Place and Scale Sparse Blocks in Large Sparse Matrices with SciPy sp.kron (Kronecker Product)

Place and scale a sparse subarray across a large sparse matrix using SciPy sp.kron. Supports arbitrary patterns efficiently without dense intermediates.

Placing the same sparse subarray at many positions inside a much larger sparse matrix is a common need: the overall shape is known, the pattern of placements is known or generated programmatically, and N can be large enough that dense operations are not an option. The question is how to do this without manually specifying every location and without materializing a mostly empty array.

Problem setup

Consider a sparse subarray whose contents and shape are known:

import scipy.sparse as sps

tile = sps.coo_array([[a, b], [c, d]])

The goal is to place this subarray repeatedly in an N×N sparse array according to an arbitrary pattern. For illustration, with a contrived pattern, if N = 4, the desired result might be:

[[a, b, a, b]
 [c, d, c, d]
 [0, 0, a, b]
 [0, 0, c, d]]

and if N = 8:

[[a, b, a, b, 0, 0, 0, 0]
 [c, d, c, d, 0, 0, 0, 0]
 [0, 0, a, b, a, b, 0, 0]
 [0, 0, c, d, c, d, 0, 0]
 [0, 0, 0, 0, a, b, a, b]
 [0, 0, 0, 0, c, d, c, d]
 [0, 0, 0, 0, 0, 0, a, b]
 [0, 0, 0, 0, 0, 0, c, d]]

N can be huge (tens of thousands), so a dense array is impractical and it’s not feasible to enumerate all placements by hand. While sp.block_array requires manual placement and sp.block_diag only covers diagonal tiling, the pattern here is arbitrary.

What’s really going on and why it’s tricky

The task is to replicate a small sparse block across a larger sparse canvas wherever a pattern matrix indicates, preferably with a way to scale each placed block. Manually assembling the large sparse structure is tedious and, for very large N, error-prone. Building any dense helper structures defeats the purpose of going sparse in the first place. The key is to use an operation that expands a sparse “pattern” into repeated blocks efficiently.

Solution: use a Kronecker product (sp.kron)

sp.kron does exactly that. Provide a sparse “pattern” matrix that holds the coefficients of where blocks should appear, and the small block itself. The resulting sparse matrix places a scaled copy of the block at every nonzero in the pattern. This approach naturally supports arbitrary patterns, including random locations, without constructing dense intermediates.

Below is a complete example tailored to the contrived pattern, including coefficients for per-block scaling:

import scipy.sparse as sps
import numpy as npy

# Subarray and big array parameters
val1, val2, val3, val4 = 1, 2, 3, 4
blk = sps.coo_array([[val1, val2], [val3, val4]])
dim = 8

# Block locations for an arbitrary pattern
r_idx = npy.hstack((npy.arange(dim/blk.shape[0], dtype=int),
                    npy.arange(dim/blk.shape[0]-1, dtype=int)))
c_idx = npy.hstack((npy.arange(dim/blk.shape[1], dtype=int),
                    npy.arange(dim/blk.shape[0]-1, dtype=int) + 1))
scales = npy.ones_like(r_idx)
pattern = sps.csc_array((scales, (r_idx, c_idx)))

# Optional: visualize where the top-left corners land
print(f"Placing block top left corners at rows {r_idx*blk.shape[0]}, cols {c_idx*blk.shape[1]}")

# Build the large sparse array in one line
big_sparse = sps.kron(pattern, blk)

print(big_sparse.toarray())

The printed dense view matches the target layout:

[[1 2 1 2 0 0 0 0]
 [3 4 3 4 0 0 0 0]
 [0 0 1 2 1 2 0 0]
 [0 0 3 4 3 4 0 0]
 [0 0 0 0 1 2 1 2]
 [0 0 0 0 3 4 3 4]
 [0 0 0 0 0 0 1 2]
 [0 0 0 0 0 0 3 4]]

Why this matters

This method scales to very large N because it never requires a dense representation of the pattern or the result. It supports any placement pattern, including random sets of locations, and it cleanly incorporates per-block coefficients. It also avoids computing per-element indices within each block: the expansion is handled by the Kronecker product directly.

Takeaways

If you need to place one sparse subarray many times into a larger sparse array according to an arbitrary pattern, construct a sparse pattern matrix of block coefficients and apply sp.kron with your subarray. This keeps memory use aligned with sparsity, avoids manual placement, and gives you a compact way to scale blocks wherever needed.