191 lines
6.6 KiB
Python
191 lines
6.6 KiB
Python
|
from itertools import product, permutations
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from numpy.testing import assert_array_less, assert_allclose
|
||
|
|
from pytest import raises as assert_raises
|
||
|
|
|
||
|
|
from scipy.linalg import inv, eigh, norm
|
||
|
|
from scipy.linalg import orthogonal_procrustes
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_ndim_too_large():
|
||
|
|
np.random.seed(1234)
|
||
|
|
A = np.random.randn(3, 4, 5)
|
||
|
|
B = np.random.randn(3, 4, 5)
|
||
|
|
assert_raises(ValueError, orthogonal_procrustes, A, B)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_ndim_too_small():
|
||
|
|
np.random.seed(1234)
|
||
|
|
A = np.random.randn(3)
|
||
|
|
B = np.random.randn(3)
|
||
|
|
assert_raises(ValueError, orthogonal_procrustes, A, B)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_shape_mismatch():
|
||
|
|
np.random.seed(1234)
|
||
|
|
shapes = ((3, 3), (3, 4), (4, 3), (4, 4))
|
||
|
|
for a, b in permutations(shapes, 2):
|
||
|
|
A = np.random.randn(*a)
|
||
|
|
B = np.random.randn(*b)
|
||
|
|
assert_raises(ValueError, orthogonal_procrustes, A, B)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_checkfinite_exception():
|
||
|
|
np.random.seed(1234)
|
||
|
|
m, n = 2, 3
|
||
|
|
A_good = np.random.randn(m, n)
|
||
|
|
B_good = np.random.randn(m, n)
|
||
|
|
for bad_value in np.inf, -np.inf, np.nan:
|
||
|
|
A_bad = A_good.copy()
|
||
|
|
A_bad[1, 2] = bad_value
|
||
|
|
B_bad = B_good.copy()
|
||
|
|
B_bad[1, 2] = bad_value
|
||
|
|
for A, B in ((A_good, B_bad), (A_bad, B_good), (A_bad, B_bad)):
|
||
|
|
assert_raises(ValueError, orthogonal_procrustes, A, B)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_scale_invariance():
|
||
|
|
np.random.seed(1234)
|
||
|
|
m, n = 4, 3
|
||
|
|
for i in range(3):
|
||
|
|
A_orig = np.random.randn(m, n)
|
||
|
|
B_orig = np.random.randn(m, n)
|
||
|
|
R_orig, s = orthogonal_procrustes(A_orig, B_orig)
|
||
|
|
for A_scale in np.square(np.random.randn(3)):
|
||
|
|
for B_scale in np.square(np.random.randn(3)):
|
||
|
|
R, s = orthogonal_procrustes(A_orig * A_scale, B_orig * B_scale)
|
||
|
|
assert_allclose(R, R_orig)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_array_conversion():
|
||
|
|
np.random.seed(1234)
|
||
|
|
for m, n in ((6, 4), (4, 4), (4, 6)):
|
||
|
|
A_arr = np.random.randn(m, n)
|
||
|
|
B_arr = np.random.randn(m, n)
|
||
|
|
As = (A_arr, A_arr.tolist(), np.matrix(A_arr))
|
||
|
|
Bs = (B_arr, B_arr.tolist(), np.matrix(B_arr))
|
||
|
|
R_arr, s = orthogonal_procrustes(A_arr, B_arr)
|
||
|
|
AR_arr = A_arr.dot(R_arr)
|
||
|
|
for A, B in product(As, Bs):
|
||
|
|
R, s = orthogonal_procrustes(A, B)
|
||
|
|
AR = A_arr.dot(R)
|
||
|
|
assert_allclose(AR, AR_arr)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes():
|
||
|
|
np.random.seed(1234)
|
||
|
|
for m, n in ((6, 4), (4, 4), (4, 6)):
|
||
|
|
# Sample a random target matrix.
|
||
|
|
B = np.random.randn(m, n)
|
||
|
|
# Sample a random orthogonal matrix
|
||
|
|
# by computing eigh of a sampled symmetric matrix.
|
||
|
|
X = np.random.randn(n, n)
|
||
|
|
w, V = eigh(X.T + X)
|
||
|
|
assert_allclose(inv(V), V.T)
|
||
|
|
# Compute a matrix with a known orthogonal transformation that gives B.
|
||
|
|
A = np.dot(B, V.T)
|
||
|
|
# Check that an orthogonal transformation from A to B can be recovered.
|
||
|
|
R, s = orthogonal_procrustes(A, B)
|
||
|
|
assert_allclose(inv(R), R.T)
|
||
|
|
assert_allclose(A.dot(R), B)
|
||
|
|
# Create a perturbed input matrix.
|
||
|
|
A_perturbed = A + 1e-2 * np.random.randn(m, n)
|
||
|
|
# Check that the orthogonal procrustes function can find an orthogonal
|
||
|
|
# transformation that is better than the orthogonal transformation
|
||
|
|
# computed from the original input matrix.
|
||
|
|
R_prime, s = orthogonal_procrustes(A_perturbed, B)
|
||
|
|
assert_allclose(inv(R_prime), R_prime.T)
|
||
|
|
# Compute the naive and optimal transformations of the perturbed input.
|
||
|
|
naive_approx = A_perturbed.dot(R)
|
||
|
|
optim_approx = A_perturbed.dot(R_prime)
|
||
|
|
# Compute the Frobenius norm errors of the matrix approximations.
|
||
|
|
naive_approx_error = norm(naive_approx - B, ord='fro')
|
||
|
|
optim_approx_error = norm(optim_approx - B, ord='fro')
|
||
|
|
# Check that the orthogonal Procrustes approximation is better.
|
||
|
|
assert_array_less(optim_approx_error, naive_approx_error)
|
||
|
|
|
||
|
|
|
||
|
|
def _centered(A):
|
||
|
|
mu = A.mean(axis=0)
|
||
|
|
return A - mu, mu
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_exact_example():
|
||
|
|
# Check a small application.
|
||
|
|
# It uses translation, scaling, reflection, and rotation.
|
||
|
|
#
|
||
|
|
# |
|
||
|
|
# a b |
|
||
|
|
# |
|
||
|
|
# d c | w
|
||
|
|
# |
|
||
|
|
# --------+--- x ----- z ---
|
||
|
|
# |
|
||
|
|
# | y
|
||
|
|
# |
|
||
|
|
#
|
||
|
|
A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float)
|
||
|
|
B_orig = np.array([[3, 2], [1, 0], [3, -2], [5, 0]], dtype=float)
|
||
|
|
A, A_mu = _centered(A_orig)
|
||
|
|
B, B_mu = _centered(B_orig)
|
||
|
|
R, s = orthogonal_procrustes(A, B)
|
||
|
|
scale = s / np.square(norm(A))
|
||
|
|
B_approx = scale * np.dot(A, R) + B_mu
|
||
|
|
assert_allclose(B_approx, B_orig, atol=1e-8)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_stretched_example():
|
||
|
|
# Try again with a target with a stretched y axis.
|
||
|
|
A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float)
|
||
|
|
B_orig = np.array([[3, 40], [1, 0], [3, -40], [5, 0]], dtype=float)
|
||
|
|
A, A_mu = _centered(A_orig)
|
||
|
|
B, B_mu = _centered(B_orig)
|
||
|
|
R, s = orthogonal_procrustes(A, B)
|
||
|
|
scale = s / np.square(norm(A))
|
||
|
|
B_approx = scale * np.dot(A, R) + B_mu
|
||
|
|
expected = np.array([[3, 21], [-18, 0], [3, -21], [24, 0]], dtype=float)
|
||
|
|
assert_allclose(B_approx, expected, atol=1e-8)
|
||
|
|
# Check disparity symmetry.
|
||
|
|
expected_disparity = 0.4501246882793018
|
||
|
|
AB_disparity = np.square(norm(B_approx - B_orig) / norm(B))
|
||
|
|
assert_allclose(AB_disparity, expected_disparity)
|
||
|
|
R, s = orthogonal_procrustes(B, A)
|
||
|
|
scale = s / np.square(norm(B))
|
||
|
|
A_approx = scale * np.dot(B, R) + A_mu
|
||
|
|
BA_disparity = np.square(norm(A_approx - A_orig) / norm(A))
|
||
|
|
assert_allclose(BA_disparity, expected_disparity)
|
||
|
|
|
||
|
|
|
||
|
|
def test_orthogonal_procrustes_skbio_example():
|
||
|
|
# This transformation is also exact.
|
||
|
|
# It uses translation, scaling, and reflection.
|
||
|
|
#
|
||
|
|
# |
|
||
|
|
# | a
|
||
|
|
# | b
|
||
|
|
# | c d
|
||
|
|
# --+---------
|
||
|
|
# |
|
||
|
|
# | w
|
||
|
|
# |
|
||
|
|
# | x
|
||
|
|
# |
|
||
|
|
# | z y
|
||
|
|
# |
|
||
|
|
#
|
||
|
|
A_orig = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], dtype=float)
|
||
|
|
B_orig = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], dtype=float)
|
||
|
|
B_standardized = np.array([
|
||
|
|
[-0.13363062, 0.6681531],
|
||
|
|
[-0.13363062, 0.13363062],
|
||
|
|
[-0.13363062, -0.40089186],
|
||
|
|
[0.40089186, -0.40089186]])
|
||
|
|
A, A_mu = _centered(A_orig)
|
||
|
|
B, B_mu = _centered(B_orig)
|
||
|
|
R, s = orthogonal_procrustes(A, B)
|
||
|
|
scale = s / np.square(norm(A))
|
||
|
|
B_approx = scale * np.dot(A, R) + B_mu
|
||
|
|
assert_allclose(B_approx, B_orig)
|
||
|
|
assert_allclose(B / norm(B), B_standardized)
|