Chapter 13 Sparse Representations

Analyzing “bigdata” in R is a challenge because the workspace is memory resident, i.e., all your objects are stored in RAM. As a rule of thumb, fitting models requires about 5 times the size of the data. This means that if you have 1 GB of data, you might need about 5 GB to fit a linear models. We will discuss how to compute out of RAM in the Memory Efficiency Chapter 14. In this chapter, we discuss efficient representations of your data, so that it takes less memory. The fundamental idea, is that if your data is sparse, i.e., there are many zero entries in your data, then a naive data.frame or matrix will consume memory for all these zeroes. If, however, you have many recurring zeroes, it is more efficient to save only the non-zero entries.

When we say data, we actually mean the model.matrix. The model.matrix is a matrix that R grows, converting all your factors to numeric variables that can be computed with. Dummy coding of your factors, for instance, is something that is done in your model.matrix. If you have a factor with many levels, you can imagine that after dummy coding it, many zeroes will be present.

The Matrix package replaces the matrix class, with several sparse representations of matrix objects.

When using sparse representation, and the Matrix package, you will need an implementation of your favorite model fitting algorithm (e.g. lm) that is adapted to these sparse representations; otherwise, R will cast the sparse matrix into a regular (non-sparse) matrix, and you will have saved nothing in RAM.

Remark. If you are familiar with MATLAB you should know that one of the great capabilities of MATLAB, is the excellent treatment of sparse matrices with the sparse function.

Before we go into details, here is a simple example. We will create a factor of letters with the letters function. Clearly, this factor can take only \(26\) values. This means that \(25/26\) of the model.matrix will be zeroes after dummy coding. We will compare the memory footprint of the naive model.matrix with the sparse representation of the same matrix.

library(magrittr)
reps <- 1e6 # number of samples
y<-rnorm(reps)
x<- letters %>% 
  sample(reps, replace=TRUE) %>% 
  factor

The object x is a factor of letters:

head(x)
## [1] n x z f a i
## Levels: a b c d e f g h i j k l m n o p q r s t u v w x y z

We dummy code x with the model.matrix function.

X.1 <- model.matrix(~x-1)
head(X.1)
##   xa xb xc xd xe xf xg xh xi xj xk xl xm xn xo xp xq xr xs xt xu xv xw xx
## 1  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0
## 2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
## 3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## 4  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## 5  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
## 6  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   xy xz
## 1  0  0
## 2  0  0
## 3  0  1
## 4  0  0
## 5  0  0
## 6  0  0

We call MatrixModels for an implementation of model.matrix that supports sparse representations.

suppressPackageStartupMessages(library(MatrixModels))
X.2<- as(x,"sparseMatrix") %>% t # Makes sparse dummy model.matrix
head(X.2)
## 6 x 26 sparse Matrix of class "dgCMatrix"
##    [[ suppressing 26 column names 'a', 'b', 'c' ... ]]
##                                                         
## [1,] . . . . . . . . . . . . . 1 . . . . . . . . . . . .
## [2,] . . . . . . . . . . . . . . . . . . . . . . . 1 . .
## [3,] . . . . . . . . . . . . . . . . . . . . . . . . . 1
## [4,] . . . . . 1 . . . . . . . . . . . . . . . . . . . .
## [5,] 1 . . . . . . . . . . . . . . . . . . . . . . . . .
## [6,] . . . . . . . . 1 . . . . . . . . . . . . . . . . .

Notice that the matrices have the same dimensions:

dim(X.1)
## [1] 1000000      26
dim(X.2)
## [1] 1000000      26

The memory footprint of the matrices, given by the pryr::object_size function, are very very different.

pryr::object_size(X.1)
## 264 MB
pryr::object_size(X.2)
## 12 MB

Things to note:

  • The sparse representation takes a whole lot less memory than the non sparse.
  • The as(,"sparseMatrix") function grows the dummy variable representation of the factor x.
  • The pryr package provides many facilities for inspecting the memory footprint of your objects and code.

With a sparse representation, we not only saved on RAM, but also on the computing time of fitting a model. Here is the timing of a non sparse representation:

system.time(lm.1 <- lm(y ~ X.1)) 
##    user  system elapsed 
##   3.828   0.120   3.946

Well actually, lm is a wrapper for the lm.fit function. If we override all the overhead of lm, and call lm.fit directly, we gain some time:

system.time(lm.1 <- lm.fit(y=y, x=X.1))
##    user  system elapsed 
##   1.080   0.008   1.088

We now do the same with the sparse representation:

system.time(lm.2 <- MatrixModels:::lm.fit.sparse(X.2,y))
##    user  system elapsed 
##   0.208   0.012   0.224

It is only left to verify that the returned coefficients are the same:

all.equal(lm.2, unname(lm.1$coefficients), tolerance = 1e-12)
## [1] TRUE

You can also visualize the non zero entries, i.e., the sparsity structure.

image(X.2[1:26,1:26])

13.1 Sparse Matrix Representations

We first distinguish between the two main goals of the efficient representation: (i) efficient writing, i.e., modification; (ii) efficient reading, i.e., access. For our purposes, we will typically want efficient reading, since the model.matrix will not change while a model is being fitted.

Representations designed for writing include the dictionary of keys, list of lists, and a coordinate list. Representations designed for efficient reading include the compressed sparse row and compressed sparse column.

13.1.1 Coordinate List Representation

A coordinate list representation, also known as COO, or triplet represantation is simply a list of the non zero entries. Each element in the list is a triplet of the column, row, and value, of each non-zero entry in the matrix.

13.1.2 Compressed Column Oriented Representation

A compressed column oriented representation, also known as compressed sparse column, or CSC, where the column index is similar to COO, but instead of saving the row indexes, we save the locations in the column index vectors where the row index has to increase. The following figure may clarify this simple idea.

The CSC data structure. From Shah and Gilbert (2004). Remember that MATLAB is written in C, where the indexing starts at 0, and not 1.

The CSC data structure. From Shah and Gilbert (2004). Remember that MATLAB is written in C, where the indexing starts at \(0\), and not \(1\).

The nature of statistical applications is such, that CSC representation is typically the most economical, justifying its popularity.

13.1.3 Compressed Row Oriented Representation

A compressed row oriented representation, also known as compressed sparse row, or CSR, is very similar to CSC, after switching the role of rows and columns. CSR is much less popular than CSC.

13.1.4 Sparse Algorithms

We will go into the details of some algorithms in the Numerical Linear Algebra Chapter 16. For our current purposes two things need to be emphasized:

  1. Working with sparse representations requires using a function that is aware of the representation you are using.

  2. A mathematician may write \(Ax=b \Rightarrow x=A^{-1}b\). This is a predicate of \(x\),i.e., a property that \(x\) satisfies, which helps with its analysis. A computer, however, would never compute \(A^{-1}\) in order to find \(x\), but rather use one of many endlessly many numerical algorithms. A computer will typically “search” various \(x\)’s until it finds the one that fulfils the predicate.

13.2 Sparse Matrices and Sparse Models in R

13.2.1 The Matrix Package

The Matrix package provides facilities to deal with real (stored as double precision), logical and so-called “pattern” (binary) dense and sparse matrices. There are provisions to provide integer and complex (stored as double precision complex) matrices.

The sparse matrix classes include:

  • TsparseMatrix: a virtual class of the various sparse matrices in triplet representation.
  • CsparseMatrix: a virtual class of the various sparse matrices in CSC representation.
  • RsparseMatrix: a virtual class of the various sparse matrices in CSR representation.

For matrices of real numbers, stored in double precision, the Matrix package provides the following (non virtual) classes:

  • dgTMatrix: a general sparse matrix of doubles, in triplet representation.
  • dgCMatrix: a general sparse matrix of doubles, in CSC representation.
  • dsCMatrix: a symmetric sparse matrix of doubles, in CSC representation.
  • dtCMatrix: a triangular sparse matrix of doubles, in CSC representation.

Why bother with distinguishing between the different shapes of the matrix? Because the more structure is assumed on a matrix, the more our (statistical) algorithms can be optimized. For our purposes dgCMatrix will be the most useful.

13.2.2 The MatrixModels Package

TODO

13.2.3 The glmnet Package

As previously stated, an efficient storage of the model.matrix is half of the story. We now need implementations of our favorite statistical algorithms that make use of this representation. At the time of writing, a very useful package that does that is the glmnet package, which allows to fit linear models, generalized linear models, with ridge, lasso, and elastic net regularization. The glmnet package allows all of this, using the sparse matrices of the Matrix package.

The following example is taken from John Myles White’s blog, and compares the runtime of fitting an OLS model, using glmnet with both sparse and dense matrix representations.

suppressPackageStartupMessages(library('glmnet'))

set.seed(1)
performance <- data.frame()
 
for (sim in 1:10){
  n <- 10000
  p <- 500
 
  nzc <- trunc(p / 10)
 
  x <- matrix(rnorm(n * p), n, p) #make a dense matrix
  iz <- sample(1:(n * p),
               size = n * p * 0.85,
               replace = FALSE)
  x[iz] <- 0 # sparsify by injecting zeroes
  sx <- Matrix(x, sparse = TRUE) # save as a sparse object
 
  beta <- rnorm(nzc)
  fx <- x[, seq(nzc)] %*% beta
 
  eps <- rnorm(n)
  y <- fx + eps # make data
 
  # Now to the actual model fitting:
  sparse.times <- system.time(fit1 <- glmnet(sx, y)) # sparse glmnet
  full.times <- system.time(fit2 <- glmnet(x, y)) # dense glmnet
  
  sparse.size <- as.numeric(object.size(sx))
  full.size <- as.numeric(object.size(x))
 
  performance <- rbind(performance, data.frame(Format = 'Sparse',
                                                UserTime = sparse.times[1],
                                               SystemTime = sparse.times[2],
                                               ElapsedTime = sparse.times[3],
                                               Size = sparse.size))
  performance <- rbind(performance, data.frame(Format = 'Full',
                                               UserTime = full.times[1],
                                               SystemTime = full.times[2],
                                               ElapsedTime = full.times[3],
                                               Size = full.size))
}

Things to note:

  • The simulation calls glmnet twice. Once with the non-sparse object x, and once with its sparse version sx.
  • The degree of sparsity of sx is \(85\%\). We know this because we “injected” zeroes in \(0.85\) of the locations of x.
  • Because y is continuous glmnet will fit a simple OLS model. We will see later how to use it to fit GLMs and use lasso, ridge, and elastic-net regularization.

We now inspect the computing time, and the memory footprint, only to discover that sparse representations make a BIG difference.

suppressPackageStartupMessages(library('ggplot2'))
ggplot(performance, aes(x = Format, y = ElapsedTime, fill = Format)) +
  stat_summary(fun.data = 'mean_cl_boot', geom = 'bar') +
  stat_summary(fun.data = 'mean_cl_boot', geom = 'errorbar') +
  ylab('Elapsed Time in Seconds') 

ggplot(performance, aes(x = Format, y = Size / 1000000, fill = Format)) +
  stat_summary(fun.data = 'mean_cl_boot', geom = 'bar') +
  stat_summary(fun.data = 'mean_cl_boot', geom = 'errorbar') +
  ylab('Matrix Size in MB') 

How do we perform other types of regression with the glmnet? We just need to use the family and alpha arguments of glmnet::glmnet. The family argument governs the type of GLM to fit: logistic, Poisson, probit, or other types of GLM. The alpha argument controls the type of regularization. Set to alpha=0 for ridge, alpha=1 for lasso, and any value in between for elastic-net regularization.

13.2.4 The SparseM Package

13.3 Bibliographic Notes

The best place to start reading on sparse representations and algorithms is the vignettes of the Matrix package. Gilbert, Moler, and Schreiber (1992) is also a great read for some general background. For the theory on solving sparse linear systems see Davis (2006). For general numerical linear algebra see Gentle (2012).

13.4 Practice Yourself

References

Shah, Viral, and John R Gilbert. 2004. “Sparse Matrices in Matlab* P: Design and Implementation.” In International Conference on High-Performance Computing, 144–55. Springer.

Gilbert, John R, Cleve Moler, and Robert Schreiber. 1992. “Sparse Matrices in Matlab: Design and Implementation.” SIAM Journal on Matrix Analysis and Applications 13 (1). SIAM: 333–56.

Davis, Timothy A. 2006. Direct Methods for Sparse Linear Systems. SIAM.

Gentle, James E. 2012. Numerical Linear Algebra for Applications in Statistics. Springer Science & Business Media.