“MatrixExtra” is an R package which extends the sparse matrix and
sparse vector classes from the Matrix package,
particularly the CSR formats, by providing optimized functions, methods,
and operators which exploit the storage order (COO, CSC, CSR) of the
inputs and work natively with different formats, such as slicing (e.g.
X[1:10,]), concatenation (e.g. rbind(X, Y),
cbind(X, Y)), matrix multiplication with a dense matrix
(X %*% Y), or elementwise multiplication
(X * Y).
A typical matrix is a 2D array which has a given number of rows and
columns, and stores tabular data inside - e.g. measures of different
variables (columns) for many observations (rows). Behind the scenes,
matrices in R are represented as a 1D array with a column-major storage,
where element (i,j) is at position
i + (j-1)*nrows. This is a straightforward concept -
performing operations on such matrices is trivial, and allows easy
exploitation of modern CPU capabilities such as SIMD and
multi-threading.
In many applications, one finds some matrices in which most of the
values are exactly zero with only a few different entries - so called
“sparse” matrices. For example, in recommender systems, one might
construct a matrix of user-item interactions, in which users are rows,
items are columns, and values denote e.g. movie ratings or hours spent
playing something. Typically, each user interacts with only a handful
items, so such a matrix will typically have >99% of the entries set
to zero. In such cases, using a typical matrix is wasteful, since it
requires creating an array which contains mostly zeros, and doing
operations on them is inefficient, since the output of
e.g. X * 2 only needs to look at the non-zero entries
rather than the full nrows*ncols entries. Similar
situations are encountered in natural language processing (e.g. word
counts by documents), social networks (e.g. connections between users),
and classification/regression with one-hot/dummy encoded features, among
others.
In such cases, it’s more efficient to use a matrix representation that stores only the non-zero values and the indices which are non-zero. In many cases it might even be impossible to represent the full matrix in a computer’s memory due it’s size (e.g. 1,000,000 users and 10,000 movies = 74.5GB, but if only 1% of the entries are non-zero, can be put down to ~1.5GB or less), and it’s thus necessary to perform operations in this sparse representation instead.
Object classes for sparse matrix representations in R are provided by
packages like Matrix or SparseM (or
igraph for more specialized topics), and those objects -
particularly the ones from Matrix - are accepted and
handled efficiently by many other packages such as rsparse or glmnet.
As a general rule, if a given matrix has <5% non-zero values, it is more efficient to do common operations on it in a sparse representation, which typically comes in one of the following formats:
The COO format is the simplest form, consisting of storing all the
triplets (row,column,value) which are non-zero.
The COO format is typically not optimal to operate with, but allows
easy conversion to CSR and CSC formats (see below). Nevertheless, some
operations such as concatenating inputs (rbind,
cbind) or elementwise multiplication with a dense matrix
(X * Y) are efficient with a COO representation.
The CSR format, instead of of storing triplets, stores the elements in a row-major format, keeping track only of the column indices and of the positions at which the column indices for a given row start and end. Typically the column indices are meant to be sorted within each row, but this is not strictly assumed by all software or all functions.
The CSR format is optimal for doing row-based operations, such as
selecting rows (X[1:1000,]), concatenating by rows
(rbind), or matrix multiplication with a vector
(CSR %*% v).
The CSC format is the same as the CSR format, but is column-major instead of row-major.
The CSC format is optimal for doing column-based operations, such as
selecting columns (X[, 1:1000]), concatenating by columns
(cbind), and matrix multiplication with a dense matrix in
column-major format (like all R’s matrices) as the LHS
(Dense %*% CSC). Typically, tree-based methods work with
CSC format.
A vector (single row or single column) can also be represented in a sparse format by keeping track of the indices which are non-zero and the values.
Sparse vectors are typically not used but some operations involving
them are fast, such as inner products or matrix multiplication with a
CSR matrix as the LHS (CSR %*% v).
The Matrix package provides S4 classes to represent all
the formats above in R. These objects are handled in a rich hierarchy of
different matrix types with multiple inheritance. In general, one should
keep in mind the following points:
TsparseMatrix.RsparseMatrix.CsparseMatrix.RsparseMatrix, but will rather have a class which inherits
from it (has RsparseMatrix as parent class), and be of a
different type depending on the type of elements
(dsparseMatrix for numeric values,
lsparseMatrix for logical values,
nsparseMatrix for binary values), and depending on whether
they are symmetric, triangular-diagonal, or regular.dgTMatrix,
dgRMatrix, and dgCMatrix; but oftentimes when
dealing with Matrix methods, one has to refer to the parent
class - e.g. as(X, "RsparseMatrix"), but not
as(X, "dgRMatrix") (which is what one usually wants to
do).Sparse matrices can be created in any of the three formats in
Matrix with the function sparseMatrix -
example:
library(Matrix)
### Will construct this Matrix
### [ 1, 0, 2 ]
### [ 0, 0, 3 ]
### [ 0, 4, 0 ]
### Non-zero coordinates are:
### [(1,1), (1,3), (2,3), (3,2)]
### Row and column coordinates go separate
row_ix <- c(1, 1, 2, 3)
col_ix <- c(1, 3, 3, 2)
values <- c(1, 2, 3, 4)
X <- Matrix::sparseMatrix(
i=row_ix, j=col_ix, x=values,
index1=TRUE, repr="T"
)
X
#> 3 x 3 sparse Matrix of class "dgTMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3
#> [3,] . 4 .They can typically be converted to other formats through
methods::as - example:
as(X, "RsparseMatrix")
#> 3 x 3 sparse Matrix of class "dgRMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3
#> [3,] . 4 .Such Matrix objects have a lot of defined operators and
functions so that they could be used as drop-in replacements of base R
matrices - e.g.:
X + X
#> 3 x 3 sparse Matrix of class "dgTMatrix"
#>
#> [1,] 2 . 4
#> [2,] . . 6
#> [3,] . 8 .The Matrix package provides most of the functions and
methods from base R which one would expect, such as +,
-, *, %*%, rbind,
cbind, [, [<-,
sqrt, norm, among many others.
However, the whole package is centered around the CSC format, with
the provided functions oftentimes converting the input to CSC if it
isn’t already, which is inefficient and loses many optimization
potentials for operations like CSR[1:100,] or
rbind(COO, CSR), to name a few. Examples:
Xr <- as(X, "RsparseMatrix")
### This will forcibly convert the matrix to triplets
Xr[1:2, ]
#> 2 x 3 sparse Matrix of class "dgTMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3### This will forcibly convert the matrix to CSC
rbind(Xr, Xr)
#> 6 x 3 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3
#> [3,] . 4 .
#> [4,] 1 . 2
#> [5,] . . 3
#> [6,] . 4 .### This will forcibly convert the matrix to CSC
X * X
#> 3 x 3 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . 4
#> [2,] . . 9
#> [3,] . 16 .Many of these methods can be woefully inefficient when dealing with real, large datasets, particularly when dealing with the CSR format:
library(microbenchmark)
set.seed(1)
X_big_csc <- Matrix::rsparsematrix(1e4, 1e4, .05, repr="C")
X_big_csr <- as(t(X_big_csc), "RsparseMatrix")
microbenchmark({X_slice <- X_big_csr[1:10, ]}, times=10L)
#> Unit: milliseconds
#> expr min lq mean median
#> { X_slice <- X_big_csr[1:10, ] } 154.7282 155.4318 167.9717 156.3491
#> uq max neval
#> 164.0112 258.7927 10Compare against what should be the mirror operation in CSC format:
microbenchmark({X_slice <- X_big_csc[, 1:10]}, times=10L)
#> Unit: milliseconds
#> expr min lq mean median
#> { X_slice <- X_big_csc[, 1:10] } 5.660606 5.759361 6.418859 5.993871
#> uq max neval
#> 6.845528 9.001482 10Some operations in Matrix, even if done natively in CSC
format with a CSC input, can still be slower than one would expect and
than what could in theory be achieved with different algorithms,
oftentimes due to making copies of the data:
microbenchmark({X_col <- X_big_csc[, 100, drop=FALSE]}, times=10L)
#> Unit: milliseconds
#> expr min lq mean
#> { X_col <- X_big_csc[, 100, drop = FALSE] } 5.327531 5.520133 5.70143
#> median uq max neval
#> 5.575491 5.651869 7.090658 10It should also be kept in mind that Matrix does not
exploit multi-threading in dense-sparse matrix multiplications, which
have substantial potential for acceleration:
set.seed(1)
Y_dense <- matrix(rnorm(1e2*nrow(X_big_csc)), nrow=1e2)
microbenchmark({Z <- Y_dense %*% X_big_csc}, times=10L)
#> Unit: milliseconds
#> expr min lq mean median uq
#> { Z <- Y_dense %*% X_big_csc } 356.6463 357.711 359.8849 360.5575 360.7445
#> max neval
#> 363.2819 10The CSR sparse format is particularly useful when dealing with machine learning applications - e.g. splitting between a train and test set, tokenizing text features, multiplying a matrix by a vector of coefficients, calculating a gradient observation-by-observation, among others. Many stochastic optimization techniques and libraries (e.g. LibSVM, VowpalWabbit) require the inputs to be in CSR format or alike (see also readsparse), which does not play well with the column-centric methods of Matrix.
In principle, one could stick with just the CSC format from Matrix and keep a mental map of the matrix as being transposed. This however gets complicated rather soon and is very prone to errors. Additionally, one might want to pass sparse matrices to another package whose code is outside of one’s control, for which the storage format can make a large difference in performance.
MatrixExtra is a package which extends the same classes
from Matrix for COO, CSR, CSC, and sparse vectors, by
providing optimized replacements for typical methods which will work
without changing the storage format of the matrices when not necessary;
and providing some faster replacements of many methods.
library(MatrixExtra)
#> 'MatrixExtra' modifies important behaviors from 'Matrix'. See ?MatrixExtra-options.MatrixExtra overrides the show method of
sparse objects with a shorter version with only summary information:
Xr
#> Sparse CSR matrix (class 'dgRMatrix')
#> Dimensions: 3 x 3
#> (4 entries, 44.44% full)This new behavior usually comes handy when one wants to examine large
sparse matrices as it will not generate so much print output, but for
the examples in here the matrices to examine are small and one would
likely want to see them in full instead. This can be controlled with a
global option in the package (see ?MatrixExtra-options for
more):
options("MatrixExtra.quick_show" = FALSE)
Xr
#> 3 x 3 sparse Matrix of class "dgRMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3
#> [3,] . 4 .The earlier examples would now become:
### This will not change the format
Xr[1:2, ]
#> 2 x 3 sparse Matrix of class "dgRMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3### This will not change the format
rbind(Xr, Xr)
#> 6 x 3 sparse Matrix of class "dgRMatrix"
#>
#> [1,] 1 . 2
#> [2,] . . 3
#> [3,] . 4 .
#> [4,] 1 . 2
#> [5,] . . 3
#> [6,] . 4 .### This will not change the format
Xr * Xr
#> 3 x 3 sparse Matrix of class "dgRMatrix"
#>
#> [1,] 1 . 4
#> [2,] . . 9
#> [3,] . 16 .Some of these operations now become much more efficient when the inputs are large:
microbenchmark({X_slice <- X_big_csr[1:10, ]}, times=10L)
#> Unit: microseconds
#> expr min lq mean median uq
#> { X_slice <- X_big_csr[1:10, ] } 347.592 349.776 22292.82 351.5795 358.924
#> max neval
#> 219742.1 10Other methods, despite having been fast before in
Matrix, will still be replaced with faster versions:
microbenchmark({X_col <- X_big_csc[, 100, drop=FALSE]}, times=10L)
#> Unit: microseconds
#> expr min lq mean
#> { X_col <- X_big_csc[, 100, drop = FALSE] } 374.963 384.801 922.3007
#> median uq max neval
#> 390.4325 407.615 5699.459 10microbenchmark({Z <- Y_dense %*% X_big_csc}, times=10L)
#> Unit: milliseconds
#> expr min lq mean median
#> { Z <- Y_dense %*% X_big_csc } 72.02418 72.19792 73.70507 72.73744
#> uq max neval
#> 73.53253 80.91364 10Conversions between sparse matrix classes also become easier:
as(Xr, "ngRMatrix")
#> 3 x 3 sparse Matrix of class "ngRMatrix"
#>
#> [1,] | . |
#> [2,] . . |
#> [3,] . | .MatrixExtra::as.csr.matrix(Xr, binary=TRUE)
#> 3 x 3 sparse Matrix of class "ngRMatrix"
#>
#> [1,] | . |
#> [2,] . . |
#> [3,] . | .Here’s a non-comprehensive list of operations which are accelerated
by MatrixExtra:
CSR %*% dense, dense %*% CSC,
tcrossprod(CSR, dense),
tcrossprod(dense, CSR), crossprod(dense, CSC),
CSR %*% vector.rbind(CSR, CSR), rbind(CSR, COO),
rbind(CSR, vector), rbind(COO, vector).cbind(CSR, CSR), cbind(CSR, vector),
cbind(CSR, COO).CSR * dense, CSR * vector,
COO * dense, COO * vector,
CSR * scalar, COO * scalar (and other
similarly-working operators like &, ^,
%, %%, %/%).CSR + CSR, CSR + COO,
CSC + CSC, CSC + COO, CSR + CSC
(and |).t(CSR), t(CSC).CSR[i,j], CSC[i,j],
COO[i,j].sqrt(CSR),
norm(CSR), diag(CSR), among others.Many of the operations with dense types in MatrixExtra
allow inputs of float32 type from the float package, which
leads to faster operations; and many of the operations with vector types
allow sparse vectors from the same Matrix package and dense
vectors from float.
In addition, it also provides utility functions which come in handy
when sparse matrices are manually constructed or output by a different
software, such as functions for sorting the indices or for removing
zero-valued and NA elements.
When one loads MatrixExtra through
library(MatrixExtra), it will modify some behaviors from
Matrix in important ways which make them more efficient,
but which can cause breakage in code or in packages if they make certain
assumptions about Matrix methods. Among others:
### Here Matrix would return a 'dgRMatrix'
t(Xr)
#> 3 x 3 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . .
#> [2,] . . 4
#> [3,] 2 3 .### Here Matrix would return a dense vector
Xr[1,]
#> sparse vector (nnz/length = 2/3) of class "dsparseVector"
#> [1] 1 . 2These behaviors can be changed to their less-optimal versions as
would be done by Matrix, either individually (see
?MatrixExtra-options) or all at once:
restore_old_matrix_behavior()
set_new_matrix_behavior()One would wonder what kind of workflows specifically does
MatrixExtra improve upon, and one obvious example would be
fitting a logistic regression with gradient-based procedures.
This example here will fit a binary logistic regression with L2 regularization using the L-BFGS-B optimizer in R. For simplicity purposes, the intercept will be calculated by concatenating a column of 1s to the data, but note that this is not the most efficient way of doing it.
The dataset used is the “Real-Simulated” data, downloaded from LibSVM datasets. This is an artificially-generated toy dataset for which it’s easy to achieve almost-perfect accuracy, but it’s nevertheless a large-ish dataset in which the improved methods and operators here become noticeable.
Loading the data:
library(readsparse)
data <- readsparse::read.sparse("real-sim")
X <- data$X
y <- as.numeric(factor(data$y))-1 ### convert to 0/1
X
#> Sparse CSR matrix (class 'dgRMatrix')
#> Dimensions: 72,309 x 20,958
#> (3,709,083 entries, 0.24% full)Adding the intercept and creating a 50-50 train-test split:
X <- cbind(rep(1, nrow(X)), X) ### Accelerated by 'MatrixExtra'
set.seed(1)
ix_train <- sample(nrow(X), floor(.5*nrow(X)), replace=FALSE)
X_train <- X[ix_train,] ### Accelerated by 'MatrixExtra'
y_train <- y[ix_train]
X_test <- X[-ix_train,] ### Accelerated by 'MatrixExtra'
y_test <- y[-ix_train]Now fitting the model:
logistic_fun <- function(coefs, X, y, lambda) {
pred <- 1 / (1 + exp(-as.numeric(X %*% coefs))) ### Accelerated by 'MatrixExtra'
ll <- mean(y * log(pred) + (1 - y) * log(1 - pred))
reg <- lambda * as.numeric(coefs %*% coefs)
### Don't regularize the intercept
reg <- reg - lambda * (coefs[1]^2)
return(-ll + reg)
}
logistic_grad <- function(coefs, X, y, lambda) {
pred <- 1 / (1 + exp(-(X %*% coefs))) ### Accelerated by 'MatrixExtra'
grad <- colMeans(X * as.numeric(pred - y)) ### Accelerated by 'MatrixExtra'
grad <- grad + 2 * lambda * as.numeric(coefs)
### Don't regularize the intercept
grad[1] <- grad[1] - 2 * lambda * coefs[1]
return(as.numeric(grad))
}
lambda <- 1e-5 ### <- Regularization parameter
res <- optim(numeric(ncol(X_train)),
logistic_fun,
logistic_grad,
method="L-BFGS-B",
X_train, y_train, lambda)
fitted_coefs <- res$parVerify that the model has good performance:
y_hat_test <- as.numeric(X_test %*% fitted_coefs)
MLmetrics::AUC(y_hat_test, y_test)
#> [1] 0.9947617Timing the optimizer:
x0 <- numeric(ncol(X_train))
microbenchmark::microbenchmark({
res <- optim(x0,
logistic_fun,
logistic_grad,
method="L-BFGS-B",
X_train, y_train, lambda)
}, times=10L)
#> Unit: seconds
#> expr
#> { res <- optim(x0, logistic_fun, logistic_grad, method = "L-BFGS-B", X_train, y_train, lambda) }
#> min lq mean median uq max neval
#> 3.907413 3.956523 4.016892 4.011786 4.035334 4.179486 10The same routine using Matrix would usually take around
7 seconds (~60% slower) in this same setup, plus some extra time in the
data preparation. The only thing that was needed to accelerate it was to
load library(MatrixExtra), with everything else remaining
the same as it would have been in base R or Matrix.