DelayedTensor 1.11.1
Authors: Koki Tsuyuzaki [aut, cre]
Last modified: 2024-05-29 20:36:38.968287
Compiled: Mon Sep 23 17:27:21 2024
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] 0.9986035 0.3563665 0.0455688
DelayedTensor::einsum('i->i', darrA)
## <3> DelayedArray object of type "double":
## [1] [2] [3]
## 0.9986035 0.3563665 0.0455688
einsum::einsum('ij->ij', arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.5645706 0.9975146 0.5309865 0.9773843
## [2,] 0.9096224 0.6687564 0.4300111 0.3405332
## [3,] 0.6854187 0.4040910 0.1880326 0.3590817
DelayedTensor::einsum('ij->ij', darrC)
## <3 x 4> DelayedArray object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.5645706 0.9975146 0.5309865 0.9773843
## [2,] 0.9096224 0.6687564 0.4300111 0.3405332
## [3,] 0.6854187 0.4040910 0.1880326 0.3590817
einsum::einsum('ijk->ijk', arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4823575 0.9242748 0.9012269 0.1836925
## [2,] 0.8287693 0.9516573 0.7710769 0.9402339
## [3,] 0.2433127 0.7061122 0.5342773 0.4857524
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3995430 0.8461261 0.2899889 0.93657130
## [2,] 0.3180940 0.6333167 0.5679007 0.55570426
## [3,] 0.5662099 0.2212268 0.3372739 0.05183431
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4307567 0.1653138 0.41341954 0.2270863
## [2,] 0.2675543 0.4600728 0.03773321 0.5851081
## [3,] 0.6184921 0.1911211 0.88326801 0.8083793
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.04907958 0.5507723 0.8841174 0.45801578
## [2,] 0.89345536 0.9487095 0.3253036 0.42293704
## [3,] 0.79382128 0.9892043 0.6624371 0.02003801
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3341407 0.1736710 0.8399110 0.6721724
## [2,] 0.1280181 0.8036354 0.3361732 0.7209466
## [3,] 0.6370535 0.1364255 0.6662824 0.3765825
DelayedTensor::einsum('ijk->ijk', darrE)
## <3 x 4 x 5> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.4823575 0.9242748 0.9012269 0.1836925
## [2,] 0.8287693 0.9516573 0.7710769 0.9402339
## [3,] 0.2433127 0.7061122 0.5342773 0.4857524
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 0.39954304 0.84612615 0.28998889 0.93657130
## [2,] 0.31809401 0.63331674 0.56790074 0.55570426
## [3,] 0.56620993 0.22122683 0.33727385 0.05183431
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.43075670 0.16531381 0.41341954 0.22708625
## [2,] 0.26755432 0.46007283 0.03773321 0.58510810
## [3,] 0.61849206 0.19112109 0.88326801 0.80837928
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.04907958 0.55077234 0.88411736 0.45801578
## [2,] 0.89345536 0.94870950 0.32530356 0.42293704
## [3,] 0.79382128 0.98920427 0.66243708 0.02003801
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.3341407 0.1736710 0.8399110 0.6721724
## [2,] 0.1280181 0.8036354 0.3361732 0.7209466
## [3,] 0.6370535 0.1364255 0.6662824 0.3765825
We can also extract the diagonal elements as follows.
einsum::einsum('ii->i', arrB)
## [1] 0.62250098 0.17311354 0.06617263
DelayedTensor::einsum('ii->i', darrB)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.62250098 0.17311354 0.06617263
einsum::einsum('iii->i', arrD)
## [1] 0.03777065 0.87545184 0.87662636
DelayedTensor::einsum('iii->i', darrD)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.03777065 0.87545184 0.87662636
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] 0.997208950 0.126997103 0.002076515
DelayedTensor::einsum('i,i->i', darrA, darrA)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 0.997208950 0.126997103 0.002076515
einsum::einsum('ij,ij->ij', arrC, arrC)
## [,1] [,2] [,3] [,4]
## [1,] 0.3187399 0.9950355 0.28194671 0.9552801
## [2,] 0.8274130 0.4472352 0.18490956 0.1159629
## [3,] 0.4697988 0.1632895 0.03535626 0.1289397
DelayedTensor::einsum('ij,ij->ij', darrC, darrC)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 0.31873994 0.99503545 0.28194671 0.95528007
## [2,] 0.82741298 0.44723515 0.18490956 0.11596287
## [3,] 0.46979877 0.16328952 0.03535626 0.12893968
einsum::einsum('ijk,ijk->ijk', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.23266876 0.8542840 0.8122099 0.03374295
## [2,] 0.68685853 0.9056516 0.5945596 0.88403975
## [3,] 0.05920108 0.4985944 0.2854523 0.23595538
##
## , , 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1596346 0.71592946 0.08409356 0.877165797
## [2,] 0.1011838 0.40109009 0.32251125 0.308807220
## [3,] 0.3205937 0.04894131 0.11375365 0.002686796
##
## , , 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.18555133 0.02732866 0.170915715 0.05156817
## [2,] 0.07158531 0.21166701 0.001423795 0.34235149
## [3,] 0.38253243 0.03652727 0.780162380 0.65347706
##
## , , 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.002408805 0.3033502 0.7816635 0.2097784515
## [2,] 0.798262488 0.9000497 0.1058224 0.1788757382
## [3,] 0.630152225 0.9785251 0.4388229 0.0004015217
##
## , , 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.11165000 0.03016160 0.7054506 0.4518157
## [2,] 0.01638863 0.64582978 0.1130124 0.5197640
## [3,] 0.40583719 0.01861192 0.4439323 0.1418144
DelayedTensor::einsum('ijk,ijk->ijk', darrE, darrE)
## <3 x 4 x 5> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4]
## [1,] 0.23266876 0.85428399 0.81220994 0.03374295
## [2,] 0.68685853 0.90565162 0.59455957 0.88403975
## [3,] 0.05920108 0.49859438 0.28545227 0.23595538
##
## ,,2
## [,1] [,2] [,3] [,4]
## [1,] 0.159634643 0.715929456 0.084093556 0.877165797
## [2,] 0.101183799 0.401090092 0.322511253 0.308807220
## [3,] 0.320593681 0.048941311 0.113753653 0.002686796
##
## ,,3
## [,1] [,2] [,3] [,4]
## [1,] 0.185551331 0.027328657 0.170915715 0.051568166
## [2,] 0.071585313 0.211667013 0.001423795 0.342351491
## [3,] 0.382532433 0.036527272 0.780162380 0.653477058
##
## ,,4
## [,1] [,2] [,3] [,4]
## [1,] 0.0024088051 0.3033501754 0.7816635099 0.2097784515
## [2,] 0.7982624884 0.9000497084 0.1058224040 0.1788757382
## [3,] 0.6301522251 0.9785250957 0.4388228799 0.0004015217
##
## ,,5
## [,1] [,2] [,3] [,4]
## [1,] 0.11165000 0.03016160 0.70545056 0.45181572
## [2,] 0.01638863 0.64582978 0.11301242 0.51976397
## [3,] 0.40583719 0.01861192 0.44393227 0.14181437
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,] 0.99720895 0.35586886 0.045505162
## [2,] 0.35586886 0.12699710 0.016239195
## [3,] 0.04550516 0.01623919 0.002076515
DelayedTensor::einsum('i,j->ij', darrA, darrA)
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.997208950 0.355868863 0.045505162
## [2,] 0.355868863 0.126997103 0.016239195
## [3,] 0.045505162 0.016239195 0.002076515
einsum::einsum('ij,klm->ijklm', arrC, arrE)
## , , 1, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2723249 0.4811587 0.25612534 0.4714486
## [2,] 0.4387632 0.3225797 0.20741909 0.1642587
## [3,] 0.3306168 0.1949163 0.09069893 0.1732058
##
## , , 2, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4678988 0.8267095 0.4400653 0.8100261
## [2,] 0.7538671 0.5542448 0.3563800 0.2822235
## [3,] 0.5680540 0.3348982 0.1558356 0.2975959
##
## , , 3, 1, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1373672 0.24270801 0.12919579 0.23781004
## [2,] 0.2213227 0.16271695 0.10462718 0.08285606
## [3,] 0.1667711 0.09832048 0.04575072 0.08736915
##
## , , 1, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5218184 0.9219777 0.4907775 0.9033717
## [2,] 0.8407411 0.6181147 0.3974485 0.3147463
## [3,] 0.6335152 0.3734911 0.1737938 0.3318902
##
## , , 2, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5372777 0.9492921 0.5053172 0.9301349
## [2,] 0.8656488 0.6364269 0.4092232 0.3240709
## [3,] 0.6522837 0.3845561 0.1789426 0.3417227
##
## , , 3, 2, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3986502 0.7043572 0.3749361 0.6901429
## [2,] 0.6422955 0.4722170 0.3036361 0.2404546
## [3,] 0.4839825 0.2853336 0.1327721 0.2535520
##
## , , 1, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5088062 0.8989870 0.4785394 0.8808450
## [2,] 0.8197762 0.6027013 0.3875376 0.3068977
## [3,] 0.6177178 0.3641777 0.1694600 0.3236141
##
## , , 2, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4353273 0.7691605 0.4094315 0.7536384
## [2,] 0.7013888 0.5156626 0.3315716 0.2625773
## [3,] 0.5285105 0.3115852 0.1449876 0.2768796
##
## , , 3, 3, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3016373 0.5329495 0.2836941 0.5221943
## [2,] 0.4859907 0.3573014 0.2297452 0.1819392
## [3,] 0.3662037 0.2158967 0.1004616 0.1918492
##
## , , 1, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1037074 0.1832360 0.09753827 0.17953820
## [2,] 0.1670909 0.1228456 0.07898983 0.06255341
## [3,] 0.1259063 0.0742285 0.03454018 0.06596063
##
## , , 2, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5308284 0.9378971 0.4992515 0.9189698
## [2,] 0.8552578 0.6287874 0.4043110 0.3201809
## [3,] 0.6444539 0.3799400 0.1767946 0.3376208
##
## , , 3, 4, 1
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2742415 0.4845451 0.25792798 0.4747668
## [2,] 0.4418513 0.3248500 0.20887893 0.1654148
## [3,] 0.3329438 0.1962882 0.09133728 0.1744248
##
## , , 1, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2255702 0.3985500 0.21215198 0.3905071
## [2,] 0.3634333 0.2671970 0.17180795 0.1360577
## [3,] 0.2738543 0.1614517 0.07512711 0.1434686
##
## , , 2, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1795865 0.3173034 0.16890364 0.3109001
## [2,] 0.2893454 0.2127274 0.13678396 0.1083216
## [3,] 0.2180276 0.1285389 0.05981204 0.1142217
##
## , , 3, 1, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3196655 0.5648027 0.3006499 0.5534047
## [2,] 0.5150373 0.3786565 0.2434766 0.1928133
## [3,] 0.3880909 0.2288003 0.1064659 0.2033156
##
## , , 1, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4776979 0.8440232 0.4492816 0.8269904
## [2,] 0.7696553 0.5658523 0.3638437 0.2881341
## [3,] 0.5799507 0.3419119 0.1590993 0.3038284
##
## , , 2, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3575520 0.6317427 0.3362827 0.6189938
## [2,] 0.5760791 0.4235346 0.2723332 0.2156654
## [3,] 0.4340871 0.2559176 0.1190842 0.2274125
##
## , , 3, 2, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1248982 0.22067700 0.11746847 0.21622363
## [2,] 0.2012329 0.14794687 0.09513000 0.07533508
## [3,] 0.1516330 0.08939577 0.04159785 0.07943851
##
## , , 1, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1637192 0.2892682 0.15398020 0.28343059
## [2,] 0.2637804 0.1939319 0.12469845 0.09875085
## [3,] 0.1987638 0.1171819 0.05452736 0.10412971
##
## , , 2, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3206201 0.5664893 0.3015477 0.5550573
## [2,] 0.5165753 0.3797873 0.2442036 0.1933891
## [3,] 0.3892498 0.2294836 0.1067838 0.2039228
##
## , , 3, 3, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1904149 0.3364356 0.17908788 0.3296462
## [2,] 0.3067919 0.2255541 0.14503151 0.1148529
## [3,] 0.2311738 0.1362893 0.06341848 0.1211089
##
## , , 1, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5287606 0.9342436 0.4973068 0.9153901
## [2,] 0.8519263 0.6263381 0.4027361 0.3189336
## [3,] 0.6419435 0.3784600 0.1761059 0.3363056
##
## , , 2, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3137343 0.5543231 0.2950715 0.5431366
## [2,] 0.5054811 0.3716308 0.2389590 0.1892358
## [3,] 0.3808901 0.2245551 0.1044905 0.1995432
##
## , , 3, 4, 2
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02926413 0.05170548 0.02752332 0.05066204
## [2,] 0.04714965 0.03466453 0.02228933 0.01765130
## [3,] 0.03552821 0.02094578 0.00974654 0.01861275
##
## , , 1, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2431926 0.4296861 0.2287260 0.4210148
## [2,] 0.3918260 0.2880713 0.1852302 0.1466870
## [3,] 0.2952487 0.1740649 0.0809963 0.1546769
##
## , , 2, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1510533 0.2668893 0.14206774 0.26150339
## [2,] 0.2433734 0.1789287 0.11505133 0.09111113
## [3,] 0.1833867 0.1081163 0.05030893 0.09607387
##
## , , 3, 1, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3491824 0.6169549 0.3284110 0.6045044
## [2,] 0.5625943 0.4136205 0.2659585 0.2106171
## [3,] 0.4239260 0.2499271 0.1162967 0.2220892
##
## , , 1, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09333132 0.16490295 0.08777941 0.16157513
## [2,] 0.15037315 0.11055467 0.07108678 0.05629484
## [3,] 0.11330918 0.06680182 0.03108438 0.05936117
##
## , , 2, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2597436 0.4589294 0.24429249 0.4496680
## [2,] 0.4184926 0.3076767 0.19783644 0.1566701
## [3,] 0.3153425 0.1859113 0.08650869 0.1652037
##
## , , 3, 2, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1079013 0.19064609 0.10148273 0.18679876
## [2,] 0.1738480 0.12781346 0.08218420 0.06508308
## [3,] 0.1309980 0.07723031 0.03593699 0.06862809
##
## , , 1, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2334045 0.4123920 0.21952021 0.4040698
## [2,] 0.3760557 0.2764770 0.17777500 0.1407831
## [3,] 0.2833655 0.1670591 0.07773635 0.1484514
##
## , , 2, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02130306 0.03763943 0.020035828 0.03687985
## [2,] 0.03432298 0.02523433 0.016225701 0.01284941
## [3,] 0.02586305 0.01524765 0.007095074 0.01354931
##
## , , 3, 3, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4986671 0.8810728 0.4690034 0.8632923
## [2,] 0.8034404 0.5906912 0.3798151 0.3007821
## [3,] 0.6054084 0.3569206 0.1660832 0.3171654
##
## , , 1, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1282062 0.22652186 0.12057974 0.22195054
## [2,] 0.2065628 0.15186539 0.09764961 0.07733041
## [3,] 0.1556492 0.09176351 0.04269962 0.08154252
##
## , , 2, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3303348 0.5836539 0.3106845 0.5718755
## [2,] 0.5322275 0.3912948 0.2516030 0.1992487
## [3,] 0.4010440 0.2364369 0.1100194 0.2101016
##
## , , 3, 4, 3
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4563872 0.8063702 0.4292385 0.7900972
## [2,] 0.7353199 0.5406088 0.3476121 0.2752800
## [3,] 0.5540783 0.3266588 0.1520017 0.2902742
##
## , , 1, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.02770889 0.04895760 0.02606060 0.04796961
## [2,] 0.04464389 0.03282228 0.02110476 0.01671323
## [3,] 0.03364006 0.01983262 0.00922856 0.01762358
##
## , , 2, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5044186 0.8912348 0.4744128 0.8732492
## [2,] 0.8127070 0.5975040 0.3841957 0.3042512
## [3,] 0.6123910 0.3610373 0.1679987 0.3208235
##
## , , 3, 1, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4481681 0.7918483 0.4215084 0.7758685
## [2,] 0.7220777 0.5308731 0.3413520 0.2703225
## [3,] 0.5440999 0.3207760 0.1492643 0.2850467
##
## , , 1, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3109499 0.5494035 0.2924527 0.5383162
## [2,] 0.5009949 0.3683325 0.2368382 0.1875563
## [3,] 0.3775097 0.2225621 0.1035632 0.1977723
##
## , , 2, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5356135 0.9463516 0.5037520 0.9272538
## [2,] 0.8629674 0.6344556 0.4079556 0.3230671
## [3,] 0.6502632 0.3833649 0.1783883 0.3406642
##
## , , 3, 2, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.5584756 0.9867457 0.5252542 0.9668327
## [2,] 0.8998024 0.6615367 0.4253688 0.3368569
## [3,] 0.6780191 0.3997285 0.1860026 0.3552052
##
## , , 1, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4991467 0.8819200 0.4694544 0.8641224
## [2,] 0.8042130 0.5912592 0.3801803 0.3010713
## [3,] 0.6059906 0.3572638 0.1662429 0.3174704
##
## , , 2, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1836568 0.3244951 0.17273181 0.3179466
## [2,] 0.2959034 0.2175488 0.13988415 0.1107767
## [3,] 0.2229691 0.1314522 0.06116767 0.1168106
##
## , , 3, 3, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3739925 0.6607907 0.3517452 0.6474556
## [2,] 0.6025676 0.4430091 0.2848553 0.2255818
## [3,] 0.4540467 0.2676848 0.1245598 0.2378690
##
## , , 1, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2585822 0.4568774 0.24320022 0.4476574
## [2,] 0.4166214 0.3063010 0.19695188 0.1559696
## [3,] 0.3139326 0.1850800 0.08612189 0.1644651
##
## , , 2, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2387778 0.4218859 0.22457388 0.4133720
## [2,] 0.3847130 0.2828419 0.18186763 0.1440241
## [3,] 0.2898889 0.1709050 0.07952595 0.1518690
##
## , , 3, 4, 4
##
## [,1] [,2] [,3] [,4]
## [1,] 0.01131287 0.019988204 0.010639911 0.019584832
## [2,] 0.01822702 0.013400545 0.008616565 0.006823606
## [3,] 0.01373442 0.008097177 0.003767798 0.007195282
##
## , , 1, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1886460 0.3333102 0.17742421 0.3265839
## [2,] 0.3039419 0.2234587 0.14368421 0.1137860
## [3,] 0.2290263 0.1350232 0.06282934 0.1199838
##
## , , 2, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.07227525 0.12769993 0.06797589 0.12512288
## [2,] 0.11644814 0.08561293 0.05504921 0.04359441
## [3,] 0.08774600 0.05173096 0.02407157 0.04596896
##
## , , 3, 1, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3596617 0.6354702 0.3382668 0.6226461
## [2,] 0.5794782 0.4260336 0.2739401 0.2169379
## [3,] 0.4366484 0.2574276 0.1197868 0.2287543
##
## , , 1, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.09804952 0.17323932 0.09221694 0.16974327
## [2,] 0.15797500 0.11614357 0.07468044 0.05914073
## [3,] 0.11903732 0.07017887 0.03265580 0.06236207
##
## , , 2, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4537089 0.8016380 0.4267196 0.7854606
## [2,] 0.7310048 0.5374363 0.3455721 0.2736645
## [3,] 0.5508267 0.3247418 0.1511096 0.2885708
##
## , , 3, 2, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.07702183 0.13608644 0.07244011 0.13334015
## [2,] 0.12409571 0.09123544 0.05866449 0.04645742
## [3,] 0.09350859 0.05512832 0.02565244 0.04898791
##
## , , 1, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4741891 0.8378236 0.4459815 0.8209159
## [2,] 0.7640019 0.5616959 0.3611711 0.2860176
## [3,] 0.5756907 0.3394005 0.1579307 0.3015967
##
## , , 2, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.1897935 0.3353377 0.17850344 0.3285704
## [2,] 0.3057907 0.2248180 0.14455821 0.1144781
## [3,] 0.2304194 0.1358446 0.06321152 0.1207137
##
## , , 3, 3, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3761635 0.6646265 0.3537870 0.6512140
## [2,] 0.6060654 0.4455807 0.2865089 0.2268913
## [3,] 0.4566824 0.2692387 0.1252828 0.2392498
##
## , , 1, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.3794888 0.6705018 0.3569145 0.6569707
## [2,] 0.6114231 0.4495196 0.2890416 0.2288970
## [3,] 0.4607195 0.2716188 0.1263903 0.2413648
##
## , , 2, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.4070252 0.7191548 0.3828129 0.7046419
## [2,] 0.6557892 0.4821377 0.3100150 0.2455063
## [3,] 0.4941503 0.2913280 0.1355615 0.2588787
##
## , , 3, 4, 5
##
## [,1] [,2] [,3] [,4]
## [1,] 0.2126074 0.3756465 0.19996023 0.3680658
## [2,] 0.3425479 0.2518420 0.16193466 0.1282388
## [3,] 0.2581167 0.1521736 0.07080978 0.1352239
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,] 0.27232486 0.48115867 0.25612534 0.47144865
## [2,] 0.43876321 0.32257968 0.20741909 0.16425875
## [3,] 0.33061684 0.19491631 0.09069893 0.17320576
##
## ,,2,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.4678988 0.8267095 0.4400653 0.8100261
## [2,] 0.7538671 0.5542448 0.3563800 0.2822235
## [3,] 0.5680540 0.3348982 0.1558356 0.2975959
##
## ,,3,1,1
## [,1] [,2] [,3] [,4]
## [1,] 0.13736721 0.24270801 0.12919579 0.23781004
## [2,] 0.22132272 0.16271695 0.10462718 0.08285606
## [3,] 0.16677109 0.09832048 0.04575072 0.08736915
##
## ...
##
## ,,1,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.3794888 0.6705018 0.3569145 0.6569707
## [2,] 0.6114231 0.4495196 0.2890416 0.2288970
## [3,] 0.4607195 0.2716188 0.1263903 0.2413648
##
## ,,2,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.4070252 0.7191548 0.3828129 0.7046419
## [2,] 0.6557892 0.4821377 0.3100150 0.2455063
## [3,] 0.4941503 0.2913280 0.1355615 0.2588787
##
## ,,3,4,5
## [,1] [,2] [,3] [,4]
## [1,] 0.21260739 0.37564654 0.19996023 0.36806581
## [2,] 0.34254788 0.25184196 0.16193466 0.12823884
## [3,] 0.25811667 0.15217358 0.07080978 0.13522389
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] 1.400539
DelayedTensor::einsum('i->', darrA)
## <1> HDF5Array object of type "double":
## [1]
## 1.400539
einsum::einsum('ij->', arrC)
## [1] 7.056003
DelayedTensor::einsum('ij->', darrC)
## <1> HDF5Array object of type "double":
## [1]
## 7.056003
einsum::einsum('ijk->', arrE)
## [1] 31.58774
DelayedTensor::einsum('ijk->', darrE)
## <1> HDF5Array object of type "double":
## [1]
## 31.58774
einsum::einsum('ij->i', arrC)
## [1] 3.070456 2.348923 1.636624
DelayedTensor::einsum('ij->i', darrC)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 3.070456 2.348923 1.636624
einsum::einsum('ij->j', arrC)
## [1] 2.159612 2.070362 1.149030 1.676999
DelayedTensor::einsum('ij->j', darrC)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 2.159612 2.070362 1.149030 1.676999
einsum::einsum('ijk->i', arrE)
## [1] 10.162238 11.496400 9.929105
DelayedTensor::einsum('ijk->i', darrE)
## <3> HDF5Array object of type "double":
## [1] [2] [3]
## 10.162238 11.496400 9.929105
einsum::einsum('ijk->j', arrE)
## [1] 6.990658 8.701640 8.450390 7.445055
DelayedTensor::einsum('ijk->j', darrE)
## <4> HDF5Array object of type "double":
## [1] [2] [3] [4]
## 6.990658 8.701640 8.450390 7.445055
einsum::einsum('ijk->k', arrE)
## [1] 7.952744 5.723790 5.088305 6.997891 5.825012
DelayedTensor::einsum('ijk->k', darrE)
## <5> HDF5Array object of type "double":
## [1] [2] [3] [4] [5]
## 7.952744 5.723790 5.088305 6.997891 5.825012
These are the same as what the modeSum
function does.
einsum::einsum('ijk->ij', arrE)
## [,1] [,2] [,3] [,4]
## [1,] 1.695878 2.660158 3.328664 2.477538
## [2,] 2.435891 3.797392 2.038188 3.224930
## [3,] 2.858890 2.244090 3.083539 1.742586
DelayedTensor::einsum('ijk->ij', darrE)
## <3 x 4> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4]
## [1,] 1.695878 2.660158 3.328664 2.477538
## [2,] 2.435891 3.797392 2.038188 3.224930
## [3,] 2.858890 2.244090 3.083539 1.742586
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.554440 1.283847 1.3168031 1.7363562 1.099212
## [2,] 2.582044 1.700670 0.8165077 2.4886861 1.113732
## [3,] 2.206581 1.195163 1.3344208 1.8718580 1.842367
## [4,] 1.609679 1.544110 1.6205736 0.9009908 1.769701
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.5544395 1.2838470 1.3168031 1.7363562 1.0992123
## [2,] 2.5820443 1.7006697 0.8165077 2.4886861 1.1137318
## [3,] 2.2065811 1.1951635 1.3344208 1.8718580 1.8423667
## [4,] 1.6096788 1.5441099 1.6205736 0.9009908 1.7697015
einsum::einsum('ijk->jk', arrE)
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.554440 1.283847 1.3168031 1.7363562 1.099212
## [2,] 2.582044 1.700670 0.8165077 2.4886861 1.113732
## [3,] 2.206581 1.195163 1.3344208 1.8718580 1.842367
## [4,] 1.609679 1.544110 1.6205736 0.9009908 1.769701
DelayedTensor::einsum('ijk->jk', darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.5544395 1.2838470 1.3168031 1.7363562 1.0992123
## [2,] 2.5820443 1.7006697 0.8165077 2.4886861 1.1137318
## [3,] 2.2065811 1.1951635 1.3344208 1.8718580 1.8423667
## [4,] 1.6096788 1.5441099 1.6205736 0.9009908 1.7697015
If we take the diagonal elements of a matrix
and add them together, we get trace
.
einsum::einsum('ii->', arrB)
## [1] 0.8617872
DelayedTensor::einsum('ii->', darrB)
## <1> HDF5Array object of type "double":
## [1]
## 0.8617872
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,] 0.622500978 0.5882665 0.93756379
## [2,] 0.377797139 0.1731135 0.85141194
## [3,] 0.004796433 0.4905398 0.06617263
DelayedTensor::einsum('ij->ji', darrB)
## <3 x 3> DelayedArray object of type "double":
## [,1] [,2] [,3]
## [1,] 0.622500978 0.588266494 0.937563786
## [2,] 0.377797139 0.173113545 0.851411941
## [3,] 0.004796433 0.490539761 0.066172630
einsum::einsum('ijk->jki', arrD)
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.03777065 0.76783791 0.4417801
## [2,] 0.48919245 0.09351242 0.4477087
## [3,] 0.11285556 0.58912246 0.4933397
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.2849856 0.3698813 0.4209580
## [2,] 0.7745016 0.8754518 0.6396971
## [3,] 0.5148250 0.4791874 0.7388468
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.5375775 0.5425131 0.9553911
## [2,] 0.5976889 0.3893509 0.3023950
## [3,] 0.6412208 0.9211341 0.8766264
DelayedTensor::einsum('ijk->jki', darrD)
## <3 x 3 x 3> DelayedArray object of type "double":
## ,,1
## [,1] [,2] [,3]
## [1,] 0.03777065 0.76783791 0.44178010
## [2,] 0.48919245 0.09351242 0.44770871
## [3,] 0.11285556 0.58912246 0.49333971
##
## ,,2
## [,1] [,2] [,3]
## [1,] 0.2849856 0.3698813 0.4209580
## [2,] 0.7745016 0.8754518 0.6396971
## [3,] 0.5148250 0.4791874 0.7388468
##
## ,,3
## [,1] [,2] [,3]
## [1,] 0.5375775 0.5425131 0.9553911
## [2,] 0.5976889 0.3893509 0.3023950
## [3,] 0.6412208 0.9211341 0.8766264
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] 1.126283
DelayedTensor::einsum('i,i->', darrA, darrA)
## <1> HDF5Array object of type "double":
## [1]
## 1.126283
einsum::einsum('ij,ij->', arrC, arrC)
## [1] 4.923907
DelayedTensor::einsum('ij,ij->', darrC, darrC)
## <1> HDF5Array object of type "double":
## [1]
## 4.923907
einsum::einsum('ijk,ijk->', arrE, arrE)
## [1] 21.38708
DelayedTensor::einsum('ijk,ijk->', darrE, darrE)
## <1> HDF5Array object of type "double":
## [1]
## 21.38708
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,] 0.9787284 0.5814121 0.6396691 1.4308235 0.5338758
## [2,] 2.2585300 1.1659609 0.2755229 2.1819250 0.6946033
## [3,] 1.6922218 0.5203585 0.9525019 1.3263088 1.2623952
## [4,] 1.1537381 1.1886598 1.0473967 0.3890557 1.1133941
DelayedTensor::einsum('ijk,ijk->jk', darrE, darrE)
## <4 x 5> HDF5Matrix object of type "double":
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.9787284 0.5814121 0.6396691 1.4308235 0.5338758
## [2,] 2.2585300 1.1659609 0.2755229 2.1819250 0.6946033
## [3,] 1.6922218 0.5203585 0.9525019 1.3263088 1.2623952
## [4,] 1.1537381 1.1886598 1.0473967 0.3890557 1.1133941
Matrix Multiplication is considered a contracted product.
einsum::einsum('ij,jk->ik', arrC, t(arrC))
## [,1] [,2] [,3]
## [1,] 2.551002 1.741802 1.2408575
## [2,] 1.741802 1.575521 1.0968460
## [3,] 1.240858 1.096846 0.7973842
DelayedTensor::einsum('ij,jk->ik', darrC, t(darrC))
## <3 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 2.5510022 1.7418023 1.2408575
## [2,] 1.7418023 1.5755206 1.0968460
## [3,] 1.2408575 1.0968460 0.7973842
Some examples of combining Multiplication and Permutation are shown below.
einsum::einsum('ij,ij->ji', arrC, arrC)
## [,1] [,2] [,3]
## [1,] 0.3187399 0.8274130 0.46979877
## [2,] 0.9950355 0.4472352 0.16328952
## [3,] 0.2819467 0.1849096 0.03535626
## [4,] 0.9552801 0.1159629 0.12893968
DelayedTensor::einsum('ij,ij->ji', darrC, darrC)
## <4 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 0.31873994 0.82741298 0.46979877
## [2,] 0.99503545 0.44723515 0.16328952
## [3,] 0.28194671 0.18490956 0.03535626
## [4,] 0.95528007 0.11596287 0.12893968
einsum::einsum('ijk,ijk->jki', arrE, arrE)
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.23266876 0.15963464 0.18555133 0.002408805 0.1116500
## [2,] 0.85428399 0.71592946 0.02732866 0.303350175 0.0301616
## [3,] 0.81220994 0.08409356 0.17091571 0.781663510 0.7054506
## [4,] 0.03374295 0.87716580 0.05156817 0.209778451 0.4518157
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6868585 0.1011838 0.071585313 0.7982625 0.01638863
## [2,] 0.9056516 0.4010901 0.211667013 0.9000497 0.64582978
## [3,] 0.5945596 0.3225113 0.001423795 0.1058224 0.11301242
## [4,] 0.8840397 0.3088072 0.342351491 0.1788757 0.51976397
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.05920108 0.320593681 0.38253243 0.6301522251 0.40583719
## [2,] 0.49859438 0.048941311 0.03652727 0.9785250957 0.01861192
## [3,] 0.28545227 0.113753653 0.78016238 0.4388228799 0.44393227
## [4,] 0.23595538 0.002686796 0.65347706 0.0004015217 0.14181437
DelayedTensor::einsum('ijk,ijk->jki', darrE, darrE)
## <4 x 5 x 3> HDF5Array object of type "double":
## ,,1
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.232668758 0.159634643 0.185551331 0.002408805 0.111649995
## [2,] 0.854283991 0.715929456 0.027328657 0.303350175 0.030161602
## [3,] 0.812209939 0.084093556 0.170915715 0.781663510 0.705450562
## [4,] 0.033742948 0.877165797 0.051568166 0.209778451 0.451815721
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.686858532 0.101183799 0.071585313 0.798262488 0.016388634
## [2,] 0.905651624 0.401090092 0.211667013 0.900049708 0.645829783
## [3,] 0.594559573 0.322511253 0.001423795 0.105822404 0.113012418
## [4,] 0.884039745 0.308807220 0.342351491 0.178875738 0.519763966
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0592010833 0.3205936811 0.3825324333 0.6301522251 0.4058371861
## [2,] 0.4985943797 0.0489413115 0.0365272725 0.9785250957 0.0186119199
## [3,] 0.2854522695 0.1137536529 0.7801623797 0.4388228799 0.4439322679
## [4,] 0.2359553846 0.0026867959 0.6534770579 0.0004015217 0.1418143680
Some examples of combining Summation and Permutation are shown below.
einsum::einsum('ijk->ki', arrE)
## [,1] [,2] [,3]
## [1,] 2.491552 3.491737 1.969455
## [2,] 2.472229 2.075016 1.176545
## [3,] 1.236576 1.350468 2.501260
## [4,] 1.941985 2.590405 2.465501
## [5,] 2.019895 1.988773 1.816344
DelayedTensor::einsum('ijk->ki', darrE)
## <5 x 3> HDF5Matrix object of type "double":
## [,1] [,2] [,3]
## [1,] 2.491552 3.491737 1.969455
## [2,] 2.472229 2.075016 1.176545
## [3,] 1.236576 1.350468 2.501260
## [4,] 1.941985 2.590405 2.465501
## [5,] 2.019895 1.988773 1.816344
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,] 0.07405726 0.05081088 0.05906003 0.0007667102 0.03553762
## [2,] 0.84885577 0.71138036 0.02715501 0.3014226533 0.02996995
## [3,] 0.22868012 0.02367679 0.04812183 0.2200796868 0.19862170
## [4,] 0.03218895 0.83676882 0.04919325 0.2001173195 0.43100781
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.20252868 0.02983529 2.110781e-02 0.235377508 0.004832390
## [2,] 0.14434243 0.06392559 3.373541e-02 0.143449598 0.102932118
## [3,] 0.03917885 0.02125207 9.382182e-05 0.006973229 0.007447019
## [4,] 0.03653319 0.01276155 1.414777e-02 0.007392091 0.021479394
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0012673866 6.863323e-03 0.008189318 1.349040e-02 0.0086882296
## [2,] 0.0037099945 3.641678e-04 0.000271796 7.281114e-03 0.0001384896
## [3,] 0.0004599042 1.832733e-04 0.001256952 7.070060e-04 0.0007152380
## [4,] 0.0013863857 1.578661e-05 0.003839587 2.359191e-06 0.0008332483
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,] 0.0740572613 0.0508108806 0.0590600282 0.0007667102 0.0355376155
## [2,] 0.8488557714 0.7113803573 0.0271550072 0.3014226533 0.0299699519
## [3,] 0.2286801249 0.0236767910 0.0481218280 0.2200796868 0.1986217046
## [4,] 0.0321889513 0.8367688229 0.0491932465 0.2001173195 0.4310078102
##
## ,,2
## [,1] [,2] [,3] [,4] [,5]
## [1,] 2.025287e-01 2.983529e-02 2.110781e-02 2.353775e-01 4.832390e-03
## [2,] 1.443424e-01 6.392559e-02 3.373541e-02 1.434496e-01 1.029321e-01
## [3,] 3.917885e-02 2.125207e-02 9.382182e-05 6.973229e-03 7.447019e-03
## [4,] 3.653319e-02 1.276155e-02 1.414777e-02 7.392091e-03 2.147939e-02
##
## ,,3
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.267387e-03 6.863323e-03 8.189318e-03 1.349040e-02 8.688230e-03
## [2,] 3.709994e-03 3.641678e-04 2.717960e-04 7.281114e-03 1.384896e-04
## [3,] 4.599042e-04 1.832733e-04 1.256952e-03 7.070060e-04 7.152380e-04
## [4,] 1.386386e-03 1.578661e-05 3.839587e-03 2.359191e-06 8.332483e-04
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 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.1 LTS
##
## Matrix products: default
## BLAS: /media/volume/teran2_disk/biocbuild/bbs-3.20-bioc/R/lib/libRblas.so
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_GB LC_COLLATE=C
## [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_1.13.1
## [3] HDF5Array_1.33.6 rhdf5_2.49.0
## [5] DelayedArray_0.31.11 SparseArray_1.5.39
## [7] S4Arrays_1.5.8 abind_1.4-8
## [9] IRanges_2.39.2 S4Vectors_0.43.2
## [11] MatrixGenerics_1.17.0 matrixStats_1.4.1
## [13] BiocGenerics_0.51.1 Matrix_1.7-0
## [15] DelayedTensor_1.11.1 BiocStyle_2.33.1
##
## loaded via a namespace (and not attached):
## [1] jsonlite_1.8.9 compiler_4.4.1 BiocManager_1.30.25
## [4] crayon_1.5.3 rsvd_1.0.5 Rcpp_1.0.13
## [7] rhdf5filters_1.17.0 parallel_4.4.1 jquerylib_0.1.4
## [10] BiocParallel_1.39.0 yaml_2.3.10 fastmap_1.2.0
## [13] lattice_0.22-6 R6_2.5.1 XVector_0.45.0
## [16] ScaledMatrix_1.13.0 knitr_1.48 bookdown_0.40
## [19] bslib_0.8.0 rlang_1.1.4 cachem_1.1.0
## [22] xfun_0.47 sass_0.4.9 cli_3.6.3
## [25] Rhdf5lib_1.27.0 BiocSingular_1.21.4 zlibbioc_1.51.1
## [28] digest_0.6.37 grid_4.4.1 irlba_2.3.5.1
## [31] rTensor_1.4.8 dqrng_0.4.1 lifecycle_1.0.4
## [34] evaluate_1.0.0 codetools_0.2-20 beachmat_2.21.6
## [37] rmarkdown_2.28 tools_4.4.1 htmltools_0.5.8.1