Operations on COO arrays


COO objects support a number of operations. They interact with scalars, Numpy arrays, other COO objects, and scipy.sparse.spmatrix objects, all following standard Python and Numpy conventions.

For example, the following Numpy expression produces equivalent results for both Numpy arrays, COO arrays, or a mix of the two:

np.log(X.dot(beta.T) + 1)

However some operations are not supported, like operations that implicitly cause dense structures, or numpy functions that are not yet implemented for sparse arrays.

np.svd(x)  # sparse svd not implemented

This page describes those valid operations, and their limitations.


This function allows you to apply any arbitrary broadcasting function to any number of arguments where the arguments can be SparseArray objects or scipy.sparse.spmatrix objects. For example, the following will add two arrays:

sparse.elemwise(np.add, x, y)


Previously, elemwise was a method of the COO class. Now, it has been moved to the sparse module.


Operations that would result in dense matrices, such as operations with Numpy arrays raises a ValueError. For example, the following will raise a ValueError if x is a numpy.ndarray:

x + y

However, all of the following are valid operations.

x + 0
x != y
x + y
x == 5
5 * x
x / 7.3
x != 0
x == 0
x + 5

We also support operations with a nonzero fill value. These are operations that map zero values to nonzero values, such as x + 1 or ~x. In these cases, they will produce an output with a fill value of 1 or True, assuming the original array has a fill value of 0 or False respectively.

If densification is needed, it must be explicit. In other words, you must call COO.todense on the COO object. If both operands are COO, both must be densified.

Operations with NumPy arrays

In certain situations, operations with NumPy arrays are also supported. For example, the following will work if x is COO and y is a NumPy array:

x * y

The following conditions must be met when performing element-wise operations with NumPy arrays:

  • The operation must produce a consistent fill-values. In other words, the resulting array must also be sparse.

  • Operating on the NumPy arrays must not increase the size when broadcasting the arrays.

Operations with scipy.sparse.spmatrix

Certain operations with scipy.sparse.spmatrix are also supported. For example, the following are all allowed if y is a scipy.sparse.spmatrix:

x + y
x - y
x * y
x > y
x < y

In general, if operating on a scipy.sparse.spmatrix is the same as operating on COO, as long as it is to the right of the operator.


Results are not guaranteed if x is a scipy.sparse.spmatrix. For this reason, we recommend that all Scipy sparse matrices should be explicitly converted to COO before any operations.


All binary operators support broadcasting. This means that (under certain conditions) you can perform binary operations on arrays with unequal shape. Namely, when the shape is missing a dimension, or when a dimension is 1. For example, performing a binary operation on two COO arrays with shapes (4,) and (5, 1) yields an object of shape (5, 4). The same happens with arrays of shape (1, 4) and (5, 1). However, (4, 1) and (5, 1) will raise a ValueError.

Element-wise Operations

COO arrays support a variety of element-wise operations. However, as with operators, operations that map zero to a nonzero value are not supported.

To illustrate, the following are all possible, and will produce another COO array:


As above, in the last three cases, an array with a nonzero fill value will be produced.

Notice that you can apply any unary or binary numpy.ufunc to COO arrays, and numpy.ndarray objects and scalars and it will work so long as the result is not dense. When applying to numpy.ndarray objects, we check that operating on the array with zero would always produce a zero.


COO objects support a number of reductions. However, not all important reductions are currently implemented (help welcome!) All of the following currently work:

np.min(x, axis=(0, 2))


If you are performing multiple reductions along the same axes, it may be beneficial to call COO.enable_caching.


This method can take an arbitrary numpy.ufunc and performs a reduction using that method. For example, the following will perform a sum:

x.reduce(np.add, axis=1)


This library currently performs reductions by grouping together all coordinates along the supplied axes and reducing those. Then, if the number in a group is deficient, it reduces an extra time with zero. As a result, if reductions can change by adding multiple zeros to it, this method won’t be accurate. However, it works in most cases.

Partial List of Supported Reductions

Although any binary numpy.ufunc should work for reductions, when calling in the form x.reduction(), the following reductions are supported:


COO arrays can be indexed just like regular numpy.ndarray objects. They support integer, slice and boolean indexing. However, currently, numpy advanced indexing is not properly supported. This means that all of the following work like in Numpy, except that they will produce COO arrays rather than numpy.ndarray objects, and will produce scalars where expected. Assume that z.shape is (5, 6, 7)

z[1, 3]
z[1, 4, 3]
z[:3, :2, 3]
z[::-1, 1, 3]

All of the following will raise an IndexError, like in Numpy 1.13 and later.

z[3, 6]
z[1, 4, 8]

Advanced Indexing

Advanced indexing (indexing arrays with other arrays) is supported, but only for indexing with a single array. Indexing a single array with multiple arrays is not supported at this time. As above, if z.shape is (5, 6, 7), all of the following will work like NumPy:

z[[0, 1, 2]]
z[1, [3]]
z[1, 4, [3, 6]]
z[:3, :2, [1, 5]]

Package Configuration

By default, when performing something like np.array(COO), we allow the array to be converted into a dense one. To prevent this and raise a RuntimeError instead, set the environment variable SPARSE_AUTO_DENSIFY to 0.

If it is desired to raise a warning if creating a sparse array that takes no less memory than an equivalent desne array, set the environment variable SPARSE_WARN_ON_TOO_DENSE to 1.

Other Operations

COO arrays support a number of other common operations. Among them are dot, tensordot, concatenate and stack, transpose and reshape. You can view the full list on the API reference page.


Some operations require zero fill-values (such as nonzero) and others (such as concatenate) require that all inputs have consistent fill-values. For details, check the API reference.