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May 2015


A Julia package to support sparse vectors.

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Sparse data has become increasingly common in machine learning and related areas. For example, in document analysis, each document is often represented as a sparse vector, which each entry represents the number of occurrences of a certain word. However, the support of sparse vectors remains quite limited in Julia base.

This package provides two types SparseVector and SparseVectorView and a series of methods to work with sparse vectors. Specifically, this package provides the following functionalities:

  • Construction of sparse vectors, either with given non-zero entries or randomly.
  • Get a view of a column in a sparse matrix (of CSC format), or a view of a range of columns.
  • Specialized arithmetic functions on sparse vectors, e.g. +, -, *, etc.
  • Specialized math functions on sparse vectors, e.g. abs, abs2, exp, sin, etc.
  • Specialized reduction functions on sparse vectors, e.g. sum, vecnorm, etc.
  • Specialized linear algebraic functions, e.g. axpy!, dot, A * x, At_mul_B, etc.

Note: Many of the functionalities implemented in this package may be migrated to Julia Base in v0.5 development cycle.


This package defines two types.

  • SparseVector: a sparse vector that owns its memory
  • SparseVectorView: a view of external data as a sparse vector.

The formal definition of these types are listed below:

immutable SparseVector{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}
    n::Int              # the number of elements
    nzind::Vector{Ti}   # the indices of nonzeros
    nzval::Vector{Tv}   # the values of nonzeros

typealias CVecView{T} ContiguousView{T,1,Vector{T}}

immutable SparseVectorView{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}
    n::Int                  # the number of elements
    nzind::CVecView{Ti}     # the indices of nonzeros
    nzval::CVecView{Tv}     # the values of nonzeros


An instance of SparseVector can be constructed as follows:

SparseVector(n, nzind, nzval)  # constructs an instance by providing all fields

Here, all inputs will be used as fields as they are. The constructor will ensure that length(nzind) == length(nzval). However, it will NOT examine the elements of nzind (e.g. the indexes are sorted, without duplication, and within the range 1:n).

The package also provides a sparsevector function to construct a sparse vector in a variety of ways:

zero sparse vector

# construct a zerp sparse vector of length `len` and element type `T`
sparsevector(T, len)  

with given lists of non-zero entries

The following methods construct a sparse vector of length len, with:

  • non-zero indices, given by an integer vector I
  • non-zero values, given by V (either a vector or a number)


  • When len is omitted, it is determined by maximum(I).
  • Multiple values are allowed to be corresponding to the same index. These values are combined with a binary function/functor combine. When it is omitted, AddFun is used by default, meaning that the values are summed.
sparsevector(I, V, len, combine)
sparsevector(I, V, combine)
sparsevector(I, V, len)
sparsevector(I, V)

from an associative collection (e.g. Dict)

sparsevector(a, len)  # a is an instance of `Associative{Ti<:Integer, Tv}`
sparsevector(a)  # length inferred as the maximum index

random sparse vector

sprand(n, p)       # construct a random sparse vector with length n and density p
                   # the non-zero values are generated using rand(nnz)

sprand(n, p, T)    # construct a random sparse vector of element type T
                   # the non-zero values are generated using rand(T, nnz)

sprandn(n, p)      # construct a random sparse vector,
                   # where values follow standard Normal distribution

sprand(n, p, rfn)  # construct a random sparse vector,
                   # where the non-zero values are generated using rfn

Basic methods

Like other array types, SparseVector and SparseVectorView support all the basic methods for arrays:

eltype(x)   # get the element type
ndims(x)    # get the number of dimensions (1)
length(x)   # get the length
size(x)     # get the size, i.e. (length(x),)

x[I]        # getindex with index(es) I,
            # where I can be either an integer or an integer array
x[i] = v    # set the i-th element of x to v

They also provide methods for extracting internal data structures:

nnz(x)          # the number of stored entries
countnz(x)      # count the actual number of nonzero entries
nonzeroinds(x)  # get the vector of indices of non-zero values  
nonzeros(x)     # get the vector of non-zero values


The package supports conversion between sparse vectors and other types of arrays.

convert(SparseVector, s)  # convert s to an instance of SparseVector
                          # s can be an instance of Vector or SparseMatrixCSC

convert(SparseVector{Tv}, s)  # convert the element-type of s to Tv
                              # where s is an instance of SparseVector

convert(SparseVector{Tv,Ti}, s)  # convert the element-type of s to Tv,
                                 # and the index-type of s to Ti,
                                 # where s is an instance of SparseVector

convert(SparseMatrixCSC, v)  # convert a sparse vector v to a sparse matrix
                             # with a single column

convert(SparseMatrixCSC{Tv}, v)
convert(SparseMatrixCSC{Tv,Ti}, v)


The package provides methods to obtain views of sparse vectors or individual columns of a sparse matrix:

view(x)   # construct a SparseVectorView instance as a view of x
          # where x is an instance of SparseVector

view(A, :, i)   # construct a view of the i-th column of X
                # where X is an instance of SparseMatrixCSC
                # returns a instance of SparseVectorView

getcol(A, i)   # get a copy of the i-th column of a sparse matrix A
               # returns an instance of SparseVector

unsafe_colrange(A, i1:i2)  # construct an unsafe view of a range of columns
                           # i.e. from the i1-th to i2-th column.
                           # returns an instance of SparseMatrixCSC

Note: unsafe_colrange uses pointer_to_array to obtain the internal vectors, and therefore the returned array should only be used within the local scope.

Math Functions

The package implement a number of specialized math functions for sparse vectors.


scale!(x, c)   # x <- x * c, where c is a scalar
scale!(c, x)   # i.e. scale!(x, c)
scale(x, c)    # returns x * c
scale(c, x)    # i.e. scale(x, c)

x * c, x .* c  # multiple x and a scalar c
c * x, c .* x  # i.e. x * c

- x            # negate x
x + y, x .+ y  # add x and y, x and y can be either dense or sparse
x - y, x .- y  # subtract y from x, x and y can be either dense or sparse
x .* y         # multiply x and y (element-wise),
               # x and y can be either dense or sparse

axpy!(a, x, y)  # y <- y + a * x
                # a: a scalar number
                # x: a sparse vector
                # y: a dense vector

Element-wise math functions

# Input: (sparse, sparse) --> Output: sparse
# Input: (sparse, dense)  --> Output: dense
# Input: (dense, sparse)  --> Output: sparse

max(x, y), min(x, y)
complex(x, y)

# zero-preserving functions
#   Input: sparse --> Output: sparse

abs(x), abs2(x)
real(x), imag(x), conj(x)
floor(x), ceil(x), trunc(x), round(x)
log1p(x), expm1(x),
sin(x), tan(x), sinpi(x), sind(x), tand(x),
asin(x), atan(x), asind(x), atand(x),
sinh(x), tanh(x), asinh(x), atanh(x)

# Non-zero-preserving functions
#   Input: sparse --> Output: dense

exp(x), exp2(x), exp10(x), log(x), log2(x), log10(x),
cos(x), csc(x), cot(x), sec(x), cospi(x),
cosd(x), cscd(x), cotd(x), secd(x),
acos(x), acot(x), acosd(x), acotd(x),
cosh(x), csch(x), coth(x), sech(x),
acsch(x), asech(x)


sum(x)      # Compute the sum of elements
sumabs(x)   # Compute the sum of absolute values
sumabs2(x)  # Compute the sum of squared absolute values
maximum(x)  # Compute the maximum of elements (including zeros)
minimum(x)  # Compute the minimum of elements (including zeros)
maxabs(x)   # Compute the maximum of absolute values
minabs(x)   # Compute the minimum of absolute values

vecnorm(x, p=2)  # Compute the p-th order vector-norm

dot(x, y)   # Compute the dot product between x and y
            # x and y can be either dense or sparse vectors

Linear Algebra: Matrix-vector product

# Note: the product is dense iff A is dense
A * x             # A * x (matrix-vector product)
At_mul_B(A, x)    # A.' * x, without explicitly transposing A
Ac_mul_B(A, x)    # A' * x, without explicitly conjugate-transposing A

# If you want to get a dense result even when both A and x are sparse
# then you can write:
densemv(A, x)               # A * x --> dense vector
densemv(A, x; trans='N')    # A * x, as above
densemv(A, x; trans='T')    # transpose(A) * x -> dense vector
densemv(A, x; trans='C')    # ctranspose(A) * x -> dense vector

# Note: the following functions are only for cases where y is a strided vector
A_mul_B!(y, A, x)         # y <- A * x
A_mul_B!(a, A, x, b, y)   # y <- a * A * x + b * y
At_mul_B(y, A, x)         # y <- A.' * x
At_mul_B!(a, A, x, b, y)  # y <- a * A.' * x + b * y
Ac_mul_B(y, A, x)         # y <- A' * x
Ac_mul_B!(a, A, x, b, y)  # y <- a * A' * x + b * y