4. Einsum operation by DelayedTensor

Koki Tsuyuzaki1 and Itoshi Nikaido1*

1Laboratory for Bioinformatics Research, RIKEN Center for Biosystems Dynamics Research

*k.t.the-answer@hotmail.co.jp

Package

DelayedTensor 1.9.0

Authors: Koki Tsuyuzaki [aut, cre]
Last modified: 2023-10-20 14:48:21.036969
Compiled: Fri Apr 19 16:51:27 2024

1 What is einsum

einsum is an easy and intuitive way to write tensor operations.

It was originally introduced by Numpy111 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 summation222 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)

2 Einsum of 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.

3 Typical operations of einsum

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 operations333 https://ajcr.net/Basic-guide-to-einsum/.

1. Multiplication: the element values of the tensors on the left side of -> are multiplied by each other
2. Summation: when comparing the left and right sides of ->, if there is a missing subscript on the right, the summation is done in the direction of the subscript
3. Permutation: the subscripts to the right of -> can be rearranged in any order

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)

3.1 No Operation

3.1.1 print, show

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] 0.8132761 0.8440048 0.1464055

DelayedTensor::einsum('i->i', darrA)

## <3> DelayedArray object of type "double":
##       [1]       [2]       [3] 
## 0.8132761 0.8440048 0.1464055

einsum::einsum('ij->ij', arrC)

##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.5663208 0.1056013 0.08936181 0.5008443
## [2,] 0.2278386 0.9665401 0.73008887 0.5803139
## [3,] 0.5336488 0.7286874 0.22242787 0.8494069

DelayedTensor::einsum('ij->ij', darrC)

## <3 x 4> DelayedArray object of type "double":
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.56632083 0.10560132 0.08936181 0.50084428
## [2,] 0.22783857 0.96654014 0.73008887 0.58031387
## [3,] 0.53364880 0.72868742 0.22242787 0.84940690

einsum::einsum('ijk->ijk', arrE)

## , , 1
## 
##            [,1]        [,2]       [,3]      [,4]
## [1,] 0.07741433 0.087557147 0.08442988 0.1392366
## [2,] 0.21531437 0.004804786 0.65532761 0.3493701
## [3,] 0.62733777 0.083445474 0.62907620 0.7989620
## 
## , , 2
## 
##           [,1]      [,2]      [,3]      [,4]
## [1,] 0.3888160 0.9541006 0.1330694 0.2480035
## [2,] 0.1955486 0.1123012 0.9478725 0.1838444
## [3,] 0.2646498 0.8959485 0.8510886 0.1442961
## 
## , , 3
## 
##           [,1]      [,2]      [,3]       [,4]
## [1,] 0.1737995 0.3953730 0.9261464 0.98845158
## [2,] 0.3462747 0.5914846 0.1823917 0.51888564
## [3,] 0.4957807 0.5671990 0.9338794 0.08552079
## 
## , , 4
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.6829760 0.1354813 0.05485244 0.9319627
## [2,] 0.8869581 0.2921841 0.66250632 0.2972975
## [3,] 0.6665800 0.9109050 0.29974918 0.9381292
## 
## , , 5
## 
##           [,1]      [,2]       [,3]       [,4]
## [1,] 0.8813173 0.1932389 0.04159738 0.86795205
## [2,] 0.3189379 0.7790131 0.03497903 0.11582314
## [3,] 0.9607333 0.8514848 0.67973276 0.05060278

DelayedTensor::einsum('ijk->ijk', darrE)

## <3 x 4 x 5> DelayedArray object of type "double":
## ,,1
##             [,1]        [,2]        [,3]        [,4]
## [1,] 0.077414331 0.087557147 0.084429882 0.139236612
## [2,] 0.215314366 0.004804786 0.655327611 0.349370135
## [3,] 0.627337774 0.083445474 0.629076196 0.798962015
## 
## ,,2
##           [,1]      [,2]      [,3]      [,4]
## [1,] 0.3888160 0.9541006 0.1330694 0.2480035
## [2,] 0.1955486 0.1123012 0.9478725 0.1838444
## [3,] 0.2646498 0.8959485 0.8510886 0.1442961
## 
## ,,3
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.17379948 0.39537296 0.92614645 0.98845158
## [2,] 0.34627466 0.59148463 0.18239166 0.51888564
## [3,] 0.49578071 0.56719903 0.93387935 0.08552079
## 
## ,,4
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.68297596 0.13548128 0.05485244 0.93196275
## [2,] 0.88695810 0.29218414 0.66250632 0.29729747
## [3,] 0.66658002 0.91090503 0.29974918 0.93812924
## 
## ,,5
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.88131729 0.19323887 0.04159738 0.86795205
## [2,] 0.31893793 0.77901313 0.03497903 0.11582314
## [3,] 0.96073331 0.85148480 0.67973276 0.05060278

3.1.2 diag

We can also extract the diagonal elements as follows.

einsum::einsum('ii->i', arrB)

## [1] 0.3388087 0.9851086 0.2574171

DelayedTensor::einsum('ii->i', darrB)

## <3> HDF5Array object of type "double":
##       [1]       [2]       [3] 
## 0.3388087 0.9851086 0.2574171

einsum::einsum('iii->i', arrD)

## [1] 0.6119565 0.3330158 0.2592453

DelayedTensor::einsum('iii->i', darrD)

## <3> HDF5Array object of type "double":
##       [1]       [2]       [3] 
## 0.6119565 0.3330158 0.2592453

3.2 Multiplication

By using multiple arrays or DelayedArray objects as input and writing “,” on the right side of ->, multiplication will be performed.

3.2.1 Hadamard Product

Hadamard Product can also be implemented in einsum, multiplying by the product of each element.

einsum::einsum('i,i->i', arrA, arrA)

## [1] 0.66141799 0.71234415 0.02143458

DelayedTensor::einsum('i,i->i', darrA, darrA)

## <3> HDF5Array object of type "double":
##        [1]        [2]        [3] 
## 0.66141799 0.71234415 0.02143458

einsum::einsum('ij,ij->ij', arrC, arrC)

##            [,1]       [,2]        [,3]      [,4]
## [1,] 0.32071929 0.01115164 0.007985533 0.2508450
## [2,] 0.05191041 0.93419984 0.533029755 0.3367642
## [3,] 0.28478105 0.53098536 0.049474156 0.7214921

DelayedTensor::einsum('ij,ij->ij', darrC, darrC)

## <3 x 4> HDF5Matrix object of type "double":
##             [,1]        [,2]        [,3]        [,4]
## [1,] 0.320719287 0.011151639 0.007985533 0.250844993
## [2,] 0.051910415 0.934199837 0.533029755 0.336764183
## [3,] 0.284781045 0.530985356 0.049474156 0.721492089

einsum::einsum('ijk,ijk->ijk', arrE, arrE)

## , , 1
## 
##             [,1]         [,2]        [,3]       [,4]
## [1,] 0.005992979 7.666254e-03 0.007128405 0.01938683
## [2,] 0.046360276 2.308597e-05 0.429454277 0.12205949
## [3,] 0.393552682 6.963147e-03 0.395736860 0.63834030
## 
## , , 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.15117792 0.91030796 0.01770746 0.06150576
## [2,] 0.03823925 0.01261157 0.89846222 0.03379877
## [3,] 0.07003954 0.80272372 0.72435173 0.02082136
## 
## , , 3
## 
##            [,1]      [,2]       [,3]        [,4]
## [1,] 0.03020626 0.1563198 0.85774724 0.977036529
## [2,] 0.11990614 0.3498541 0.03326672 0.269242305
## [3,] 0.24579851 0.3217147 0.87213064 0.007313806
## 
## , , 4
## 
##           [,1]       [,2]       [,3]       [,4]
## [1,] 0.4664562 0.01835518 0.00300879 0.86855456
## [2,] 0.7866947 0.08537157 0.43891462 0.08838578
## [3,] 0.4443289 0.82974798 0.08984957 0.88008647
## 
## , , 5
## 
##           [,1]       [,2]        [,3]        [,4]
## [1,] 0.7767202 0.03734126 0.001730342 0.753340757
## [2,] 0.1017214 0.60686146 0.001223533 0.013415000
## [3,] 0.9230085 0.72502637 0.462036619 0.002560642

DelayedTensor::einsum('ijk,ijk->ijk', darrE, darrE)

## <3 x 4 x 5> HDF5Array object of type "double":
## ,,1
##              [,1]         [,2]         [,3]         [,4]
## [1,] 5.992979e-03 7.666254e-03 7.128405e-03 1.938683e-02
## [2,] 4.636028e-02 2.308597e-05 4.294543e-01 1.220595e-01
## [3,] 3.935527e-01 6.963147e-03 3.957369e-01 6.383403e-01
## 
## ,,2
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.15117792 0.91030796 0.01770746 0.06150576
## [2,] 0.03823925 0.01261157 0.89846222 0.03379877
## [3,] 0.07003954 0.80272372 0.72435173 0.02082136
## 
## ,,3
##             [,1]        [,2]        [,3]        [,4]
## [1,] 0.030206258 0.156319780 0.857747239 0.977036529
## [2,] 0.119906139 0.349854070 0.033266717 0.269242305
## [3,] 0.245798512 0.321714739 0.872130643 0.007313806
## 
## ,,4
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.46645616 0.01835518 0.00300879 0.86855456
## [2,] 0.78669467 0.08537157 0.43891462 0.08838578
## [3,] 0.44432892 0.82974798 0.08984957 0.88008647
## 
## ,,5
##             [,1]        [,2]        [,3]        [,4]
## [1,] 0.776720169 0.037341261 0.001730342 0.753340757
## [2,] 0.101721402 0.606861457 0.001223533 0.013415000
## [3,] 0.923008496 0.725026365 0.462036619 0.002560642

3.2.2 Outer Product

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,] 0.6614180 0.6864089 0.11906812
## [2,] 0.6864089 0.7123441 0.12356697
## [3,] 0.1190681 0.1235670 0.02143458

DelayedTensor::einsum('i,j->ij', darrA, darrA)

## <3 x 3> HDF5Matrix object of type "double":
##            [,1]       [,2]       [,3]
## [1,] 0.66141799 0.68640894 0.11906812
## [2,] 0.68640894 0.71234415 0.12356697
## [3,] 0.11906812 0.12356697 0.02143458

einsum::einsum('ij,klm->ijklm', arrC, arrE)

## , , 1, 1, 1
## 
##            [,1]        [,2]        [,3]       [,4]
## [1,] 0.04384135 0.008175055 0.006917885 0.03877252
## [2,] 0.01763797 0.074824058 0.056519341 0.04492461
## [3,] 0.04131206 0.056410849 0.017219104 0.06575627
## 
## , , 2, 1, 1
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.12193701 0.02273748 0.01924088 0.1078390
## [2,] 0.04905692 0.20810998 0.15719862 0.1249499
## [3,] 0.11490225 0.15689687 0.04789192 0.1828895
## 
## , , 3, 1, 1
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.3552745 0.0662477 0.05606004 0.3141985
## [2,] 0.1429317 0.6063471 0.45801233 0.3640528
## [3,] 0.3347781 0.4571331 0.13953740 0.5328650
## 
## , , 1, 2, 1
## 
##            [,1]       [,2]        [,3]       [,4]
## [1,] 0.04958544 0.00924615 0.007824265 0.04385250
## [2,] 0.01994890 0.08462750 0.063924498 0.05081063
## [3,] 0.04672477 0.06380179 0.019475150 0.07437165
## 
## , , 2, 2, 1
## 
##             [,1]         [,2]         [,3]        [,4]
## [1,] 0.002721050 0.0005073917 0.0004293644 0.002406450
## [2,] 0.001094716 0.0046440186 0.0035079208 0.002788284
## [3,] 0.002564068 0.0035011872 0.0010687183 0.004081218
## 
## , , 3, 2, 1
## 
##            [,1]        [,2]        [,3]       [,4]
## [1,] 0.04725691 0.008811952 0.007456838 0.04179319
## [2,] 0.01901210 0.080653400 0.060922612 0.04842457
## [3,] 0.04453058 0.060805667 0.018560599 0.07087916
## 
## , , 1, 3, 1
## 
##            [,1]        [,2]        [,3]       [,4]
## [1,] 0.04781440 0.008915907 0.007544807 0.04228622
## [2,] 0.01923638 0.081604869 0.061641317 0.04899583
## [3,] 0.04505591 0.061522993 0.018779559 0.07171532
## 
## , , 2, 3, 1
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3711257 0.06920346 0.05856126 0.3282171
## [2,] 0.1493089 0.63340044 0.47844739 0.3802957
## [3,] 0.3497148 0.47752899 0.14576312 0.5566398
## 
## , , 3, 3, 1
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3562590 0.06643128 0.05621539 0.3150692
## [2,] 0.1433278 0.60802739 0.45928153 0.3650616
## [3,] 0.3357058 0.45839991 0.13992408 0.5343417
## 
## , , 1, 4, 1
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.07885259 0.01470357 0.01244244 0.06973586
## [2,] 0.03172347 0.13457777 0.10165510 0.08080094
## [3,] 0.07430345 0.10145997 0.03097010 0.11826854
## 
## , , 2, 4, 1
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.19785559 0.03689395 0.03122035 0.1749800
## [2,] 0.07959999 0.33768026 0.25507125 0.2027443
## [3,] 0.18644095 0.25458162 0.07770965 0.2967574
## 
## , , 3, 4, 1
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4524688 0.08437144 0.07139669 0.4001556
## [2,] 0.1820344 0.77222886 0.58331327 0.4636487
## [3,] 0.4263651 0.58219357 0.17771142 0.6786439
## 
## , , 1, 1, 2
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.22019463 0.04105949 0.03474531 0.1947363
## [2,] 0.08858729 0.37580631 0.28387027 0.2256353
## [3,] 0.20749122 0.28332536 0.08648352 0.3302630
## 
## , , 2, 1, 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.11074324 0.02065019 0.01747458 0.09793939
## [2,] 0.04455351 0.18900555 0.14276784 0.11347955
## [3,] 0.10435427 0.14249379 0.04349545 0.16610032
## 
## , , 3, 1, 2
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.14987672 0.02794737 0.02364959 0.1325484
## [2,] 0.06029744 0.25579470 0.19321790 0.1535800
## [3,] 0.14123007 0.19284701 0.05886550 0.2247954
## 
## , , 1, 2, 2
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.5403270 0.1007543 0.08526016 0.4778558
## [2,] 0.2173809 0.9221765 0.69657823 0.5536778
## [3,] 0.5091546 0.6952411 0.21221856 0.8104196
## 
## , , 2, 2, 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.06359853 0.01185916 0.01003544 0.05624543
## [2,] 0.02558655 0.10854364 0.08198988 0.06516996
## [3,] 0.05992942 0.08183249 0.02497892 0.09538944
## 
## , , 3, 2, 2
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5073943 0.09461334 0.08006358 0.4487307
## [2,] 0.2041316 0.86597019 0.65412203 0.5199313
## [3,] 0.4781218 0.65286641 0.19928392 0.7610248
## 
## , , 1, 3, 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.07535996 0.01405230 0.01189132 0.06664703
## [2,] 0.03031833 0.12861689 0.09715247 0.07722200
## [3,] 0.07101231 0.09696598 0.02959834 0.11303004
## 
## , , 2, 3, 2
## 
##           [,1]      [,2]      [,3]      [,4]
## [1,] 0.5367999 0.1000966 0.0847036 0.4747365
## [2,] 0.2159619 0.9161568 0.6920311 0.5500635
## [3,] 0.5058310 0.6907027 0.2108333 0.8051294
## 
## , , 3, 3, 2
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4819892 0.08987607 0.07605481 0.4262628
## [2,] 0.1939108 0.82261125 0.62137028 0.4938985
## [3,] 0.4541824 0.62017752 0.18930581 0.7229205
## 
## , , 1, 4, 2
## 
##            [,1]      [,2]       [,3]      [,4]
## [1,] 0.14044958 0.0261895 0.02216205 0.1242112
## [2,] 0.05650477 0.2397054 0.18106463 0.1439199
## [3,] 0.13234680 0.1807171 0.05516290 0.2106559
## 
## , , 2, 4, 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.10411493 0.01941421 0.01642867 0.09207743
## [2,] 0.04188685 0.17769302 0.13422277 0.10668747
## [3,] 0.09810836 0.13396512 0.04089212 0.15615872
## 
## , , 3, 4, 2
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.08171787 0.01523786 0.01289456 0.07226986
## [2,] 0.03287621 0.13946794 0.10534895 0.08373701
## [3,] 0.07700342 0.10514673 0.03209547 0.12256608
## 
## , , 1, 1, 3
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.09842626 0.01835345 0.01553104 0.08704647
## [2,] 0.03959822 0.16798417 0.12688906 0.10085825
## [3,] 0.09274788 0.12664549 0.03865785 0.14762648
## 
## , , 2, 1, 3
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.19610255 0.03656706 0.03094373 0.1734297
## [2,] 0.07889472 0.33468836 0.25281127 0.2009480
## [3,] 0.18478906 0.25232599 0.07702113 0.2941281
## 
## , , 3, 1, 3
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.2807709 0.0523551 0.04430386 0.2483089
## [2,] 0.1129580 0.4791920 0.36196398 0.2877084
## [3,] 0.2645728 0.3612692 0.11027545 0.4211196
## 
## , , 1, 2, 3
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.22390795 0.04175191 0.03533124 0.1980203
## [2,] 0.09008121 0.38214384 0.28865740 0.2294404
## [3,] 0.21099031 0.28810330 0.08794197 0.3358325
## 
## , , 2, 2, 3
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3349701 0.06246156 0.05285614 0.2962417
## [2,] 0.1347630 0.57169364 0.43183635 0.3432467
## [3,] 0.3156451 0.43100741 0.13156267 0.5024111
## 
## , , 3, 2, 3
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3212166 0.05989697 0.05068593 0.2840784
## [2,] 0.1292298 0.54822063 0.41410570 0.3291535
## [3,] 0.3026851 0.41331080 0.12616087 0.4817828
## 
## , , 1, 3, 3
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5244960 0.09780229 0.08276212 0.4638552
## [2,] 0.2110119 0.89515771 0.67616921 0.5374556
## [3,] 0.4942369 0.67487126 0.20600078 0.7866752
## 
## , , 2, 3, 3
## 
##            [,1]      [,2]       [,3]       [,4]
## [1,] 0.10329220 0.0192608 0.01629885 0.09134982
## [2,] 0.04155585 0.1762889 0.13316212 0.10584441
## [3,] 0.09733309 0.1329065 0.04056899 0.15492473
## 
## , , 3, 3, 3
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5288753 0.09861889 0.08345315 0.4677281
## [2,] 0.2127737 0.90263188 0.68181492 0.5419431
## [3,] 0.4983636 0.68050614 0.20772079 0.7932436
## 
## , , 1, 4, 3
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.5597807 0.1043818 0.08832982 0.4950603
## [2,] 0.2252074 0.9553781 0.72165750 0.5736122
## [3,] 0.5274860 0.7202722 0.21985918 0.8395976
## 
## , , 2, 4, 3
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.2938557 0.05479501 0.04636856 0.2598809
## [2,] 0.1182222 0.50152380 0.37883263 0.3011165
## [3,] 0.2769027 0.37810544 0.11541463 0.4407450
## 
## , , 3, 4, 3
## 
##            [,1]        [,2]        [,3]       [,4]
## [1,] 0.04843221 0.009031108 0.007642293 0.04283260
## [2,] 0.01948494 0.082659278 0.062437778 0.04962890
## [3,] 0.04563807 0.062317926 0.019022207 0.07264195
## 
## , , 1, 1, 4
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3867835 0.07212316 0.06103197 0.3420646
## [2,] 0.1556083 0.66012367 0.49863314 0.3963404
## [3,] 0.3644693 0.49767599 0.15191289 0.5801245
## 
## , , 2, 1, 4
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5023029 0.09366395 0.07926018 0.4442279
## [2,] 0.2020833 0.85728060 0.64755823 0.5147141
## [3,] 0.4733241 0.64631521 0.19728420 0.7533883
## 
## , , 3, 1, 4
## 
##           [,1]       [,2]      [,3]      [,4]
## [1,] 0.3774982 0.07039173 0.0595668 0.3338528
## [2,] 0.1518726 0.64427634 0.4866626 0.3868256
## [3,] 0.3557196 0.48572847 0.1482660 0.5661977
## 
## , , 1, 2, 4
## 
##            [,1]      [,2]       [,3]       [,4]
## [1,] 0.07672587 0.0143070 0.01210685 0.06785502
## [2,] 0.03086786 0.1309481 0.09891337 0.07862167
## [3,] 0.07229942 0.0987235 0.03013481 0.11507874
## 
## , , 2, 2, 4
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.16546996 0.03085503 0.02611010 0.1463388
## [2,] 0.06657082 0.28240770 0.21332039 0.1695585
## [3,] 0.15592372 0.21291091 0.06498989 0.2481832
## 
## , , 3, 2, 4
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5158645 0.09619277 0.08140012 0.4562216
## [2,] 0.2075393 0.88042627 0.66504162 0.5286108
## [3,] 0.4861034 0.66376504 0.20261066 0.7737290
## 
## , , 1, 3, 4
## 
##            [,1]       [,2]        [,3]       [,4]
## [1,] 0.03106408 0.00579249 0.004901713 0.02747253
## [2,] 0.01249750 0.05301708 0.040047154 0.03183163
## [3,] 0.02927194 0.03997028 0.012200711 0.04659204
## 
## , , 2, 3, 4
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3751911 0.06996154 0.05920276 0.3318125
## [2,] 0.1509445 0.64033895 0.48368849 0.3844616
## [3,] 0.3535457 0.48276002 0.14735987 0.5627374
## 
## , , 3, 3, 4
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.16975421 0.03165391 0.02678613 0.1501277
## [2,] 0.06829443 0.28971962 0.21884354 0.1739486
## [3,] 0.15996079 0.21842346 0.06667257 0.2546090
## 
## , , 1, 4, 4
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.5277899 0.0984165 0.08328188 0.4667682
## [2,] 0.2123371 0.9007794 0.68041563 0.5408309
## [3,] 0.4973408 0.6791095 0.20729449 0.7916156
## 
## , , 2, 4, 4
## 
##            [,1]      [,2]       [,3]      [,4]
## [1,] 0.16836575 0.0313950 0.02656704 0.1488997
## [2,] 0.06773583 0.2873499 0.21705357 0.1725258
## [3,] 0.15865244 0.2166369 0.06612724 0.2525265
## 
## , , 3, 4, 4
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.5312821 0.09906769 0.08383293 0.4698567
## [2,] 0.2137420 0.90673956 0.68491771 0.5444094
## [3,] 0.5006315 0.68360297 0.20866609 0.7968535
## 
## , , 1, 1, 5
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4991083 0.09306827 0.07875611 0.4414027
## [2,] 0.2007981 0.85182854 0.64343994 0.5114406
## [3,] 0.4703139 0.64220482 0.19602953 0.7485970
## 
## , , 2, 1, 5
## 
##            [,1]       [,2]       [,3]      [,4]
## [1,] 0.18062119 0.03368027 0.02850087 0.1597382
## [2,] 0.07266636 0.30826631 0.23285303 0.1850841
## [3,] 0.17020084 0.23240606 0.07094068 0.2709081
## 
## , , 3, 1, 5
## 
##           [,1]      [,2]       [,3]      [,4]
## [1,] 0.5440833 0.1014547 0.08585287 0.4811778
## [2,] 0.2188921 0.9285873 0.70142070 0.5575269
## [3,] 0.5126942 0.7000743 0.21369386 0.8160535
## 
## , , 1, 2, 5
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.10943520 0.02040628 0.01726818 0.09678258
## [2,] 0.04402727 0.18677312 0.14108155 0.11213920
## [3,] 0.10312169 0.14081073 0.04298171 0.16413843
## 
## , , 2, 2, 5
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4411714 0.08226481 0.06961402 0.3901643
## [2,] 0.1774892 0.75294746 0.56874881 0.4520721
## [3,] 0.4157194 0.56765707 0.17327423 0.6616991
## 
## , , 3, 2, 5
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4822136 0.08991792 0.07609022 0.4264613
## [2,] 0.1940011 0.82299424 0.62165957 0.4941284
## [3,] 0.4543938 0.62046626 0.18939395 0.7232571
## 
## , , 1, 3, 5
## 
##             [,1]        [,2]        [,3]       [,4]
## [1,] 0.023557463 0.004392738 0.003717217 0.02083381
## [2,] 0.009477488 0.040205537 0.030369784 0.02413954
## [3,] 0.022198392 0.030311487 0.009252416 0.03533310
## 
## , , 2, 3, 5
## 
##             [,1]        [,2]        [,3]       [,4]
## [1,] 0.019809355 0.003693832 0.003125790 0.01751905
## [2,] 0.007969573 0.033808640 0.025537803 0.02029882
## [3,] 0.018666519 0.025488782 0.007780312 0.02971143
## 
## , , 3, 3, 5
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.3849468 0.07178068 0.06074215 0.3404403
## [2,] 0.1548693 0.65698899 0.49626532 0.3944583
## [3,] 0.3627386 0.49531271 0.15119151 0.5773697
## 
## , , 1, 4, 5
## 
##           [,1]       [,2]       [,3]      [,4]
## [1,] 0.4915393 0.09165688 0.07756176 0.4347088
## [2,] 0.1977530 0.83891049 0.63368213 0.5036846
## [3,] 0.4631816 0.63246574 0.19305672 0.7372445
## 
## , , 2, 4, 5
## 
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.06559306 0.01223108 0.01035017 0.05800936
## [2,] 0.02638898 0.11194772 0.08456119 0.06721378
## [3,] 0.06180888 0.08439887 0.02576229 0.09838098
## 
## , , 3, 4, 5
## 
##            [,1]        [,2]        [,3]       [,4]
## [1,] 0.02865741 0.005343721 0.004521956 0.02534411
## [2,] 0.01152927 0.048909620 0.036944528 0.02936550
## [3,] 0.02700411 0.036873611 0.011255469 0.04298235

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,] 0.043841348 0.008175055 0.006917885 0.038772525
## [2,] 0.017637971 0.074824058 0.056519341 0.044924609
## [3,] 0.041312065 0.056410849 0.017219104 0.065756267
## 
## ,,2,1,1
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.12193701 0.02273748 0.01924088 0.10783897
## [2,] 0.04905692 0.20810998 0.15719862 0.12494991
## [3,] 0.11490225 0.15689687 0.04789192 0.18288951
## 
## ,,3,1,1
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.35527445 0.06624770 0.05606004 0.31419854
## [2,] 0.14293174 0.60634714 0.45801233 0.36405281
## [3,] 0.33477805 0.45713314 0.13953740 0.53286504
## 
## ...
## 
## ,,1,4,5
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.49153933 0.09165688 0.07756176 0.43470882
## [2,] 0.19775295 0.83891049 0.63368213 0.50368461
## [3,] 0.46318157 0.63246574 0.19305672 0.73724446
## 
## ,,2,4,5
##            [,1]       [,2]       [,3]       [,4]
## [1,] 0.06559306 0.01223108 0.01035017 0.05800936
## [2,] 0.02638898 0.11194772 0.08456119 0.06721378
## [3,] 0.06180888 0.08439887 0.02576229 0.09838098
## 
## ,,3,4,5
##             [,1]        [,2]        [,3]        [,4]
## [1,] 0.028657410 0.005343721 0.004521956 0.025344114
## [2,] 0.011529266 0.048909620 0.036944528 0.029365496
## [3,] 0.027004114 0.036873611 0.011255469 0.042982353

3.3 Summation

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] 1.803686

DelayedTensor::einsum('i->', darrA)

## <1> HDF5Array object of type "double":
##      [1] 
## 1.803686

einsum::einsum('ij->', arrC)

## [1] 6.101081

DelayedTensor::einsum('ij->', darrC)

## <1> HDF5Array object of type "double":
##      [1] 
## 6.101081

einsum::einsum('ijk->', arrE)

## [1] 27.812

DelayedTensor::einsum('ijk->', darrE)

## <1> HDF5Array object of type "double":
##    [1] 
## 27.812

3.3.1 Row-wise / Column-wise Summation

einsum::einsum('ij->i', arrC)

## [1] 1.262128 2.504781 2.334171

DelayedTensor::einsum('ij->i', darrC)

## <3> HDF5Array object of type "double":
##      [1]      [2]      [3] 
## 1.262128 2.504781 2.334171

einsum::einsum('ij->j', arrC)

## [1] 1.327808 1.800829 1.041879 1.930565

DelayedTensor::einsum('ij->j', darrC)

## <4> HDF5Array object of type "double":
##      [1]      [2]      [3]      [4] 
## 1.327808 1.800829 1.041879 1.930565

3.3.2 Mode-wise Vectorization

einsum::einsum('ijk->i', arrE)

## [1]  8.385776  7.691119 11.735101

DelayedTensor::einsum('ijk->i', darrE)

## <3> HDF5Array object of type "double":
##       [1]       [2]       [3] 
##  8.385776  7.691119 11.735101

einsum::einsum('ijk->j', arrE)

## [1] 7.182438 6.854522 7.116699 6.658338

DelayedTensor::einsum('ijk->j', darrE)

## <4> HDF5Array object of type "double":
##      [1]      [2]      [3]      [4] 
## 7.182438 6.854522 7.116699 6.658338

einsum::einsum('ijk->k', arrE)

## [1] 3.752276 5.319539 6.205187 6.759582 5.775412

DelayedTensor::einsum('ijk->k', darrE)

## <5> HDF5Array object of type "double":
##      [1]      [2]      [3]      [4]      [5] 
## 3.752276 5.319539 6.205187 6.759582 5.775412

3.3.3 Mode-wise Summation

These are the same as what the modeSum function does.

einsum::einsum('ijk->ij', arrE)

##          [,1]     [,2]     [,3]     [,4]
## [1,] 2.204323 1.765751 1.240096 3.175607
## [2,] 1.963034 1.779788 2.483077 1.465221
## [3,] 3.015082 3.308983 3.393526 2.017511

DelayedTensor::einsum('ijk->ij', darrE)

## <3 x 4> HDF5Matrix object of type "double":
##          [,1]     [,2]     [,3]     [,4]
## [1,] 2.204323 1.765751 1.240096 3.175607
## [2,] 1.963034 1.779788 2.483077 1.465221
## [3,] 3.015082 3.308983 3.393526 2.017511

einsum::einsum('ijk->jk', arrE)

##           [,1]      [,2]     [,3]     [,4]      [,5]
## [1,] 0.9200665 0.8490145 1.015855 2.236514 2.1609885
## [2,] 0.1758074 1.9623503 1.554057 1.338570 1.8237368
## [3,] 1.3688337 1.9320304 2.042417 1.017108 0.7563092
## [4,] 1.2875688 0.5761440 1.592858 2.167389 1.0343780

DelayedTensor::einsum('ijk->jk', darrE)

## <4 x 5> HDF5Matrix object of type "double":
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9200665 0.8490145 1.0158548 2.2365141 2.1609885
## [2,] 0.1758074 1.9623503 1.5540566 1.3385704 1.8237368
## [3,] 1.3688337 1.9320304 2.0424175 1.0171079 0.7563092
## [4,] 1.2875688 0.5761440 1.5928580 2.1673895 1.0343780

einsum::einsum('ijk->jk', arrE)

##           [,1]      [,2]     [,3]     [,4]      [,5]
## [1,] 0.9200665 0.8490145 1.015855 2.236514 2.1609885
## [2,] 0.1758074 1.9623503 1.554057 1.338570 1.8237368
## [3,] 1.3688337 1.9320304 2.042417 1.017108 0.7563092
## [4,] 1.2875688 0.5761440 1.592858 2.167389 1.0343780

DelayedTensor::einsum('ijk->jk', darrE)

## <4 x 5> HDF5Matrix object of type "double":
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] 0.9200665 0.8490145 1.0158548 2.2365141 2.1609885
## [2,] 0.1758074 1.9623503 1.5540566 1.3385704 1.8237368
## [3,] 1.3688337 1.9320304 2.0424175 1.0171079 0.7563092
## [4,] 1.2875688 0.5761440 1.5928580 2.1673895 1.0343780

3.3.4 Trace

If we take the diagonal elements of a matrix and add them together, we get trace.

einsum::einsum('ii->', arrB)

## [1] 1.581334

DelayedTensor::einsum('ii->', darrB)

## <1> HDF5Array object of type "double":
##      [1] 
## 1.581334

3.4 Permutation

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,] 0.3388087 0.07643278 0.4705341
## [2,] 0.4566356 0.98510864 0.9706010
## [3,] 0.5833659 0.23154603 0.2574171

DelayedTensor::einsum('ij->ji', darrB)

## <3 x 3> DelayedArray object of type "double":
##            [,1]       [,2]       [,3]
## [1,] 0.33880871 0.07643278 0.47053410
## [2,] 0.45663561 0.98510864 0.97060103
## [3,] 0.58336587 0.23154603 0.25741709

einsum::einsum('ijk->jki', arrD)

## , , 1
## 
##           [,1]       [,2]      [,3]
## [1,] 0.6119565 0.94279386 0.6368407
## [2,] 0.9846637 0.96663758 0.6357265
## [3,] 0.8821443 0.06458412 0.5043059
## 
## , , 2
## 
##           [,1]      [,2]      [,3]
## [1,] 0.2691625 0.2219062 0.1964229
## [2,] 0.9319940 0.3330158 0.7411989
## [3,] 0.5088740 0.7530398 0.5271144
## 
## , , 3
## 
##            [,1]      [,2]       [,3]
## [1,] 0.27711173 0.4775506 0.20419603
## [2,] 0.03461886 0.2658436 0.07866031
## [3,] 0.26164131 0.2748289 0.25924531

DelayedTensor::einsum('ijk->jki', darrD)

## <3 x 3 x 3> DelayedArray object of type "double":
## ,,1
##            [,1]       [,2]       [,3]
## [1,] 0.61195647 0.94279386 0.63684072
## [2,] 0.98466372 0.96663758 0.63572653
## [3,] 0.88214425 0.06458412 0.50430585
## 
## ,,2
##           [,1]      [,2]      [,3]
## [1,] 0.2691625 0.2219062 0.1964229
## [2,] 0.9319940 0.3330158 0.7411989
## [3,] 0.5088740 0.7530398 0.5271144
## 
## ,,3
##            [,1]       [,2]       [,3]
## [1,] 0.27711173 0.47755065 0.20419603
## [2,] 0.03461886 0.26584356 0.07866031
## [3,] 0.26164131 0.27482891 0.25924531

3.5 Multiplication + Summation

Some examples of combining Multiplication and Summation are shown below.

3.5.1 Inner Product (Squared Frobenius Norm)

Inner Product first calculate Hadamard Product and collapses it to 0D tensor (norm).

einsum::einsum('i,i->', arrA, arrA)

## [1] 1.395197

DelayedTensor::einsum('i,i->', darrA, darrA)

## <1> HDF5Array object of type "double":
##      [1] 
## 1.395197

einsum::einsum('ij,ij->', arrC, arrC)

## [1] 4.033338

DelayedTensor::einsum('ij,ij->', darrC, darrC)

## <1> HDF5Array object of type "double":
##      [1] 
## 4.033338

einsum::einsum('ijk,ijk->', arrE, arrE)

## [1] 19.45969

DelayedTensor::einsum('ijk,ijk->', darrE, darrE)

## <1> HDF5Array object of type "double":
##      [1] 
## 19.45969

3.5.2 Contracted Product

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,] 0.44590594 0.2594567 0.3959109 1.6974797 1.8014501
## [2,] 0.01465249 1.7256433 0.8278886 0.9334747 1.3692291
## [3,] 0.83231954 1.6405214 1.7631446 0.5317730 0.4649905
## [4,] 0.77978663 0.1161259 1.2535926 1.8370268 0.7693164

DelayedTensor::einsum('ijk,ijk->jk', darrE, darrE)

## <4 x 5> HDF5Matrix object of type "double":
##            [,1]       [,2]       [,3]       [,4]       [,5]
## [1,] 0.44590594 0.25945670 0.39591091 1.69747974 1.80145007
## [2,] 0.01465249 1.72564325 0.82788859 0.93347472 1.36922908
## [3,] 0.83231954 1.64052141 1.76314460 0.53177299 0.46499049
## [4,] 0.77978663 0.11612589 1.25359264 1.83702681 0.76931640

3.5.3 Matrix Multiplication

Matrix Multiplication is considered a contracted product.

einsum::einsum('ij,jk->ik', arrC, t(arrC))

##           [,1]      [,2]      [,3]
## [1,] 0.5907015 0.5869866 0.8244639
## [2,] 0.5869866 1.8559042 1.4812061
## [3,] 0.8244639 1.4812061 1.5867326

DelayedTensor::einsum('ij,jk->ik', darrC, t(darrC))

## <3 x 3> HDF5Matrix object of type "double":
##           [,1]      [,2]      [,3]
## [1,] 0.5907015 0.5869866 0.8244639
## [2,] 0.5869866 1.8559042 1.4812061
## [3,] 0.8244639 1.4812061 1.5867326

3.6 Multiplication + Permutation

Some examples of combining Multiplication and Permutation are shown below.

einsum::einsum('ij,ij->ji', arrC, arrC)

##             [,1]       [,2]       [,3]
## [1,] 0.320719287 0.05191041 0.28478105
## [2,] 0.011151639 0.93419984 0.53098536
## [3,] 0.007985533 0.53302976 0.04947416
## [4,] 0.250844993 0.33676418 0.72149209

DelayedTensor::einsum('ij,ij->ji', darrC, darrC)

## <4 x 3> HDF5Matrix object of type "double":
##             [,1]        [,2]        [,3]
## [1,] 0.320719287 0.051910415 0.284781045
## [2,] 0.011151639 0.934199837 0.530985356
## [3,] 0.007985533 0.533029755 0.049474156
## [4,] 0.250844993 0.336764183 0.721492089

einsum::einsum('ijk,ijk->jki', arrE, arrE)

## , , 1
## 
##             [,1]       [,2]       [,3]       [,4]        [,5]
## [1,] 0.005992979 0.15117792 0.03020626 0.46645616 0.776720169
## [2,] 0.007666254 0.91030796 0.15631978 0.01835518 0.037341261
## [3,] 0.007128405 0.01770746 0.85774724 0.00300879 0.001730342
## [4,] 0.019386834 0.06150576 0.97703653 0.86855456 0.753340757
## 
## , , 2
## 
##              [,1]       [,2]       [,3]       [,4]        [,5]
## [1,] 4.636028e-02 0.03823925 0.11990614 0.78669467 0.101721402
## [2,] 2.308597e-05 0.01261157 0.34985407 0.08537157 0.606861457
## [3,] 4.294543e-01 0.89846222 0.03326672 0.43891462 0.001223533
## [4,] 1.220595e-01 0.03379877 0.26924230 0.08838578 0.013415000
## 
## , , 3
## 
##             [,1]       [,2]        [,3]       [,4]        [,5]
## [1,] 0.393552682 0.07003954 0.245798512 0.44432892 0.923008496
## [2,] 0.006963147 0.80272372 0.321714739 0.82974798 0.725026365
## [3,] 0.395736860 0.72435173 0.872130643 0.08984957 0.462036619
## [4,] 0.638340301 0.02082136 0.007313806 0.88008647 0.002560642

DelayedTensor::einsum('ijk,ijk->jki', darrE, darrE)

## <4 x 5 x 3> HDF5Array object of type "double":
## ,,1
##             [,1]        [,2]        [,3]        [,4]        [,5]
## [1,] 0.005992979 0.151177917 0.030206258 0.466456157 0.776720169
## [2,] 0.007666254 0.910307961 0.156319780 0.018355177 0.037341261
## [3,] 0.007128405 0.017707457 0.857747239 0.003008790 0.001730342
## [4,] 0.019386834 0.061505760 0.977036529 0.868554559 0.753340757
## 
## ,,2
##              [,1]         [,2]         [,3]         [,4]         [,5]
## [1,] 4.636028e-02 3.823925e-02 1.199061e-01 7.866947e-01 1.017214e-01
## [2,] 2.308597e-05 1.261157e-02 3.498541e-01 8.537157e-02 6.068615e-01
## [3,] 4.294543e-01 8.984622e-01 3.326672e-02 4.389146e-01 1.223533e-03
## [4,] 1.220595e-01 3.379877e-02 2.692423e-01 8.838578e-02 1.341500e-02
## 
## ,,3
##             [,1]        [,2]        [,3]        [,4]        [,5]
## [1,] 0.393552682 0.070039540 0.245798512 0.444328917 0.923008496
## [2,] 0.006963147 0.802723724 0.321714739 0.829747975 0.725026365
## [3,] 0.395736860 0.724351729 0.872130643 0.089849573 0.462036619
## [4,] 0.638340301 0.020821356 0.007313806 0.880086468 0.002560642

3.7 Summation + Permutation

Some examples of combining Summation and Permutation are shown below.

einsum::einsum('ijk->ki', arrE)

##          [,1]     [,2]     [,3]
## [1,] 0.388638 1.224817 2.138821
## [2,] 1.723990 1.439567 2.155983
## [3,] 2.483770 1.639037 2.082380
## [4,] 1.805272 2.138946 2.815363
## [5,] 1.984106 1.248753 2.542554

DelayedTensor::einsum('ijk->ki', darrE)

## <5 x 3> HDF5Matrix object of type "double":
##          [,1]     [,2]     [,3]
## [1,] 0.388638 1.224817 2.138821
## [2,] 1.723990 1.439567 2.155983
## [3,] 2.483770 1.639037 2.082380
## [4,] 1.805272 2.138946 2.815363
## [5,] 1.984106 1.248753 2.542554

3.8 Multiplication + Summation + Permutation

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,] 1.563169e-03 0.0394322389 0.007878799 1.216673e-01 2.025945e-01
## [2,] 6.952803e-05 0.0082559115 0.001417721 1.664697e-04 3.386614e-04
## [3,] 4.629502e-05 0.0001150001 0.005570590 1.954041e-05 1.123761e-05
## [4,] 3.955035e-03 0.0125475584 0.199321543 1.771905e-01 1.536862e-01
## 
## , , 2
## 
##              [,1]        [,2]        [,3]       [,4]         [,5]
## [1,] 2.031166e-03 0.001675362 0.005253405 0.03446717 0.0044566832
## [2,] 1.820257e-05 0.009943831 0.275849149 0.06731285 0.4784915505
## [3,] 1.932028e-01 0.404199904 0.014966020 0.19745877 0.0005504425
## [4,] 3.469304e-02 0.009606645 0.076526901 0.02512194 0.0038129535
## 
## , , 3
## 
##              [,1]        [,2]         [,3]         [,4]         [,5]
## [1,] 0.0164085965 0.002920195 0.0102482051 0.0185256364 0.0384834728
## [2,] 0.0005413094 0.062403094 0.0250098438 0.0645039371 0.0563629638
## [3,] 0.0028664368 0.005246690 0.0063170952 0.0006508065 0.0033466652
## [4,] 0.0674281612 0.002199369 0.0007725605 0.0929639129 0.0002704817

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,] 1.563169e-03 3.943224e-02 7.878799e-03 1.216673e-01 2.025945e-01
## [2,] 6.952803e-05 8.255912e-03 1.417721e-03 1.664697e-04 3.386614e-04
## [3,] 4.629502e-05 1.150001e-04 5.570590e-03 1.954041e-05 1.123761e-05
## [4,] 3.955035e-03 1.254756e-02 1.993215e-01 1.771905e-01 1.536862e-01
## 
## ,,2
##              [,1]         [,2]         [,3]         [,4]         [,5]
## [1,] 2.031166e-03 1.675362e-03 5.253405e-03 3.446717e-02 4.456683e-03
## [2,] 1.820257e-05 9.943831e-03 2.758491e-01 6.731285e-02 4.784916e-01
## [3,] 1.932028e-01 4.041999e-01 1.496602e-02 1.974588e-01 5.504425e-04
## [4,] 3.469304e-02 9.606645e-03 7.652690e-02 2.512194e-02 3.812954e-03
## 
## ,,3
##              [,1]         [,2]         [,3]         [,4]         [,5]
## [1,] 0.0164085965 0.0029201949 0.0102482051 0.0185256364 0.0384834728
## [2,] 0.0005413094 0.0624030937 0.0250098438 0.0645039371 0.0563629638
## [3,] 0.0028664368 0.0052466896 0.0063170952 0.0006508065 0.0033466652
## [4,] 0.0674281612 0.0021993688 0.0007725605 0.0929639129 0.0002704817

4 Create your original function by 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 functions444 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

Session information

## R version 4.4.0 beta (2024-04-15 r86425)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 22.04.4 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_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_1.11.0
##  [3] HDF5Array_1.31.6          rhdf5_2.47.7             
##  [5] DelayedArray_0.29.9       SparseArray_1.3.5        
##  [7] S4Arrays_1.3.7            abind_1.4-5              
##  [9] IRanges_2.37.1            S4Vectors_0.41.6         
## [11] MatrixGenerics_1.15.1     matrixStats_1.3.0        
## [13] BiocGenerics_0.49.1       Matrix_1.7-0             
## [15] DelayedTensor_1.9.0       BiocStyle_2.31.0         
## 
## loaded via a namespace (and not attached):
##  [1] jsonlite_1.8.8      compiler_4.4.0      BiocManager_1.30.22
##  [4] crayon_1.5.2        rsvd_1.0.5          Rcpp_1.0.12        
##  [7] rhdf5filters_1.15.5 parallel_4.4.0      jquerylib_0.1.4    
## [10] BiocParallel_1.37.1 yaml_2.3.8          fastmap_1.1.1      
## [13] lattice_0.22-6      R6_2.5.1            XVector_0.43.1     
## [16] ScaledMatrix_1.11.1 knitr_1.46          bookdown_0.39      
## [19] bslib_0.7.0         rlang_1.1.3         cachem_1.0.8       
## [22] xfun_0.43           sass_0.4.9          cli_3.6.2          
## [25] Rhdf5lib_1.25.3     BiocSingular_1.19.0 zlibbioc_1.49.3    
## [28] digest_0.6.35       grid_4.4.0          irlba_2.3.5.1      
## [31] rTensor_1.4.8       dqrng_0.3.2         lifecycle_1.0.4    
## [34] evaluate_0.23       codetools_0.2-20    beachmat_2.19.4    
## [37] rmarkdown_2.26      tools_4.4.0         htmltools_0.5.8.1