]
## 0.2629702 0.3360933 0.6081032
einsum::einsum('ij->ij', arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.8290624 0.39677844 0.41593749 0.05485368
## [2,] 0.7372489 0.41709692 0.08648592 0.39089161
## [3,] 0.7781028 0.02044262 0.73236600 0.37622064
DelayedTensor::einsum('ij->ij', darrC)
## <3 x 4> DelayedArray object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.82906245 0.39677844 0.41593749 0.05485368
## [2,] 0.73724889 0.41709692 0.08648592 0.39089161
## [3,] 0.77810281 0.02044262 0.73236600 0.37622064
einsum::einsum('ijk->ijk', arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3812977 0.1820969 0.29514239 0.02201983
## [2,] 0.4951155 0.9298326 0.02841801 0.16154655
## [3,] 0.9580172 0.7969576 0.78999521 0.94168207
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.001764826 0.3050367 0.6319875 0.64951137
## [2,] 0.068582661 0.1938585 0.9245301 0.90416308
## [3,] 0.676292656 0.9683193 0.6530972 0.07505918
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.6659755 0.8974535 0.6199101 0.1178649
## [2,] 0.4696335 0.2288339 0.8546451 0.5555089
## [3,] 0.3395294 0.0824663 0.4795878 0.6554943
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.9942740 0.1437270 0.3784346 0.5031835
## [2,] 0.8651440 0.5018223 0.6877795 0.9812351
## [3,] 0.4540285 0.7141506 0.2000684 0.5044849
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.03893093 0.49430781 0.4779090 0.5484045
## [2,] 0.08125681 0.48140985 0.4302264 0.9145417
## [3,] 0.01819715 0.02147867 0.8920896 0.9907613
DelayedTensor::einsum('ijk->ijk', darrE)
## <3 x 4 x 5> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.38129769 0.18209688 0.29514239 0.02201983
## [2,] 0.49511546 0.92983259 0.02841801 0.16154655
## [3,] 0.95801721 0.79695758 0.78999521 0.94168207
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 0.001764826 0.305036654 0.631987484 0.649511375
## [2,] 0.068582661 0.193858461 0.924530105 0.904163080
## [3,] 0.676292656 0.968319304 0.653097193 0.075059182
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.6659755 0.8974535 0.6199101 0.1178649
## [2,] 0.4696335 0.2288339 0.8546451 0.5555089
## [3,] 0.3395294 0.0824663 0.4795878 0.6554943
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.9942740 0.1437270 0.3784346 0.5031835
## [2,] 0.8651440 0.5018223 0.6877795 0.9812351
## [3,] 0.4540285 0.7141506 0.2000684 0.5044849
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.03893093 0.49430781 0.47790905 0.54840449
## [2,] 0.08125681 0.48140985 0.43022637 0.91454170
## [3,] 0.01819715 0.02147867 0.89208962 0.99076131
We can also extract the diagonal elements as follows.
einsum::einsum('ii->i', arrB)
## [1] 0.4200485 0.9008985 0.6553774
DelayedTensor::einsum('ii->i', darrB)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.4200485 0.9008985 0.6553774
einsum::einsum('iii->i', arrD)
## [1] 0.7135298 0.5345016 0.7826825
DelayedTensor::einsum('iii->i', darrD)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.7135298 0.5345016 0.7826825
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] 0.06915333 0.11295868 0.36978951
DelayedTensor::einsum('i,i->i', darrA, darrA)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.06915333 0.11295868 0.36978951
einsum::einsum('ij,ij->ij', arrC, arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.6873445 0.1574331270 0.173003993 0.003008926
## [2,] 0.5435359 0.1739698433 0.007479815 0.152796252
## [3,] 0.6054440 0.0004179006 0.536359956 0.141541972
DelayedTensor::einsum('ij,ij->ij', darrC, darrC)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.6873445394 0.1574331270 0.1730039933 0.0030089258
## [2,] 0.5435359201 0.1739698433 0.0074798148 0.1527962520
## [3,] 0.6054439761 0.0004179006 0.5363599564 0.1415419723
einsum::einsum('ijk,ijk->ijk', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1453879 0.03315927 0.0871090323 0.0004848728
## [2,] 0.2451393 0.86458864 0.0008075832 0.0260972892
## [3,] 0.9177970 0.63514138 0.6240924296 0.8867651118
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 3.114611e-06 0.09304736 0.3994082 0.421865026
## [2,] 4.703581e-03 0.03758110 0.8547559 0.817510874
## [3,] 4.573718e-01 0.93764228 0.4265359 0.005633881
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4435234 0.80542276 0.3842885 0.01389213
## [2,] 0.2205556 0.05236496 0.7304182 0.30859012
## [3,] 0.1152802 0.00680069 0.2300045 0.42967284
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.9885807 0.02065745 0.14321277 0.2531936
## [2,] 0.7484741 0.25182559 0.47304061 0.9628224
## [3,] 0.2061419 0.51001104 0.04002736 0.2545050
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.0015156174 0.2443402080 0.2283971 0.3007475
## [2,] 0.0066026690 0.2317554480 0.1850947 0.8363865
## [3,] 0.0003311362 0.0004613333 0.7958239 0.9816080
DelayedTensor::einsum('ijk,ijk->ijk', darrE, darrE)
## <3 x 4 x 5> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.1453879260 0.0331592720 0.0871090323 0.0004848728
## [2,] 0.2451393138 0.8645886394 0.0008075832 0.0260972892
## [3,] 0.9177969702 0.6351413793 0.6240924296 0.8867651118
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 3.114611e-06 9.304736e-02 3.994082e-01 4.218650e-01
## [2,] 4.703581e-03 3.758110e-02 8.547559e-01 8.175109e-01
## [3,] 4.573718e-01 9.376423e-01 4.265359e-01 5.633881e-03
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.44352338 0.80542276 0.38428848 0.01389213
## [2,] 0.22055563 0.05236496 0.73041817 0.30859012
## [3,] 0.11528021 0.00680069 0.23000447 0.42967284
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.98858073 0.02065745 0.14321277 0.25319362
## [2,] 0.74847412 0.25182559 0.47304061 0.96282242
## [3,] 0.20614189 0.51001104 0.04002736 0.25450503
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.0015156174 0.2443402080 0.2283970582 0.3007474840
## [2,] 0.0066026690 0.2317554480 0.1850947266 0.8363865200
## [3,] 0.0003311362 0.0004613333 0.7958238926 0.9816079810
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,] 0.06915333 0.08838252 0.1599130
## [2,] 0.08838252 0.11295868 0.2043794
## [3,] 0.15991303 0.20437939 0.3697895
DelayedTensor::einsum('i,j->ij', darrA, darrA)
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.06915333 0.08838252 0.15991303
## [2,] 0.08838252 0.11295868 0.20437939
## [3,] 0.15991303 0.20437939 0.36978951
einsum::einsum('ij,klm->ijklm', arrC, arrE)
## , , 1, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3161196 0.151290700 0.15859600 0.02091558
## [2,] 0.2811113 0.159038092 0.03297688 0.14904607
## [3,] 0.2966888 0.007794722 0.27924946 0.14345206
##
## , , 2, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4104816 0.19645114 0.20593708 0.0271589
## [2,] 0.3650233 0.20651113 0.04282052 0.1935365
## [3,] 0.3852507 0.01012146 0.36260572 0.1862727
##
## , , 3, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7942561 0.38012057 0.3984753 0.05255077
## [2,] 0.7062971 0.39958603 0.0828550 0.37448089
## [3,] 0.7454359 0.01958438 0.7016192 0.36042585
##
## , , 1, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1509697 0.072252113 0.07574092 0.009988683
## [2,] 0.1342507 0.075952046 0.01574882 0.071180141
## [3,] 0.1416901 0.003722537 0.13336156 0.068508604
##
## , , 2, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7708893 0.36893752 0.38675223 0.05100474
## [2,] 0.6855180 0.38783031 0.08041743 0.36346376
## [3,] 0.7235053 0.01900821 0.68097777 0.34982221
##
## , , 3, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.6607276 0.3162156 0.33148453 0.04371605
## [2,] 0.5875561 0.3324086 0.06892561 0.31152403
## [3,] 0.6201149 0.0162919 0.58366463 0.29983189
##
## , , 1, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2446915 0.117106137 0.12276079 0.01618965
## [2,] 0.2175934 0.123102984 0.02552566 0.11536869
## [3,] 0.2296511 0.006033483 0.21615225 0.11103866
##
## , , 2, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02356030 0.0112756531 0.011820115 0.001558832
## [2,] 0.02095115 0.0118530641 0.002457758 0.011108361
## [3,] 0.02211213 0.0005809385 0.020812384 0.010691442
##
## , , 3, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.6549554 0.31345306 0.32858862 0.04333414
## [2,] 0.5824231 0.32950457 0.06832346 0.30880250
## [3,] 0.6146975 0.01614957 0.57856563 0.29721251
##
## , , 1, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01825581 0.0087369931 0.009158872 0.001207869
## [2,] 0.01623409 0.0091844027 0.001904405 0.008607366
## [3,] 0.01713369 0.0004501429 0.016126574 0.008284314
##
## , , 2, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1339322 0.064098189 0.06719327 0.008861422
## [2,] 0.1191000 0.067380571 0.01397150 0.063147193
## [3,] 0.1256998 0.003302434 0.11831120 0.060777148
##
## , , 3, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7807132 0.37363914 0.39168087 0.05165472
## [2,] 0.6942541 0.39277269 0.08144224 0.36809562
## [3,] 0.7327255 0.01925045 0.68965593 0.35428023
##
## , , 1, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.001463151 7.002449e-04 0.0007340573 9.680719e-05
## [2,] 0.001301116 7.361035e-04 0.0001526326 6.898557e-04
## [3,] 0.001373216 3.607766e-05 0.0012924986 6.639640e-04
##
## , , 2, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.05685931 0.027212121 0.028526100 0.003762011
## [2,] 0.05056249 0.028605617 0.005931435 0.026808387
## [3,] 0.05336436 0.001402009 0.050227609 0.025802213
##
## , , 3, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5606888 0.26833834 0.28129547 0.03709714
## [2,] 0.4985960 0.28207959 0.05848979 0.26435713
## [3,] 0.5262252 0.01382519 0.49529375 0.25443526
##
## , , 1, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2528944 0.121031966 0.12687618 0.01673238
## [2,] 0.2248879 0.127229850 0.02638138 0.11923627
## [3,] 0.2373499 0.006235747 0.22339847 0.11476109
##
## , , 2, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1607208 0.076918857 0.08063300 0.01063385
## [2,] 0.1429219 0.080857768 0.01676603 0.07577765
## [3,] 0.1508418 0.003962974 0.14197535 0.07293356
##
## , , 3, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8027972 0.38420822 0.40276030 0.05311587
## [2,] 0.7138923 0.40388300 0.08374599 0.37850789
## [3,] 0.7534520 0.01979498 0.70916413 0.36430171
##
## , , 1, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5239571 0.25075901 0.26286729 0.03466684
## [2,] 0.4659321 0.26360004 0.05465802 0.24703861
## [3,] 0.4917512 0.01291948 0.46284615 0.23776674
##
## , , 2, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7664932 0.36683361 0.38454673 0.05071388
## [2,] 0.6816088 0.38561866 0.07995884 0.36139106
## [3,] 0.7193795 0.01889981 0.67709441 0.34782731
##
## , , 3, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5414584 0.25913488 0.27164761 0.03582478
## [2,] 0.4814952 0.27240483 0.05648371 0.25529021
## [3,] 0.5081768 0.01335102 0.47830618 0.24570865
##
## , , 1, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5384855 0.25771211 0.27015613 0.03562809
## [2,] 0.4788515 0.27090920 0.05617359 0.25388855
## [3,] 0.5053866 0.01327771 0.47568005 0.24435959
##
## , , 2, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7496077 0.35875241 0.37607532 0.04959667
## [2,] 0.6665932 0.37712364 0.07819738 0.35342976
## [3,] 0.7035318 0.01848346 0.66217830 0.34016482
##
## , , 3, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06222875 0.029781865 0.031219928 0.004117272
## [2,] 0.05533730 0.031306954 0.006491563 0.029340005
## [3,] 0.05840376 0.001534406 0.054970793 0.028238814
##
## , , 1, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5521353 0.26424472 0.27700418 0.03653121
## [2,] 0.4909897 0.27777634 0.05759751 0.26032424
## [3,] 0.5181974 0.01361428 0.48773782 0.25055374
##
## , , 2, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3893555 0.186340447 0.19533818 0.02576112
## [2,] 0.3462368 0.195882689 0.04061669 0.18357580
## [3,] 0.3654231 0.009600538 0.34394361 0.17668582
##
## , , 3, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2814911 0.134717943 0.14122300 0.01862444
## [2,] 0.2503177 0.141616666 0.02936451 0.13271919
## [3,] 0.2641888 0.006940869 0.24865979 0.12773797
##
## , , 1, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7440450 0.3560902 0.37328455 0.04922862
## [2,] 0.6616466 0.3743251 0.07761709 0.35080704
## [3,] 0.6983111 0.0183463 0.65726442 0.33764053
##
## , , 2, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1897176 0.090796360 0.09518060 0.01255238
## [2,] 0.1687075 0.095445919 0.01979091 0.08944925
## [3,] 0.1780563 0.004677964 0.16759017 0.08609204
##
## , , 3, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06836971 0.032720849 0.034300825 0.00452358
## [2,] 0.06079819 0.034396439 0.007132174 0.03223538
## [3,] 0.06416726 0.001685827 0.060395512 0.03102552
##
## , , 1, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5139442 0.24596694 0.25784383 0.03400435
## [2,] 0.4570280 0.25856258 0.05361349 0.24231764
## [3,] 0.4823538 0.01267258 0.45400105 0.23322296
##
## , , 2, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7085541 0.33910473 0.35547892 0.04688042
## [2,] 0.6300861 0.35646982 0.07391477 0.33407358
## [3,] 0.6650017 0.01747118 0.62591298 0.32153511
##
## , , 3, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3976082 0.19029010 0.19947855 0.02630715
## [2,] 0.3535756 0.20003460 0.04147759 0.18746685
## [3,] 0.3731686 0.00980403 0.35123381 0.18043084
##
## , , 1, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09771734 0.046766241 0.04902442 0.006465322
## [2,] 0.08689575 0.049161077 0.01019365 0.046072391
## [3,] 0.09171099 0.002409466 0.08632023 0.044343199
##
## , , 2, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4605516 0.22041394 0.2310570 0.0304717
## [2,] 0.4095483 0.23170104 0.0480437 0.2171438
## [3,] 0.4322430 0.01135605 0.4068358 0.2089939
##
## , , 3, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5434457 0.26008602 0.27264467 0.03595627
## [2,] 0.4832625 0.27340467 0.05669103 0.25622724
## [3,] 0.5100420 0.01340002 0.48006177 0.24661050
##
## , , 1, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8243152 0.39450647 0.4135558 0.05453958
## [2,] 0.7330274 0.41470861 0.0859907 0.38865335
## [3,] 0.7736474 0.02032556 0.7281724 0.37406639
##
## , , 2, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7172584 0.34327048 0.35984582 0.04745633
## [2,] 0.6378264 0.36084889 0.07482278 0.33817753
## [3,] 0.6731710 0.01768581 0.63360204 0.32548503
##
## , , 3, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3764180 0.180148723 0.18884748 0.02490513
## [2,] 0.3347320 0.189373895 0.03926707 0.17747594
## [3,] 0.3532809 0.009281531 0.33251504 0.17081490
##
## , , 1, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1191587 0.057027776 0.05978145 0.007883955
## [2,] 0.1059626 0.059948092 0.01243036 0.056181681
## [3,] 0.1118344 0.002938156 0.10526077 0.054073067
##
## , , 2, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4160420 0.19911225 0.20872669 0.0275268
## [2,] 0.3699679 0.20930852 0.04340056 0.1961581
## [3,] 0.3904693 0.01025856 0.36751757 0.1887959
##
## , , 3, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5920754 0.28335955 0.29704199 0.03917378
## [2,] 0.5265067 0.29787001 0.06176397 0.27915547
## [3,] 0.5556826 0.01459911 0.52301960 0.26867819
##
## , , 1, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3137459 0.150154699 0.15740515 0.02075853
## [2,] 0.2790005 0.157843918 0.03272927 0.14792692
## [3,] 0.2944610 0.007736194 0.27715265 0.14237492
##
## , , 2, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5702121 0.27289607 0.28607327 0.03772723
## [2,] 0.5070647 0.28687070 0.05948324 0.26884723
## [3,] 0.5351631 0.01406001 0.50370631 0.25875684
##
## , , 3, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1658692 0.079382827 0.08321595 0.01097449
## [2,] 0.1475002 0.083447914 0.01730310 0.07820506
## [3,] 0.1556738 0.004089922 0.14652329 0.07526986
##
## , , 1, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4171705 0.19965236 0.20929287 0.02760146
## [2,] 0.3709715 0.20987628 0.04351829 0.19669020
## [3,] 0.3915285 0.01028639 0.36851447 0.18930801
##
## , , 2, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8135052 0.38933295 0.40813248 0.05382436
## [2,] 0.7234145 0.40927016 0.08486303 0.38355659
## [3,] 0.7635018 0.02005901 0.71862326 0.36916092
##
## , , 3, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4182495 0.20016873 0.20983419 0.02767285
## [2,] 0.3719309 0.21041910 0.04363084 0.19719892
## [3,] 0.3925411 0.01031299 0.36946760 0.18979764
##
## , , 1, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.03227617 0.0154469538 0.016192833 0.002135505
## [2,] 0.02870179 0.0162379714 0.003366977 0.015217774
## [3,] 0.03029227 0.0007958501 0.028511690 0.014646620
##
## , , 2, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.06736697 0.032240950 0.03379775 0.004457235
## [2,] 0.05990649 0.033891965 0.00702757 0.031762605
## [3,] 0.06322615 0.001661102 0.05950972 0.030570489
##
## , , 3, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01508657 0.0072202360 0.007568876 0.0009981805
## [2,] 0.01341583 0.0075899745 0.001573797 0.0071131125
## [3,] 0.01415925 0.0003719973 0.013326973 0.0068461428
##
## , , 1, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4098120 0.19613068 0.20560115 0.0271146
## [2,] 0.3644279 0.20617427 0.04275067 0.1932208
## [3,] 0.3846223 0.01010494 0.36201423 0.1859688
##
## , , 2, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3991188 0.191013049 0.20023641 0.0264071
## [2,] 0.3549189 0.200794569 0.04163518 0.1881791
## [3,] 0.3745864 0.009841277 0.35256821 0.1811163
##
## , , 3, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01780716 0.0085222738 0.008933785 0.001178184
## [2,] 0.01583513 0.0089586879 0.001857603 0.008395833
## [3,] 0.01671261 0.0004390802 0.015730249 0.008080720
##
## , , 1, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3962164 0.189624004 0.19878029 0.02621507
## [2,] 0.3523379 0.199334393 0.04133241 0.18681064
## [3,] 0.3718624 0.009769711 0.35000434 0.17979925
##
## , , 2, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3566845 0.170704545 0.17894727 0.0235995
## [2,] 0.3171839 0.179446094 0.03720852 0.1681719
## [3,] 0.3347603 0.008794953 0.31508316 0.1618600
##
## , , 3, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7395980 0.35396192 0.37105352 0.0489344
## [2,] 0.6576921 0.37208784 0.07715319 0.3487103
## [3,] 0.6941374 0.01823665 0.65333611 0.3356225
##
## , , 1, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4546616 0.21759508 0.22810199 0.0300820
## [2,] 0.4043106 0.22873783 0.04742927 0.2143667
## [3,] 0.4267151 0.01121082 0.40163280 0.2063211
##
## , , 2, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.7582122 0.36287042 0.38039218 0.05016597
## [2,] 0.6742448 0.38145253 0.07909498 0.35748668
## [3,] 0.7116075 0.01869563 0.66977925 0.34406947
##
## , , 3, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.8214030 0.39311272 0.41209477 0.0543469
## [2,] 0.7304377 0.41324350 0.08568691 0.3872803
## [3,] 0.7709142 0.02025375 0.72559990 0.3727449
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,] 0.316119593 0.151290700 0.158596002 0.020915580
## [2,] 0.281111295 0.159038092 0.032976882 0.149046067
## [3,] 0.296688800 0.007794722 0.279249461 0.143452061
##
## ,,2,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.41048163 0.19645114 0.20593708 0.02715890
## [2,] 0.36502332 0.20651113 0.04282052 0.19353648
## [3,] 0.38525072 0.01012146 0.36260572 0.18627265
##
## ,,3,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.79425609 0.38012057 0.39847527 0.05255077
## [2,] 0.70629712 0.39958603 0.08285500 0.37448089
## [3,] 0.74543588 0.01958438 0.70161923 0.36042585
##
## ...
##
## ,,1,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.45466157 0.21759508 0.22810199 0.03008200
## [2,] 0.40431060 0.22873783 0.04742927 0.21436671
## [3,] 0.42671507 0.01121082 0.40163280 0.20632109
##
## ,,2,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.75821218 0.36287042 0.38039218 0.05016597
## [2,] 0.67424485 0.38145253 0.07909498 0.35748668
## [3,] 0.71160746 0.01869563 0.66977925 0.34406947
##
## ,,3,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.82140300 0.39311272 0.41209477 0.05434690
## [2,] 0.73043768 0.41324350 0.08568691 0.38728029
## [3,] 0.77091416 0.02025375 0.72559990 0.37274486
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] 1.207167
DelayedTensor::einsum('i->', darrA)
## <1> HDF5Array object of type "double":
## [1]
## 1.207167
einsum::einsum('ij->', arrC)
## [1] 5.235487
DelayedTensor::einsum('ij->', darrC)
## <1> HDF5Array object of type "double":
## [1]
## 5.235487
einsum::einsum('ijk->', arrE)
## [1] 30.31907
DelayedTensor::einsum('ijk->', darrE)
## <1> HDF5Array object of type "double":
## [1]
## 30.31907
einsum::einsum('ij->i', arrC)
## [1] 1.696632 1.631723 1.907132
DelayedTensor::einsum('ij->i', darrC)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 1.696632 1.631723 1.907132
einsum::einsum('ij->j', arrC)
## [1] 2.3444141 0.8343180 1.2347894 0.8219659
DelayedTensor::einsum('ij->j', darrC)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 2.3444141 0.8343180 1.2347894 0.8219659
einsum::einsum('ijk->i', arrE)
## [1] 8.349232 10.758084 11.211757
DelayedTensor::einsum('ijk->i', darrE)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 8.349232 10.758084 11.211757
einsum::einsum('ijk->j', arrE)
## [1] 6.508040 6.941751 8.343821 8.525461
DelayedTensor::einsum('ijk->j', darrE)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 6.508040 6.941751 8.343821 8.525461
einsum::einsum('ijk->k', arrE)
## [1] 5.982121 6.052203 5.966903 6.928332 5.389514
DelayedTensor::einsum('ijk->k', darrE)
## <5> HDF5Array object of type "double":
## [1] [2] [3] [4] [5]
## 5.982121 6.052203 5.966903 6.928332 5.389514
These are the same as what the modeSum
function does.
einsum::einsum('ijk->ij', arrE)
## [,1] [,2] [,3] [,4]
## [1,] 2.082243 2.022622 2.403384 1.840984
## [2,] 1.979732 2.335757 2.925599 3.516995
## [3,] 2.446065 2.583372 3.014838 3.167482
DelayedTensor::einsum('ijk->ij', darrE)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 2.082243 2.022622 2.403384 1.840984
## [2,] 1.979732 2.335757 2.925599 3.516995
## [3,] 2.446065 2.583372 3.014838 3.167482
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.834430 0.7466401 1.475138 2.313446 0.1383849
## [2,] 1.908887 1.4672144 1.208754 1.359700 0.9971963
## [3,] 1.113556 2.2096148 1.954143 1.266283 1.8002250
## [4,] 1.125248 1.6287336 1.328868 1.988904 2.4537075
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.8344303 0.7466401 1.4751384 2.3134465 0.1383849
## [2,] 1.9088870 1.4672144 1.2087537 1.3596998 0.9971963
## [3,] 1.1135556 2.2096148 1.9541429 1.2662825 1.8002250
## [4,] 1.1252484 1.6287336 1.3288681 1.9889035 2.4537075
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.834430 0.7466401 1.475138 2.313446 0.1383849
## [2,] 1.908887 1.4672144 1.208754 1.359700 0.9971963
## [3,] 1.113556 2.2096148 1.954143 1.266283 1.8002250
## [4,] 1.125248 1.6287336 1.328868 1.988904 2.4537075
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.8344303 0.7466401 1.4751384 2.3134465 0.1383849
## [2,] 1.9088870 1.4672144 1.2087537 1.3596998 0.9971963
## [3,] 1.1135556 2.2096148 1.9541429 1.2662825 1.8002250
## [4,] 1.1252484 1.6287336 1.3288681 1.9889035 2.4537075
If we take the diagonal elements of a matrix
and add them together, we get trace
.
einsum::einsum('ii->', arrB)
## [1] 1.976324
DelayedTensor::einsum('ii->', darrB)
## <1> HDF5Array object of type "double":
## [1]
## 1.976324
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,] 0.4200485 0.8837633 0.6857667
## [2,] 0.5573708 0.9008985 0.1107852
## [3,] 0.1543078 0.3227742 0.6553774
DelayedTensor::einsum('ij->ji', darrB)
## <3 x 3> DelayedArray object of type "double":
## [,1] [,2] [,3]
## [1,] 0.4200485 0.8837633 0.6857667
## [2,] 0.5573708 0.9008985 0.1107852
## [3,] 0.1543078 0.3227742 0.6553774
einsum::einsum('ijk->jki', arrD)
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.7135298 0.10558662 0.14789722
## [2,] 0.7324247 0.91817988 0.02967713
## [3,] 0.5964449 0.03823761 0.41915364
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.8250967 0.8156074 0.4303639
## [2,] 0.7977488 0.5345016 0.3899781
## [3,] 0.5893655 0.5896236 0.1305676
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.19081569 0.1247063 0.4195805
## [2,] 0.03621812 0.6413666 0.6800148
## [3,] 0.26878162 0.6911105 0.7826825
DelayedTensor::einsum('ijk->jki', darrD)
## <3 x 3 x 3> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3]
## [1,] 0.71352982 0.10558662 0.14789722
## [2,] 0.73242472 0.91817988 0.02967713
## [3,] 0.59644491 0.03823761 0.41915364
##
## ,,2
## [,1] [,2] [,3]
## [1,] 0.8250967 0.8156074 0.4303639
## [2,] 0.7977488 0.5345016 0.3899781
## [3,] 0.5893655 0.5896236 0.1305676
##
## ,,3
## [,1] [,2] [,3]
## [1,] 0.19081569 0.12470633 0.41958055
## [2,] 0.03621812 0.64136657 0.68001477
## [3,] 0.26878162 0.69111047 0.78268250
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] 0.5519015
DelayedTensor::einsum('i,i->', darrA, darrA)
## <1> HDF5Array object of type "double":
## [1]
## 0.5519015
einsum::einsum('ij,ij->', arrC, arrC)
## [1] 3.182336
DelayedTensor::einsum('ij,ij->', darrC, darrC)
## <1> HDF5Array object of type "double":
## [1]
## 3.182336
einsum::einsum('ijk,ijk->', arrE, arrE)
## [1] 21.329
DelayedTensor::einsum('ijk,ijk->', darrE, darrE)
## <1> HDF5Array object of type "double":
## [1]
## 21.329
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,] 1.3083242 0.4620785 0.7793592 1.9431967 0.008449423
## [2,] 1.5328893 1.0682707 0.8645884 0.7824941 0.476556989
## [3,] 0.7120090 1.6807000 1.3447111 0.6562807 1.209315677
## [4,] 0.9133473 1.2450098 0.7521551 1.4705211 2.118741985
DelayedTensor::einsum('ijk,ijk->jk', darrE, darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.308324210 0.462078453 0.779359221 1.943196735 0.008449423
## [2,] 1.532889291 1.068270738 0.864588404 0.782494081 0.476556989
## [3,] 0.712009045 1.680700039 1.344711128 0.656280744 1.209315677
## [4,] 0.913347274 1.245009781 0.752155081 1.470521058 2.118741985
Matrix Multiplication is considered a contracted product.
einsum::einsum('ij,jk->ik', arrC, t(arrC))
## [,1] [,2] [,3]
## [1,] 1.0207906 0.8341350 0.9784626
## [2,] 0.8341350 0.8777818 0.7925828
## [3,] 0.9784626 0.7925828 1.2837638
DelayedTensor::einsum('ij,jk->ik', darrC, t(darrC))
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 1.0207906 0.8341350 0.9784626
## [2,] 0.8341350 0.8777818 0.7925828
## [3,] 0.9784626 0.7925828 1.2837638
Some examples of combining Multiplication and Permutation are shown below.
einsum::einsum('ij,ij->ji', arrC, arrC)
## [,1] [,2] [,3]
## [1,] 0.687344539 0.543535920 0.6054439761
## [2,] 0.157433127 0.173969843 0.0004179006
## [3,] 0.173003993 0.007479815 0.5363599564
## [4,] 0.003008926 0.152796252 0.1415419723
DelayedTensor::einsum('ij,ij->ji', darrC, darrC)
## <4 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.6873445394 0.5435359201 0.6054439761
## [2,] 0.1574331270 0.1739698433 0.0004179006
## [3,] 0.1730039933 0.0074798148 0.5363599564
## [4,] 0.0030089258 0.1527962520 0.1415419723
einsum::einsum('ijk,ijk->jki', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.1453879260 3.114611e-06 0.44352338 0.98858073 0.001515617
## [2,] 0.0331592720 9.304736e-02 0.80542276 0.02065745 0.244340208
## [3,] 0.0871090323 3.994082e-01 0.38428848 0.14321277 0.228397058
## [4,] 0.0004848728 4.218650e-01 0.01389213 0.25319362 0.300747484
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2451393138 0.004703581 0.22055563 0.7484741 0.006602669
## [2,] 0.8645886394 0.037581103 0.05236496 0.2518256 0.231755448
## [3,] 0.0008075832 0.854755915 0.73041817 0.4730406 0.185094727
## [4,] 0.0260972892 0.817510874 0.30859012 0.9628224 0.836386520
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.9177970 0.457371757 0.11528021 0.20614189 0.0003311362
## [2,] 0.6351414 0.937642275 0.00680069 0.51001104 0.0004613333
## [3,] 0.6240924 0.426535944 0.23000447 0.04002736 0.7958238926
## [4,] 0.8867651 0.005633881 0.42967284 0.25450503 0.9816079810
DelayedTensor::einsum('ijk,ijk->jki', darrE, darrE)
## <4 x 5 x 3> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.453879e-01 3.114611e-06 4.435234e-01 9.885807e-01 1.515617e-03
## [2,] 3.315927e-02 9.304736e-02 8.054228e-01 2.065745e-02 2.443402e-01
## [3,] 8.710903e-02 3.994082e-01 3.842885e-01 1.432128e-01 2.283971e-01
## [4,] 4.848728e-04 4.218650e-01 1.389213e-02 2.531936e-01 3.007475e-01
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2451393138 0.0047035814 0.2205556273 0.7484741168 0.0066026690
## [2,] 0.8645886394 0.0375811031 0.0523649571 0.2518255895 0.2317554480
## [3,] 0.0008075832 0.8547559146 0.7304181733 0.4730406126 0.1850947266
## [4,] 0.0260972892 0.8175108745 0.3085901167 0.9628224151 0.8363865200
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.9177969702 0.4573717572 0.1152802108 0.2061418898 0.0003311362
## [2,] 0.6351413793 0.9376422751 0.0068006902 0.5100110398 0.0004613333
## [3,] 0.6240924296 0.4265359438 0.2300044710 0.0400273646 0.7958238926
## [4,] 0.8867651118 0.0056338808 0.4296728359 0.2545050263 0.9816079810
Some examples of combining Summation and Permutation are shown below.
einsum::einsum('ijk->ki', arrE)
## [,1] [,2] [,3]
## [1,] 0.8805568 1.614913 3.486652
## [2,] 1.5883003 2.091134 2.372768
## [3,] 2.3012039 2.108621 1.557078
## [4,] 2.0196191 3.035981 1.872732
## [5,] 1.5595523 1.907435 1.922527
DelayedTensor::einsum('ijk->ki', darrE)
## <5 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.8805568 1.6149126 3.4866521
## [2,] 1.5883003 2.0911343 2.3727683
## [3,] 2.3012039 2.1086213 1.5570779
## [4,] 2.0196191 3.0359809 1.8727324
## [5,] 1.5595523 1.9074347 1.9225268
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,] 2.627903e-02 5.629695e-07 8.016736e-02 0.1786870933 0.0002739496
## [2,] 1.372801e-03 3.852181e-03 3.334468e-02 0.0008552231 0.0101157391
## [3,] 3.963016e-03 1.817103e-02 1.748316e-02 0.0065154501 0.0103909006
## [4,] 3.836594e-07 3.338040e-04 1.099226e-05 0.0002003414 0.0002379688
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 4.478175e-02 0.0008592444 0.040290829 0.136730326 0.0012061674
## [2,] 5.055258e-02 0.0021973706 0.003061784 0.014724266 0.0135507626
## [3,] 2.030196e-06 0.0021487841 0.001836210 0.001189184 0.0004653125
## [4,] 1.340195e-03 0.0419822830 0.015847273 0.049444582 0.0429516190
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3379075341 0.1683916679 4.244299e-02 0.0758957590 1.219152e-04
## [2,] 0.0001614064 0.0002382799 1.728237e-06 0.0001296074 1.172371e-07
## [3,] 0.2035553654 0.1391199056 7.501877e-02 0.0130554136 2.595677e-01
## [4,] 0.0763257594 0.0004849201 3.698285e-02 0.0219057890 8.448909e-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,] 2.627903e-02 5.629695e-07 8.016736e-02 1.786871e-01 2.739496e-04
## [2,] 1.372801e-03 3.852181e-03 3.334468e-02 8.552231e-04 1.011574e-02
## [3,] 3.963016e-03 1.817103e-02 1.748316e-02 6.515450e-03 1.039090e-02
## [4,] 3.836594e-07 3.338040e-04 1.099226e-05 2.003414e-04 2.379688e-04
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 4.478175e-02 8.592444e-04 4.029083e-02 1.367303e-01 1.206167e-03
## [2,] 5.055258e-02 2.197371e-03 3.061784e-03 1.472427e-02 1.355076e-02
## [3,] 2.030196e-06 2.148784e-03 1.836210e-03 1.189184e-03 4.653125e-04
## [4,] 1.340195e-03 4.198228e-02 1.584727e-02 4.944458e-02 4.295162e-02
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.379075e-01 1.683917e-01 4.244299e-02 7.589576e-02 1.219152e-04
## [2,] 1.614064e-04 2.382799e-04 1.728237e-06 1.296074e-04 1.172371e-07
## [3,] 2.035554e-01 1.391199e-01 7.501877e-02 1.305541e-02 2.595677e-01
## [4,] 7.632576e-02 4.849201e-04 3.698285e-02 2.190579e-02 8.448909e-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 Under development (unstable) (2023-10-22 r85388)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 22.04.3 LTS
##
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.19-bioc/R/lib/libRblas.so
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: America/New_York
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats4 stats graphics grDevices utils datasets methods
## [8] base
##
## other attached packages:
## [1] einsum_0.1.2 DelayedRandomArray_1.11.0
## [3] HDF5Array_1.31.0 rhdf5_2.47.0
## [5] DelayedArray_0.29.0 SparseArray_1.3.0
## [7] S4Arrays_1.3.0 abind_1.4-5
## [9] IRanges_2.37.0 S4Vectors_0.41.0
## [11] MatrixGenerics_1.15.0 matrixStats_1.0.0
## [13] BiocGenerics_0.49.0 Matrix_1.6-1.1
## [15] DelayedTensor_1.9.0 BiocStyle_2.31.0
##
## loaded via a namespace (and not attached):
## [1] jsonlite_1.8.7 compiler_4.4.0 BiocManager_1.30.22
## [4] crayon_1.5.2 rsvd_1.0.5 Rcpp_1.0.11
## [7] rhdf5filters_1.15.0 parallel_4.4.0 jquerylib_0.1.4
## [10] BiocParallel_1.37.0 yaml_2.3.7 fastmap_1.1.1
## [13] lattice_0.22-5 R6_2.5.1 XVector_0.43.0
## [16] ScaledMatrix_1.11.0 knitr_1.44 bookdown_0.36
## [19] bslib_0.5.1 rlang_1.1.1 cachem_1.0.8
## [22] xfun_0.40 sass_0.4.7 cli_3.6.1
## [25] Rhdf5lib_1.25.0 BiocSingular_1.19.0 zlibbioc_1.49.0
## [28] digest_0.6.33 grid_4.4.0 irlba_2.3.5.1
## [31] rTensor_1.4.8 dqrng_0.3.1 evaluate_0.22
## [34] codetools_0.2-19 beachmat_2.19.0 rmarkdown_2.25
## [37] tools_4.4.0 htmltools_0.5.6.1