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