Author: Cesar Lobato-Fernandez, Juan A. Ferrer-Bonsoms & Ángel Rubio

# Get started with GPUmatrix package

### Seamlessly harness the power of GPU computing in R

#### Cesar Lobato-Fernandez, Juan A. Ferrer-Bonsoms & Ángel Rubio

#### February 2023

## Abstract

GPUs are great resources for data analysis, especially in statistics and linear algebra. Unfortunately, very few packages connect R to the GPU, and none of them are transparent enough to run the computations on the GPU without substantial changes to the code. The maintenance of these packages is cumbersome: several of the earlier attempts have been removed from their respective repositories. It would be desirable to have a properly maintained R package that takes advantage of the GPU with minimal changes to the existing code.

We have developed the GPUmatrix package (available on CRAN). GPUmatrix mimics the behavior of the Matrix package and extends R to use the GPU for computations. It includes single(FP32) and double(FP64) precision data types, and provides support for sparse matrices. It is easy to learn, and requires very few code changes to perform the operations on the GPU. GPUmatrix relies on either the Torch or Tensorflow R packages to perform the GPU operations.

We have demonstrated its usefulness for several statistical applications and machine learning applications: non-negative matrix factorization, logistic regression and general linear models. We have also included a comparison of GPU and CPU performance on different matrix operations.

Before starting, please be advised that this R package is designed to have the lowest learning curve for the R user to perform algebraic operations using the GPU. Therefore, this tutorial will mostly cover procedures that will go beyond the operations that the user can already perform with R’s CPU matrices.

# 0 Installation

## 0.1 Dependences

GPUmatrix is an R package that utilizes tensors through the torch or tensorflow packages (see Advanced Users section for more information). One or the other must be installed for the use of GPUmatrix. Both packages are hosted in CRAN and have specific installation instructions. In both cases, it is necessary to have an NVIDIA® GPU card with the latest drivers installed in order to use the packages, as well as a version of Python 3. The NVIDIA card must be compatible; please see the list of capable cards here. If there is no compatible graphics card or not graphic card at all, you can still install tensorFlow and torch, but only with the CPU version, which means that GPUmatrix will only be able to run in CPU mode.

### For torch: (Link installation here)

    install.packages("torch")
    library(torch)
    install_torch() # In some cases is required.

- MUST INSTALL:

  - Before installing torch is required for GPU computing to install CUDA and cuDNN. Links for them and instructions for intallation are in the torch documentation.

### For Tensorflow: (Link installation here)

The installation of TensorFlow allows the selection to install the GPU, CPU, or both versions (in Windows the GPU version is no longer supported). This will depend on the version of TensorFlow that we install with the `install_tensorflow()` function. The mode in which the tensors are created using GPUmatrix, if we choose to use TensorFlow, will depend on the installation mode. The options to switch from CPU to GPU are not enabled when using GPUmatrix with TensorFlow for this precise reason. To install the GPU version, it is not necessary to specify the version since if it detects that the CUDA dependencies are met, it will automatically install using the GPU mode. If you want to install the CPU version, you need to specify it as follows:

`install_tensorflow(version="nightly-cpu")`

    install.packages("tensorflow")
    library(tensorflow)
    install_tensorflow(version = "nightly-gpu")

- MUST INSTALL:

  - Before installing tensorflow is required for GPU computing to install CUDA and cuDNN. Links for them and instructions for intallation are in the tensorflow documentation. ### For Apple Silicon

Great news! No additional software required to work on Apple Silicon (either the CPU or the GPU). ## 0.2 GPUmatrix installation

Once the dependencies for Torch or TensorFlow are installed, the GPUmatrix package, being a package hosted on CRAN, can be easily installed using:

    install.packages("GPUmarix")

Alternatively, it is possible to install the package from GitHub ot get the last version of the package.

    devtools::install_github(" ceslobfer/GPUmatrix")

# 1 Initialization GPUmatrix

The GPUmatrix package is based on S4 objects in R and we have created a constructor function that acts similarly to the default `matrix()` constructor in R for CPU matrices. The constructor function is `gpu.matrix()` and accepts the same parameters as `matrix()`:

    library(GPUmatrix)

    ## Torch tensors allowed

    ## Tensorflow tensors allowed

    ## Your Torch installation have CUDA tensors available.

    if (installTorch()) {
      #R matrix initialization
      m <- matrix(c(1:20)+40,10,2)
      #Show CPU matrix
      m
      #GPU matrix initialization
      Gm <- gpu.matrix(c(1:20)+40,10,2)
      #Show GPU matrix
      Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  51
    ##  42  52
    ##  43  53
    ##  44  54
    ##  45  55
    ##  46  56
    ##  47  57
    ##  48  58
    ##  49  59
    ##  50  60
    ## [ CUDADoubleType{10,2} ]

Although the indexing of tensors in both torch and tensorflow is 0-based, the indexing of GPUmatrix objects is 1-based, making it as close as possible to working with native R matrices and more convenient for the user. In the previous example, a normal R CPU matrix called `m` and its GPU counterpart `Gm` are created. Just like regular matrices, the created GPU matrices allow for indexing of its elements and assignment of values. The concatenation operators `rbind()` and `cbind()` work independently of the type of matrices that are to be concatenated, resulting in a gpu.matrix:

    if (installTorch()) {
    Gm[c(2,3),1]

    Gm[,2]

    Gm2 <- cbind(Gm[c(1,2),], Gm[c(6,7),])
    Gm2

    Gm2[1,3] <- 0
    Gm2
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  51   0  56
    ##  42  52  47  57
    ## [ CUDADoubleType{2,4} ]

It is also possible to initialize the data with NaN values:

    if (installTorch()) {
    Gm3 <- gpu.matrix(nrow = 2,ncol=3)
    Gm3[,2]
    Gm3[1,2] <- 1
    Gm3
    Gm3[1,3] <- 0
    Gm3
    }

    ## GPUmatrix
    ## torch_tensor
    ## nan  1  0
    ## nan nan nan
    ## [ CUDADoubleType{2,3} ]

These examples demonstrate that, contrary to standard R, subsetting a gpu.matrix —even when selecting only one column or row— still results in a gpu.matrix. This behavior is analogous to using ‘drop=F’ in standard R. The default standard matrices in R have limitations. The only allowed numeric data types are int and float64. It neither natively allows the creation or handling of sparse matrices. To make up for this lack of functionality, other R packages hosted in CRAN have been created to manage these types.

## 1.1 Using GPUMatrix on CPU

In the GPUmatrix constructor, we can specify the location of the matrix, i.e., we can decide to host it on the GPU or in RAM memory to use it with the CPU. As a package, as its name suggests, oriented towards algebraic operations in R using the GPU, it will by default be hosted on the GPU, but it allows the same functionalities using the CPU. To do this, we use the `device` attribute of the constructor and assign it the value “cpu”.

    if (installTorch()) {
    #GPUmatrix initialization with CPU option
    Gm <- gpu.matrix(c(1:20)+40,10,2,device="cpu")
    #Show CPU matrix from GPUmatrix
    Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  51
    ##  42  52
    ##  43  53
    ##  44  54
    ##  45  55
    ##  46  56
    ##  47  57
    ##  48  58
    ##  49  59
    ##  50  60
    ## [ CPUDoubleType{10,2} ]

## 1.2 Using GPUMatrix on Apple Silicon

The GPUmatrix package also supports Apple Silicon devices, allowing GPU operations to work seamlessly. By using the `device = "mds"` option in most functions, users can leverage the GPU on Apple Silicon machines out of the box. This ensures optimal performance on devices equipped with the M1 or M2 chips.

    if (installTorch()) {
      # GPUmatrix initialization for Apple Silicon GPU
      Gm <- gpu.matrix(c(1:20)+40,10,2,device="mds")
      # Show Apple Silicon GPU matrix from GPUmatrix
      Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  51
    ##  42  52
    ##  43  53
    ##  44  54
    ##  45  55
    ##  46  56
    ##  47  57
    ##  48  58
    ##  49  59
    ##  50  60
    ## [ CPUDoubleType{10,2} ]

## 1.3 Using GPUMatrix with Tensorflow

As commented in the introduction and dependency section, GPUmatrix can be used with both TensorFlow and Torch. By default, the GPU matrix constructor is initialized with Torch tensors because, in our opinion, it provides an advantage in terms of installation and usage compared to TensorFlow. Additionally, it allows the use of GPUmatrix not only with GPU tensors but also with CPU tensors. To use GPUmatrix with TensorFlow, simply use the `type` attribute in the constructor function and assign it the value “tensorflow” as shown in the following example:

    # library(GPUmatrix)
    tensorflowGPUmatrix <- gpu.matrix(c(1:20)+40,10,2, type = "tensorflow") tensorflowGPUmatrix

####

# 2 Cast GPU matrices and data types

The default matrices in R have limitations. The numeric data types it allows are int and float64, with float64 being the type used generally in R by default. It also does not natively allow for the creation and handling of sparse matrices. To make up for this lack of functionality, other R packages hosted in CRAN have been created that allow for programming these types of functionality in R. The problem with these packages is that in most cases they are not compatible with each other, meaning we can have a sparse matrix with float64 and a non-sparse matrix with float32, but not a sparse matrix with float32.

## 2.1 Cast from other packages

GPUmatrix allows for compatibility with sparse matrices and different data types such as float32. For this reason, casting operations between different matrix types from multiple packages to GPUmatrix type have been implemented:

Table 1. Cast options from other packages. If back cast is TRUE, then it is possible to convert a gpu.matrix to this object and vice versa. If is FALSE, it is possible to convert these objects to gpu.matrix but not vice versa.
Matrix class | Package | Data type default | SPARSE | Back cast
--- | --- | --- | --- | ---
matrix | base | float64 | FALSE | TRUE
data.frame | base | float64 | FALSE | TRUE
integer | base | float64 | FALSE | TRUE
numeric | base | float64 | FALSE | TRUE
dgeMatrix | Matrix | float64 | FALSE | FALSE
ddiMatrix | Matrix | float64 | TRUE | FALSE
dpoMatrix | Matrix | float64 | FALSE | FALSE
dgCMatrix | Matrix | float64 | TRUE | FALSE
float32 | float | float32 | FALSE | FALSE
torch_tensor | torch | float64 | Depends of tensor type | TRUE
tensorflow.tensor | tensorflow | float64 | Depends of tensor type | TRUE

There are two functions for casting to create a gpu.matrix: `as.gpu.matrix()` and the `gpu.matrix()` constructor itself. Both have the same input parameters for casting: the object to be cast and extra parameters for creating a GPUmatrix.

reate ‘Gm’ from ‘m’ matrix R-base:

    if (installTorch()) {
    m <- matrix(c(1:10)+40,5,2)
    Gm <- gpu.matrix(m)
    Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  46
    ##  42  47
    ##  43  48
    ##  44  49
    ##  45  50
    ## [ CUDADoubleType{5,2} ]

Create ‘Gm’ from ‘M’ with Matrix package:

    if (installTorch()) {
    library(Matrix)
    M <- Matrix(c(1:10)+40,5,2)
    Gm <- gpu.matrix(M)
    Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  46
    ##  42  47
    ##  43  48
    ##  44  49
    ##  45  50
    ## [ CUDADoubleType{5,2} ]

Create ‘Gm’ from ‘mfloat32’ with float package:

    if (installTorch()) {
    library(float)
    mfloat32 <- fl(m)
    Gm <- gpu.matrix(mfloat32)
    Gm
    }

    ## GPUmatrix
    ## torch_tensor
    ##  41  46
    ##  42  47
    ##  43  48
    ##  44  49
    ##  45  50
    ## [ CUDAFloatType{5,2} ]

Interestingly, GPUmatrix returns a float32 data type matrix if the input is a float matrix.

It is also possible to a gpu.matrix create ‘Gms’ type sparse from ‘Ms’ type sparse dgCMatrix, dgeMatrix, ddiMatrix or dpoMatrix with Matrix package:

    if (installTorch()) {
    Ms <- Matrix(sample(0:1, 10, replace = TRUE), nrow=5, ncol=2, sparse=TRUE)
    Ms

    Gms <- gpu.matrix(Ms)
    Gms
    }

    ## GPUmatrix
    ## torch_tensor
    ## [ SparseCUDAFloatType{}
    ## indices:
    ##  0  0  1  2  2  3
    ##  0  1  1  0  1  1
    ## [ CUDALongType{2,6} ]
    ## values:
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ## [ CUDAFloatType{6} ]
    ## size:
    ## [5, 2]
    ## ]

## 2.2 Data type and sparsity

The data types allowed by GPUmatrix are: float64, float32, int, bool or logical, complex64 and complex32. We can create a GPU matrix with a specific data type using the `dtype` parameter of the `gpu.matrix()` constructor function. It is also possible change the data type of a previously created GPU matrix using the the data type of a previously created GPU matrix using the `dtype()` function. The same applies to GPU sparse matrices, we can create them from the constructor using the `sparse` parameter, which will obtain a Boolean value of `TRUE`/`FALSE` depending on whether we want the resulting matrix to be sparse or not. We can also modify the sparsity of an existing GPU matrix with the functions `to_dense()`, if we want it to go from sparse to dense, and `to_sparse()`, if we want it to go from dense to sparse.

    if (installTorch()) {
    #Creating a float32 matrix
    Gm32 <- gpu.matrix(c(1:20)+40,10,2, dtype = "float32")
    Gm32

    #Creating a non sparse martix with data type float32 from a sparse matrix type float64
    Ms <- Matrix(sample(0:1, 20, replace = TRUE), nrow=10, ncol=2, sparse=TRUE)
    Gm32 <- gpu.matrix(Ms, dtype = "float32", sparse = F)
    Gm32

    #Convert Gm32 in sparse matrix Gms32
    Gms32 <- to_sparse(Gm32)
    Gms32

    ##Convert data type Gms32 in float64
    Gms64 <- Gms32
    dtype(Gms64) <- "float64"
    Gms64
    }

    ## GPUmatrix
    ## torch_tensor
    ## [ SparseCUDADoubleType{}
    ## indices:
    ##  2  3  4  5  6  7  8  1  4  5  6  7  8
    ##  0  0  0  0  0  0  0  1  1  1  1  1  1
    ## [ CUDALongType{2,13} ]
    ## values:
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ##  1
    ## [ CUDADoubleType{13} ]
    ## size:
    ## [10, 2]
    ## ]

# 3 GPUmatrix functions

## 3.1 Arithmetic and comparison operators

GPUmatrix supports all basic arithmetic operators in R: `+`, `-`, `*`, `^`, `/`, `%*%` and `%%`. Its usage is the same as for basic R matrices, and it allows compatibility with other matrix objects from the packages mentioned above.

    if (installTorch()) {
    (Gm + Gm) == (m + m)

    (Gm + M) == (mfloat32 + Gm)

    (M + M) == (mfloat32 + Gm)

    (M + M) > (Gm + Gm)*2
    }

    ##       [,1]  [,2]
    ## [1,] FALSE FALSE
    ## [2,] FALSE FALSE
    ## [3,] FALSE FALSE
    ## [4,] FALSE FALSE
    ## [5,] FALSE FALSE

As seen in the previous example, the comparison operators (`==`, `!=`, `>`, `<`, `>=`, `<=`) also work following the same dynamic as the arithmetic operators.

## 3.2 Math operators

Similarly to arithmetic operators, mathematical operators follow the same operation they would perform on regular matrices of R. `Gm` is a gpu.matrix variable:

Table 2. Mathematical operators that accept a gpu.matrix as input
Mathematical operators | Usage
--- | ---
`log` | `log(Gm)`
`log2` | `log2(Gm)`
`log10` | `log10(Gm)`
`cos` | `cos(Gm)`
`cosh` | `cosh(Gm)`
`acos` | `acos(Gm)`
`acosh` | `acosh(Gm)`
`sin` | `sin(Gm)`
`sinh` | `sinh(Gm)`
`asin` | `asin(Gm)`
`asinh` | `asinh(Gm)`
`tan` | `tan(Gm)`
`atan` | `atan(Gm)`
`tanh` | `tanh(Gm)`
`atanh` | `atanh(Gm)`
`sqrt` | `sqrt(Gm)`
`abs` | `abs(Gm)`
`sign` | `sign(Gm)`
`ceiling` | `ceiling(Gm)`
`floor` | `floor(Gm)`
`cumsum` | `cumsum(Gm)`
`cumprod` | `cumprod(Gm)`
`exp` | `exp(Gm)`
`expm1` | `expm1(Gm)`

## 3.2 Complex operators

There are certain functions only applicable to numbers of complex type. In R these functions are grouped as complex operators and all of them are available for GPUmatrix matrices with the same functionality as in R base

Table 3. Complex operators that accept a gpu.matrix with complex type data as input
Mathematical operators | Usage
--- | ---
`Re` | `Re(Gm)`
`Im` | `Im(Gm)`
`Conj` | `Conj(Gm)`
`Arg` | `Arg(Gm)`
`Mod` | `Mod(Gm)`

## 3.3 Other functions

We can find a multitude of functions that can be applied to gpu.matrix type matrices. Most of the functions are functions from the base R package that can be used on gpu.matrix matrices in the same way they would be applied to regular matrices of R. There are other functions from other packages like Matrix or matrixStats that have been implemented due to their widespread use within the user community, such as `rowVars` or `colMaxs`. The output of these functions, which originally produced R default matrix type objects, will now return gpu.matrix type matrices if the input type of the function is gpu.matrix.

    if (installTorch()) {
    m <- matrix(c(1:20)+40,10,2)
    Gm <- gpu.matrix(c(1:20)+40,10,2)

    head(tcrossprod(m),1)

    head(tcrossprod(Gm),1)

    Gm <- tail(Gm,3)
    rownames(Gm) <- c("a","b","c")
    tail(Gm,2)

    colMaxs(Gm)
    }

    ## [1] 50 60

There is a wide variety of functions implemented in GPUmatrix, and they are adapted to be used just like regular R matrices.

Table 4. Functions that accept one or several gpu.matrix matrices as input
Functions | Usage | Package
--- | --- | ---
`determinant` | `determinant(Gm, logarithm=T)` | `base`
`fft` | `fft(Gm)` | `base`
`sort` | `sort(Gm,decreasing=F)` | `base`
`round` | `round(Gm, digits=0)` | `base`
`show` | `show(Gm)` | `base`
`length` | `length(Gm)` | `base`
`dim` | `dim(Gm)` | `base`
`dim<-` | `dim(Gm) <- c(...,...)` | `base`
`rownames` | `rownames(Gm)` | `base`
`rownames<-` | `rownames(Gm) <- c(...)` | `base`
`row.names` | `row.names(Gm)` | `base`
`row.names<-` | `row.names(Gm) <- c(...)` | `base`
`colnames` | `colnames(Gm)` | `base`
`colnames<-` | `colnames(Gm) <- c(...)` | `base`
`rowSums` | `rowSums(Gm)` | `Matrix`
`colSums` | `colSums(Gm)` | `Matrix`
`cbind` | `cbind(Gm,...)` | `base`
`rbind` | `rbind(Gm,...)` | `base`
`head` | `head(Gm,...)` | `base`
`tail` | `tail(Gm,...)` | `base`
`nrow` | `nrow(Gm)` | `base`
`ncol` | `ncol(Gm)` | `base`
`t` | `t(Gm)` | `base`
`crossprod` | `crossprod(Gm,...)` | `base`
`tcrossprod` | `tcrossprod(Gm,…)` | `base`
`%x%` | `Gm %x% … || … %x% Gm` | `base`
`%^%` | `Gm %^% … || … %^% Gm` | `base`
`diag` | `diag(Gm)` | `base`
`diag<-` | `diag(Gm) <- c(…)` | `base`
`solve` | `solve(Gm, …)` | `base`
`qr` | `qr(Gm)` | `base`
`qr.Q` | `qr.Q``(…)` | `base`
`qr.R` | `qr.R``(…)` | `base`
`qr.X` | `qr.X``(…)` | `base`
`qr.solve` | `qr.solve``(…)` | `base`
`qr.coef` | `qr.coef``(…)` | `base`
`qr.qy` | `qr.qy``(…)` | `base`
`qr.qty` | `qr.qty``(…)` | `base`
`qr.resid` | `qr.resid``(…)` | `base`
`eigen` | `eigen(Gm)` | `base`
`svd` | `svd(Gm)` | `base`
`pinv` | `pinv(Gm, tol = sqrt(.Machine$double.eps))` | `MASS`
`chol` | `chol(Gm)` | `base`
`chol_solve` | `chol_solve(Gm, …)` | `GPUmatrix`
`mean` | `mean(Gm)` | `base`
`density` | `density(Gm)` | `base`
`hist` | `hist(Gm)` | `base`
`colMeans` | `colMeans(Gm)` | `Matrix`
`rowMeans` | `rowMeans(Gm)` | `Matrix`
`sum` | `sum(Gm)` | `base`
`min` | `min(Gm)` | `base`
`max` | `max(Gm)` | `base`
`which.max` | `which.max(Gm)` | `base`
`which.min` | `which.min(Gm)` | `base`
`aperm` | `aperm(Gm)` | `base`
`apply` | `apply(Gm, MARGIN, FUN, …, simplify=TRUE)` | `base`
`cov` | `cov(Gm)` | `stats`
`cov2cor` | `cov2cor(Gm)` | `stats`
`cor` | `cor(Gm, …)` | `stats`
`rowVars` | `rowVars(Gm)` | `matrixStats`
`colVars` | `colVars(Gm)` | `matrixStats`
`colMaxs` | `colMaxs(Gm)` | `matrixStats`
`rowMaxs` | `rowMaxs(Gm)` | `matrixStats`
`rowRanks` | `rowRanks(Gm)` | `matrixStats`
`colRanks` | `colRanks(Gm)` | `matrixStats`
`colMins` | `colMins(Gm)` | `matrixStats`
`rowMins` | `rowMins` | `matrixStats`
`dtype` | `dtype(Gm)` | `GPUmatrix`
`dtype<-` | `dtype(Gm)` | `GPUmatrix`
`to_dense` | `to_dense(Gm)` | `GPUmatrix`
`to_sparse` | `to_sparse(Gm)` | `GPUmatrix`

## 3.4 Function time comparison

We have compared the computation time for different matrix functions, different precision and running the operations either on the GPU or on the CPU (using both GPUmatrix and plain R).

The functions that we tested are ‘`*`’ (Hadamard or element-wise product of matrices), ‘`exp`’, (exponential of each element of a matrix), ‘`rowMeans`’ (means of the rows of a matrix), ‘`%*%`’ (standard product of matrices), ‘`solve`’ (inverse of a matrix) and ‘`svd`’ (singular value decomposition of a matrix).

These functions were tested on square matrices whose row sizes are 500, 700, 1000, 1400, 2000, 2800 and 4000.

Figure 1: Computation time (in seconds) where MKL-R solid green, solid lines for CPU, dashed lines for GPU with CUDA, pink lines for GPUmatrix with float64 and blue lines for GPUmatrix with float32. All calculations are performed on square matrices. The x-axis represents the number of rows in the matrices. The operations are the element-wise or Hadamard product of two matrices, the exponential of each element of a matrix, the mean of the rows of a matrix, the standard matrix product, the inverse of a matrix, and the singular value decomposition of a matrix (SVD).

Figure 1 compares the different computational architectures, namely, CPU -standard R matrices running on MKL on FP64-, CPU64 -GPUmatrix matrices computed on the CPU with FP64-, CPU32 -similar to the previous using FP32-, GPU64 -GPUmatrix matrices stored and computed on the GPU with FP64- and GPU32 -identical to the previous using FP32-.

It is important to note that the y-axis is in logarithmic scale. For example, the element-wise product of a matrix on GPU32 is around five times faster than the same operation on CPU. The results show that the GPU is particularly effective for element-wise operations (Hadamard product, exponential of a matrix). For these operations, it is easier to fully utilize the huge number of cores of a GPU. R-MKL seems to use a single core to perform element-wise operations. The torch implementation is much faster, but not as much as using the GPU. rowMeans is also faster on the GPU than on the CPU. In this case, the GPUmatrix CPU implementation is on par with the GPU. When the operations become more complex, as in the standard product of matrices and computing the inverse, CPU and GPU (using double precision) are closer to each other. However, GPU32 is still much faster than its CPU32 counterpart. Finally, it is not advisable -in terms of speed- to use the GPU for even more complex operations such as the SVD. In this case, GPU64 is the slowest method. GPU32 hardly stands up the comparison with CPU32.

Figure 2: Computation time (in seconds) for the Matrix package (solid violet), yellow lines for GPUmatrix with float32, orange lines for GPUmatrix with float64, solid lines for CPU and dashed lines for GPU with CUDA. Time shown in y-axis is in logarithmic scale. The small model is a random square matrix of size 2,000 x 2,000. The proportion of non-zero elements is either 0.001 or 0.01. The large model is a 20,000 x 20,000 matrix with the same proportion of non-zero elements. The element-wise multiplication is performed on the sparse matrix. The right panel shows the time required to multiply these matrices by dense matrices whose sizes are 2,000 x 500 and 20,000 x 500, respectively.

Regarding sparse matrices, Torch or Tensorflow for R have very limited support for them. In many cases, it is necessary to cast the sparse matrix to a full matrix and perform the operations on the full matrix. However, there are two operations that do not require this casting: element-wise multiplication of matrices and multiplication of a sparse matrix and a full matrix. We created four sparse matrices. The size of two of them (small ones) is 2,000 x 2,000. The fraction of nonzero entries is 0.01 and 0.001. The size of the other two (large ones) is 20,000 x 20,000. The fraction of non-null entries is also 0.01 and 0.001. Figure 2 shows the results for these matrices. The element-wise multiplication is faster using GPUmatrix. On the contrary, the time to multiply a sparse and a full matrices is similar using Matrix and GPUmatrix. As mentioned earlier, the implementation of sparse operations in torch or tensorflow for R is far from perfect, probably, because the only storage mode is “coo” in their respective R packages.

GPUmatrix clearly outperforms CPU computation for the element-wise matrix product, since it is not possible to use MKL in these cases.

In all cases, the larger the matrix, the more advantageous it is to compute on the GPU. For the small matrices 500 x 500, the CPU is among the fastest methods for all operations except the exponential computation.

# 4. Statistical functions applications

The main advantage of GPUmatrix is its versatility: R code needs only minor changes to adapt it to work in the CPU. We are showing here three statistical applications where its advantages are more apparent.

## 4.1 Non negative factorization of a matrix

The non-negative factorization (NMF) of a matrix is an approximate factorization were an initial matrix V is approximated by the product of two matrix W and H so that,

\[ \mathbf{V}_{m \times n} \approx \mathbf{W}_{m \times k}\mathbf{H}_{k \times n} \]

We have implemented our own non-negative matrix factorization function (`NMFgpumatrix()`) using the Lee and Seung multiplicative update rule.

The rules are

\[ \mathbf{W}_{[i, j]}^{n+1} \leftarrow \mathbf{W}_{[i, j]}^n \frac{\left(\mathbf{V}\left(\mathbf{H}^{n+1}\right)^T\right)_{[i, j]}}{\left(\mathbf{W}^n \mathbf{H}^{n+1}\left(\mathbf{H}^{n+1}\right)^T\right)_{[i, j]}} \] and \[ \mathbf{H}_{[i, j]}^{n+1} \leftarrow \mathbf{H}_{[i, j]}^n \frac{\left(\left(\mathbf{W}^n\right)^T \mathbf{V}\right)_{[i, j]}}{\left(\left(\mathbf{W}^n\right)^T \mathbf{W}^n \mathbf{H}^n\right)_{[i, j]}} \] to update the W and H respectively.

Figure 3: Computation time (in seconds) of non-negative factorization for MKL-R (i.e. R with the optimized MKL BLAS library, solid green), solid lines for CPU, dashed lines for GPU with CUDA, pink lines for GPUmatrix with float64, and blue lines for GPUmatrix with float32. Time shown in y-axis is in logarithmic scale. All calculations are performed on square matrices. The x-axis represents the number of rows in the matrices. The internal size of the factorization is 10.

The implemented function is `NMFgpumatrix`. This function operates in the same way with basic R matrices as with GPUmatrix matrices, and it does not require any additional changes beyond initializing the input matrix as a GPUmatrix. Indeed, the input matrices W, H, and V can be either gpu.matrix or R base matrices interchangeably. Figure 3 shows that using GPUmatrix boosts the performance improvement using both GPU and CPU. This improvement is especially apparent with float32 matrices.

## 4.2 Logistic regression of large models

Logistic regression is a widespread statistical analysis technique that is the “first to test” method for classification problems where the outcome is binary. R base implements this analysis in the `glm` function. However, `glm` can be very slow for big models and, in addition, does not accept sparse coefficient matrices as input. In this example, we have implemented a logistic regression solver that accepts as input both dense or sparse matrices.

Figure 4: Computation time (in seconds) of the logistic regression using the conjugate gradient method for MKL-R (i.e. R with the optimized MKL BLAS library, solid green), solid lines for CPU, dashed lines for GPU with CUDA, pink lines for GPUmatrix with float64, and blue lines for GPUmatrix with float32. Time shown in y-axis is in logarithmic scale. The calculations are performed on random matrices whose size are n x (n/100). Therefore, the leftmost part of the graph shows the computing time for a 10,000 x 100 matrix and the rightmost part a 50,000 x 500 matrix.

The developed function performs the logistic regression using the Conjugate Gradient method (CG) Figure 4. This method has shown to be very effective for logistic regression of big models (Minka 2003). The code is general enough to accommodate standard R matrices, sparse matrices from the Matrix package and, more interestingly, GPUmatrices from the GPUmatrix package.

We would like to stress several points here. Firstly, the conjugate gradient is an efficient technique that outperforms `glm.fit` in this case. Secondly, this code runs on matrix, Matrix or GPUmatrix objects without the need of carefully selecting the type of input. Thirdly, using the GPUmatrix accelerates the computation time two-fold if compared to standard R (and more than ten fold if compared to `glm.fit` function)

Figure 5: Computation time (in seconds) of the logistic regression using the conjugate gradient method in a sparse matrix. Solid green for MKL-R dense case (i.e. the computation is performed without any consideration of the sparsity of the matrix). Solid lines for CPU, dashed lines for GPU with CUDA, pink lines for GPUmatrix dense with float64, blue lines for GPUmatrix dense with float32, yellow lines for GPUmatrix sparse with float32, orange lines for GPUmatrix sparse with float64. Violet line, using Matrix package(that implicitly considers the matrix to be sparse). Time shown in y-axis is in logarithmic scale. The calculations are performed on random matrices whose size are n x (n/100). Therefore, the leftmost part of the graph shows the computing time for a 10,000 x 100 matrix and the rightmost part a 50,000 x 500 matrix.

We have tested the function also using a sparse input. In this case, the memory requirements are much smaller. However, there are no advantages in execution time (despite the sparsity is 90 \%).

Torch (and Tensorflow) for R only provides a type of sparse matrices: the “coo” coding where each element is described by its position (i and j) and its value. Matrix (and torch and tensorflow for Python) include other storage models (column compressed format, for example) where the matrix computations can be better optimized. It seems that there is still room for improvement in the sparse matrix algebra in Torch for R.

Sparse Matrix -that includes the column compressed form- performs extraordinary well in this case Figure 5. In all the other cases, GPUmatrix in CPU was faster than R-MKL. In this case, despite Matrix is single-threaded, it is roguhly ten times faster than GPUmatrix.

## 4.3 General Linear Models

One of the most frequently used functions in R is “glm”, that stands for generalized linear models. In turn, “glm” relies on the “glm.fit”. This function uses the iteratively reweighted least squares algorithm to provide the solution of a generalized linear model. glm.fit, subsequently, calls a C function to do the “hard work” for solving the linear models, including the least squares solution of the intermediate linear systems.

Since glm.fit calls a C function is not easy to develop a “plug-in” substitute using GPUmatrix. Specifically, the qr algorithm in torch or tensorflow for R and in the C function (that is similar to base R) substantially differ: in torch the function returns directly the Q and R matrices whereas the C function called by glm.fit returns a matrix and several auxiliary vectors that can be used to reconstruct the Q and R matrices.

One side-effect of the dependency of base glm on a C function is that, even if R is compiled to use an optimized linear algebra library, the function does not exploit that and the execution time is much longer than expected for this task.

In order to ameliorate this problem, there are other packages (fastlm and speedglm) that mimic the behavior of the base glm. fastlm relies on using multithreaded RcppEigen functions to boost the performance. On the contrary, speedglm is written in plain R and the performance is increased especially if R runs using an optimized BLAS library.

We have used speedglm as starting point for our glm implementation in the GPU since all the code is written in R and includes no calls to external functions. Specifically, the most time-consuming task for large general linear models, is the solution of the intermediate least squares problems. In the GPUmatrix implementation, we used most of the code of speed.glm. The only changes are casts on the matrix of the least squares problem and the corresponding independent term to solve the problem using the GPU. Timing results of the last section show that `GPUglm` outperforms speed.glm for large models.

The following script illustrates how to use the `GPUglm` function in toy examples.

    # Standard glm
    if (installTorch()) {
    counts <- c(18,17,15,20,10,20,25,13,12)
    outcome <- gl(3,1,9)
    treatment <- gl(3,3)
    glm.D93 <- glm(counts ~ outcome + treatment, family = poisson())
    summary(glm.D93)

    # Using speedglm
    # library(speedglm)
    # sglm.D93 <- speedglm(counts ~ outcome + treatment, family = poisson())
    # summary(sglm.D93)

    # GPU glm
    library(GPUmatrix)
    gpu.glm.D93 <- GPUglm(counts ~ outcome + treatment, family = poisson())
    summary(gpu.glm.D93)
    }

    ## $call
    ## glm(formula = ..1, family = ..2, method = "glm.fit.GPU")
    ##
    ## $terms
    ## counts ~ outcome + treatment
    ## attr(,"variables")
    ## list(counts, outcome, treatment)
    ## attr(,"factors")
    ##           outcome treatment
    ## counts          0         0
    ## outcome         1         0
    ## treatment       0         1
    ## attr(,"term.labels")
    ## [1] "outcome"   "treatment"
    ## attr(,"order")
    ## [1] 1 1
    ## attr(,"intercept")
    ## [1] 1
    ## attr(,"response")
    ## [1] 1
    ## attr(,".Environment")
    ## <environment: R_GlobalEnv>
    ## attr(,"predvars")
    ## list(counts, outcome, treatment)
    ## attr(,"dataClasses")
    ##    counts   outcome treatment
    ## "numeric"  "factor"  "factor"
    ##
    ## $family
    ##
    ## Family: poisson
    ## Link function: log
    ##
    ##
    ## $deviance
    ## [1] 5.129141
    ##
    ## $aic
    ## [1] 56.76132
    ##
    ## $df
    ## [1] 4
    ##
    ## $nulldev
    ## [1] 10.58145
    ##
    ## $nulldf
    ## [1] 8
    ##
    ## $iter
    ## [1] 4
    ##
    ## $tol
    ## [1] 2.140131e-14
    ##
    ## $n
    ## [1] 9
    ##
    ## $convergence
    ## [1] TRUE
    ##
    ## $ngoodobs
    ## [1] 9
    ##
    ## $logLik
    ## [1] -23.38066
    ##
    ## $RSS
    ## [1] 5.173202
    ##
    ## $rank
    ## [1] 5
    ##
    ## $coefficients
    ##                  Estimate Std. Error       z value     Pr(>|z|)
    ## (Intercept)  3.044522e+00  0.1708987  1.781478e+01 5.426767e-71
    ## outcome2    -4.542553e-01  0.2021708 -2.246889e+00 2.464711e-02
    ## outcome3    -2.929871e-01  0.1927423 -1.520097e+00 1.284865e-01
    ## treatment2   2.300896e-15  0.2000000  1.150448e-14 1.000000e+00
    ## treatment3   1.122370e-15  0.2000000  5.611849e-15 1.000000e+00
    ##
    ## $dispersion
    ## [1] 1
    ##
    ## $correlation
    ## [1] FALSE
    ##
    ## $cov.unscaled
    ##             (Intercept)      outcome2      outcome3    treatment2    treatment3
    ## (Intercept)  0.02920635 -1.587302e-02 -1.587302e-02 -2.000000e-02 -2.000000e-02
    ## outcome2    -0.01587302  4.087301e-02  1.587302e-02 -5.808405e-18 -2.904202e-18
    ## outcome3    -0.01587302  1.587302e-02  3.714961e-02 -2.255691e-18 -1.127846e-18
    ## treatment2  -0.02000000 -9.840453e-18 -6.023175e-18  4.000000e-02  2.000000e-02
    ## treatment3  -0.02000000 -1.096830e-17 -8.662814e-18  2.000000e-02  4.000000e-02
    ##
    ## $cov.scaled
    ##             (Intercept)      outcome2      outcome3    treatment2    treatment3
    ## (Intercept)  0.03777258 -2.052858e-02 -2.052858e-02 -2.586601e-02 -2.586601e-02
    ## outcome2    -0.02052858  5.286109e-02  2.052858e-02 -7.512012e-18 -3.756006e-18
    ## outcome3    -0.02052858  2.052858e-02  4.804561e-02 -2.917286e-18 -1.458643e-18
    ## treatment2  -0.02586601 -1.272666e-17 -7.789775e-18  5.173202e-02  2.586601e-02
    ## treatment3  -0.02586601 -1.418530e-17 -1.120362e-17  2.586601e-02  5.173202e-02
    ##
    ## attr(,"class")
    ## [1] "summary.GPUglm"

Results are identical in this test.

Figure 6: Computation time (in seconds) of general linear model using speedglm funciton with MKL-R matrix (i.e. R with the optimized MKL BLAS library, solid green), solid black line for glm function, solid lines for CPU, dashed lines for GPU with CUDA, pink lines for GPUmatrix with float64, and blue lines for GPUmatrix with float32. Time shown in y-axis is in logarithmic scale. The calculations are performed on random matrices whose size are n x (n/10). Therefore, the leftmost part of the graph shows the computing time for a 1,000 x 100 matrix and the rightmost part a 10,000 x 1000 matrix.

Figure 6 compares the timing of the implementation using speedglm, glm and the implementation of glm in GPUmatrix.

#### References

Bates D, Maechler M, Jagan M (2022). Matrix: Sparse and Dense Matrix Classes and Methods. R package version 1.5-3, https://CRAN.R-project.org/package=Matrix.

Schmidt D (2022). “float: 32-Bit Floats.” R package version 0.3-0, https://cran.r-project.org/package=float.

Falbel D, Luraschi J (2022). torch: Tensors and Neural Networks with ‘GPU’ Acceleration. R package version 0.9.0, https://CRAN.R-project.org/package=torch.

Allaire J, Tang Y (2022). tensorflow: R Interface to ‘TensorFlow’. R package version 2.11.0, https://CRAN.R-project.org/package=tensorflow.

Minka, Thomas P. 2003. “A Comparison of Numerical Optimizers for Logistic Regression.” https://tminka.github.io/papers/logreg/minka-logreg.pdf.