Convolve1d rewrites

[ricardoV94]

Description

Follow up to #1318

Some of these may apply to higher-dimensional convolutions.

Sum of convolution as product of sum of inputs

If we have convolve1d(x, y, mode="full").sum(), we can rewrite it as x.sum() * y.sum()

import numpy as np

rng = np.random.default_rng(37)
x = rng.normal(size=(5))
y = rng.normal(size=(9))
np.testing.assert_allclose(np.convolve(x, y).sum(), x.sum() * y.sum())

convolve constant kernels

If we have two sequential applications of convolve, with constant inputs we can convolve the constant inputs first, reducing the number of runtime convolutions

import numpy as np

rng = np.random.default_rng(37)
x = rng.normal(size=(5))
y1 = rng.normal(size=(9))
y2 = rng.normal(size=(3,))

r1 = np.convolve(np.convolve(x, y1), y2)
r2 = np.convolve(x, np.convolve(y1, y2))
np.testing.assert_allclose(r1, r2)

Merge convolutions with flipped inputs

Convolutions give the same output regardless of order of inputs, so if we see two with the same inputs but in different order we can merge them. In general, we may want to have an Op property that tells us when the output is invariant to the order of inputs to apply such merge automatically. This Applies to Add, Mul, ...

import numpy as np

rng = np.random.default_rng(37)
x = rng.normal(size=(5))
y = rng.normal(size=(9))

r1 = np.convolve(x, y)
r2 = np.convolve(y, x)
np.testing.assert_allclose(r1, r2)