Associativity and matrix multiplication

The matrix multiplication m1 @ m2 @ m3 can be done in two different ways: (m1 @ m2) @ m3 or m1 @ (m2 @ m3). Are these two orders equivalent or is there a better order?

import pprint
import numpy
import matplotlib.pyplot as plt
from pandas import DataFrame
from tqdm import tqdm
from teachcompute.ext_test_case import measure_time

First try

m1 = numpy.random.rand(100, 100)
m2 = numpy.random.rand(100, 10)
m3 = numpy.random.rand(10, 100)

m = m1 @ m2 @ m3

print(m.shape)

mm1 = (m1 @ m2) @ m3
mm2 = m1 @ (m2 @ m3)

print(mm1.shape, mm2.shape)

t1 = measure_time(lambda: (m1 @ m2) @ m3, context={}, number=50, repeat=50)
pprint.pprint(t1)

t2 = measure_time(lambda: m1 @ (m2 @ m3), context={}, number=50, repeat=50)
pprint.pprint(t2)

(100, 100)
(100, 100) (100, 100)
{'average': 7.918572000035058e-05,
 'context_size': 64,
 'deviation': 7.472517220805028e-05,
 'max_exec': 0.0002890740000020742,
 'min_exec': 2.662199999576842e-05,
 'number': 50,
 'repeat': 50,
 'ttime': 0.003959286000017529,
 'warmup_time': 0.00012929999957123073}
{'average': 0.003669156359999761,
 'context_size': 64,
 'deviation': 0.0058324362524603416,
 'max_exec': 0.021446986000000834,
 'min_exec': 7.507799999984854e-05,
 'number': 50,
 'repeat': 50,
 'ttime': 0.18345781799998806,
 'warmup_time': 0.025857000000087282}

With different sizes

obs = []
for i in tqdm([50, 100, 125, 150, 175, 200]):
    m1 = numpy.random.rand(i, i)
    m2 = numpy.random.rand(i, 10)
    m3 = numpy.random.rand(10, i)

    t1 = measure_time(lambda: (m1 @ m2) @ m3, context={}, number=50, repeat=50)
    t1["formula"] = "(m1 @ m2) @ m3"
    t1["size"] = i
    obs.append(t1)
    t2 = measure_time(lambda: m1 @ (m2 @ m3), context={}, number=50, repeat=50)
    t2["formula"] = "m1 @ (m2 @ m3)"
    t2["size"] = i
    obs.append(t2)

df = DataFrame(obs)
piv = df.pivot(index="size", columns="formula", values="average")
piv

  0%|          | 0/6 [00:00<?, ?it/s]
 17%|█▋        | 1/6 [00:00<00:00,  5.72it/s]
 33%|███▎      | 2/6 [00:02<00:04,  1.16s/it]
 50%|█████     | 3/6 [00:02<00:02,  1.12it/s]
 67%|██████▋   | 4/6 [00:04<00:02,  1.10s/it]
 83%|████████▎ | 5/6 [00:04<00:01,  1.04s/it]
100%|██████████| 6/6 [00:06<00:00,  1.26s/it]
100%|██████████| 6/6 [00:06<00:00,  1.11s/it]

formula	(m1 @ m2) @ m3	m1 @ (m2 @ m3)
size
50	0.000027	0.000041
100	0.000040	0.000692
125	0.000070	0.000160
150	0.000081	0.000488
175	0.000086	0.000285
200	0.000131	0.000540

Graph

fig, ax = plt.subplots(1, 2, figsize=(12, 4))
piv.plot(
    logx=True,
    logy=True,
    ax=ax[0],
    title=f"{m1.shape!r} @ {m2.shape!r} @ " f"{m3.shape!r}".replace("200", "size"),
)
piv["ratio"] = piv["m1 @ (m2 @ m3)"] / piv["(m1 @ m2) @ m3"]
piv[["ratio"]].plot(ax=ax[1])

<Axes: xlabel='size'>