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# Copyright © 2023 Apple Inc.
import itertools
import math
import unittest
import mlx.core as mx
import mlx_tests
import numpy as np
class TestLinalg(mlx_tests.MLXTestCase):
def test_norm(self):
vector_ords = [None, 0.5, 0, 1, 2, 3, -1, float("inf"), -float("inf")]
matrix_ords = [None, "fro", "nuc", -1, 1, -2, 2, float("inf"), -float("inf")]
for shape in [(3,), (2, 3), (2, 3, 3)]:
x_mx = mx.arange(1, math.prod(shape) + 1, dtype=mx.float32).reshape(shape)
x_np = np.arange(1, math.prod(shape) + 1, dtype=np.float32).reshape(shape)
# Test when at least one axis is provided
for num_axes in range(1, len(shape)):
if num_axes == 1:
ords = vector_ords
else:
ords = matrix_ords
for axis in itertools.combinations(range(len(shape)), num_axes):
for keepdims in [True, False]:
for o in ords:
stream = (
mx.cpu if o in ["nuc", -2, 2] else mx.default_device()
)
out_np = np.linalg.norm(
x_np, ord=o, axis=axis, keepdims=keepdims
)
out_mx = mx.linalg.norm(
x_mx, ord=o, axis=axis, keepdims=keepdims, stream=stream
)
with self.subTest(
shape=shape, ord=o, axis=axis, keepdims=keepdims
):
self.assertTrue(
np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6)
)
# Test only ord provided
for shape in [(3,), (2, 3)]:
x_mx = mx.arange(1, math.prod(shape) + 1).reshape(shape)
x_np = np.arange(1, math.prod(shape) + 1).reshape(shape)
for o in [None, 1, -1, float("inf"), -float("inf")]:
for keepdims in [True, False]:
out_np = np.linalg.norm(x_np, ord=o, keepdims=keepdims)
out_mx = mx.linalg.norm(x_mx, ord=o, keepdims=keepdims)
with self.subTest(shape=shape, ord=o, keepdims=keepdims):
self.assertTrue(
np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6)
)
# Test no ord and no axis provided
for shape in [(3,), (2, 3), (2, 3, 3)]:
x_mx = mx.arange(1, math.prod(shape) + 1).reshape(shape)
x_np = np.arange(1, math.prod(shape) + 1).reshape(shape)
for keepdims in [True, False]:
out_np = np.linalg.norm(x_np, keepdims=keepdims)
out_mx = mx.linalg.norm(x_mx, keepdims=keepdims)
with self.subTest(shape=shape, keepdims=keepdims):
self.assertTrue(np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6))
def test_complex_norm(self):
for shape in [(3,), (2, 3), (2, 3, 3)]:
x_np = np.random.uniform(size=shape).astype(
np.float32
) + 1j * np.random.uniform(size=shape).astype(np.float32)
x_mx = mx.array(x_np)
out_np = np.linalg.norm(x_np)
out_mx = mx.linalg.norm(x_mx)
with self.subTest(shape=shape):
self.assertTrue(np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6))
for num_axes in range(1, len(shape)):
for axis in itertools.combinations(range(len(shape)), num_axes):
out_np = np.linalg.norm(x_np, axis=axis)
out_mx = mx.linalg.norm(x_mx, axis=axis)
with self.subTest(shape=shape, axis=axis):
self.assertTrue(
np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6)
)
x_np = np.random.uniform(size=(4, 4)).astype(
np.float32
) + 1j * np.random.uniform(size=(4, 4)).astype(np.float32)
x_mx = mx.array(x_np)
out_np = np.linalg.norm(x_np, ord="fro")
out_mx = mx.linalg.norm(x_mx, ord="fro")
self.assertTrue(np.allclose(out_np, out_mx, atol=1e-5, rtol=1e-6))
def test_qr_factorization(self):
with self.assertRaises(ValueError):
mx.linalg.qr(mx.array(0.0))
with self.assertRaises(ValueError):
mx.linalg.qr(mx.array([0.0, 1.0]))
with self.assertRaises(ValueError):
mx.linalg.qr(mx.array([[0, 1], [1, 0]]))
A = mx.array([[2.0, 3.0], [1.0, 2.0]])
Q, R = mx.linalg.qr(A, stream=mx.cpu)
out = Q @ R
self.assertTrue(mx.allclose(out, A))
out = Q.T @ Q
self.assertTrue(mx.allclose(out, mx.eye(2), rtol=1e-5, atol=1e-7))
self.assertTrue(mx.allclose(mx.tril(R, -1), mx.zeros_like(R)))
self.assertEqual(Q.dtype, mx.float32)
self.assertEqual(R.dtype, mx.float32)
# Multiple matrices
B = mx.array([[-1.0, 2.0], [-4.0, 1.0]])
A = mx.stack([A, B])
Q, R = mx.linalg.qr(A, stream=mx.cpu)
for a, q, r in zip(A, Q, R):
out = q @ r
self.assertTrue(mx.allclose(out, a))
out = q.T @ q
self.assertTrue(mx.allclose(out, mx.eye(2), rtol=1e-5, atol=1e-7))
self.assertTrue(mx.allclose(mx.tril(r, -1), mx.zeros_like(r)))
# Non square matrices
for shape in [(4, 8), (8, 4)]:
A = mx.random.uniform(shape=shape)
Q, R = mx.linalg.qr(A, stream=mx.cpu)
out = Q @ R
self.assertTrue(mx.allclose(out, A, rtol=1e-4, atol=1e-6))
out = Q.T @ Q
self.assertTrue(
mx.allclose(out, mx.eye(min(A.shape)), rtol=1e-4, atol=1e-6)
)
def test_svd_decomposition(self):
A = mx.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]], dtype=mx.float32)
U, S, Vt = mx.linalg.svd(A, compute_uv=True, stream=mx.cpu)
self.assertTrue(
mx.allclose(U[:, : len(S)] @ mx.diag(S) @ Vt, A, rtol=1e-5, atol=1e-7)
)
S = mx.linalg.svd(A, compute_uv=False, stream=mx.cpu)
self.assertTrue(
mx.allclose(
mx.linalg.norm(S), mx.linalg.norm(A, ord="fro"), rtol=1e-5, atol=1e-7
)
)
# Multiple matrices
B = A + 10.0
AB = mx.stack([A, B])
Us, Ss, Vts = mx.linalg.svd(AB, compute_uv=True, stream=mx.cpu)
for M, U, S, Vt in zip([A, B], Us, Ss, Vts):
self.assertTrue(
mx.allclose(U[:, : len(S)] @ mx.diag(S) @ Vt, M, rtol=1e-5, atol=1e-7)
)
Ss = mx.linalg.svd(AB, compute_uv=False, stream=mx.cpu)
for M, S in zip([A, B], Ss):
self.assertTrue(
mx.allclose(
mx.linalg.norm(S),
mx.linalg.norm(M, ord="fro"),
rtol=1e-5,
atol=1e-7,
)
)
def test_inverse(self):
A = mx.array([[1, 2, 3], [6, -5, 4], [-9, 8, 7]], dtype=mx.float32)
A_inv = mx.linalg.inv(A, stream=mx.cpu)
self.assertTrue(mx.allclose(A @ A_inv, mx.eye(A.shape[0]), rtol=0, atol=1e-6))
# Multiple matrices
B = A - 100
AB = mx.stack([A, B])
invs = mx.linalg.inv(AB, stream=mx.cpu)
for M, M_inv in zip(AB, invs):
self.assertTrue(
mx.allclose(M @ M_inv, mx.eye(M.shape[0]), rtol=0, atol=1e-5)
)
def test_tri_inverse(self):
for upper in (False, True):
A = mx.array([[1, 0, 0], [6, -5, 0], [-9, 8, 7]], dtype=mx.float32)
B = mx.array([[7, 0, 0], [3, -2, 0], [1, 8, 3]], dtype=mx.float32)
if upper:
A = A.T
B = B.T
AB = mx.stack([A, B])
invs = mx.linalg.tri_inv(AB, upper=upper, stream=mx.cpu)
for M, M_inv in zip(AB, invs):
self.assertTrue(
mx.allclose(M @ M_inv, mx.eye(M.shape[0]), rtol=0, atol=1e-5)
)
# Ensure that tri_inv will 0-out the supposedly 0 triangle
x = mx.random.normal((2, 8, 8))
y1 = mx.linalg.tri_inv(x, upper=True, stream=mx.cpu)
y2 = mx.linalg.tri_inv(x, upper=False, stream=mx.cpu)
self.assertTrue(mx.all(y1 == mx.triu(y1)))
self.assertTrue(mx.all(y2 == mx.tril(y2)))
def test_cholesky(self):
sqrtA = mx.array(
[[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]], dtype=mx.float32
)
A = sqrtA.T @ sqrtA / 81
L = mx.linalg.cholesky(A, stream=mx.cpu)
U = mx.linalg.cholesky(A, upper=True, stream=mx.cpu)
self.assertTrue(mx.allclose(L @ L.T, A, rtol=1e-5, atol=1e-7))
self.assertTrue(mx.allclose(U.T @ U, A, rtol=1e-5, atol=1e-7))
# Multiple matrices
B = A + 1 / 9
AB = mx.stack([A, B])
Ls = mx.linalg.cholesky(AB, stream=mx.cpu)
for M, L in zip(AB, Ls):
self.assertTrue(mx.allclose(L @ L.T, M, rtol=1e-5, atol=1e-7))
def test_pseudo_inverse(self):
A = mx.array([[1, 2, 3], [6, -5, 4], [-9, 8, 7]], dtype=mx.float32)
A_plus = mx.linalg.pinv(A, stream=mx.cpu)
self.assertTrue(mx.allclose(A @ A_plus @ A, A, rtol=0, atol=1e-5))
# Multiple matrices
B = A - 100
AB = mx.stack([A, B])
pinvs = mx.linalg.pinv(AB, stream=mx.cpu)
for M, M_plus in zip(AB, pinvs):
self.assertTrue(mx.allclose(M @ M_plus @ M, M, rtol=0, atol=1e-3))
# Test singular matrix
A = mx.array([[4.0, 1.0], [4.0, 1.0]])
A_plus = mx.linalg.pinv(A, stream=mx.cpu)
self.assertTrue(mx.allclose(A @ A_plus @ A, A))
def test_cholesky_inv(self):
mx.random.seed(7)
sqrtA = mx.array(
[[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]], dtype=mx.float32
)
A = sqrtA.T @ sqrtA / 81
N = 3
A = mx.random.uniform(shape=(N, N))
A = A @ A.T
for upper in (False, True):
L = mx.linalg.cholesky(A, upper=upper, stream=mx.cpu)
A_inv = mx.linalg.cholesky_inv(L, upper=upper, stream=mx.cpu)
self.assertTrue(mx.allclose(A @ A_inv, mx.eye(N), atol=1e-4))
# Multiple matrices
B = A + 1 / 9
AB = mx.stack([A, B])
Ls = mx.linalg.cholesky(AB, stream=mx.cpu)
for upper in (False, True):
Ls = mx.linalg.cholesky(AB, upper=upper, stream=mx.cpu)
AB_inv = mx.linalg.cholesky_inv(Ls, upper=upper, stream=mx.cpu)
for M, M_inv in zip(AB, AB_inv):
self.assertTrue(mx.allclose(M @ M_inv, mx.eye(N), atol=1e-4))
def test_cross_product(self):
a = mx.array([1.0, 2.0, 3.0])
b = mx.array([4.0, 5.0, 6.0])
result = mx.linalg.cross(a, b)
expected = np.cross(a, b)
self.assertTrue(np.allclose(result, expected))
# Test with negative values
a = mx.array([-1.0, -2.0, -3.0])
b = mx.array([4.0, -5.0, 6.0])
result = mx.linalg.cross(a, b)
expected = np.cross(a, b)
self.assertTrue(np.allclose(result, expected))
# Test with integer values
a = mx.array([1, 2, 3])
b = mx.array([4, 5, 6])
result = mx.linalg.cross(a, b)
expected = np.cross(a, b)
self.assertTrue(np.allclose(result, expected))
# Test with 2D arrays and axis parameter
a = mx.array([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]])
b = mx.array([[4.0, 5.0, 6.0], [1.0, 2.0, 3.0]])
result = mx.linalg.cross(a, b, axis=1)
expected = np.cross(a, b, axis=1)
self.assertTrue(np.allclose(result, expected))
# Test with broadcast
a = mx.random.uniform(shape=(2, 1, 3))
b = mx.random.uniform(shape=(1, 2, 3))
result = mx.linalg.cross(a, b)
expected = np.cross(a, b)
self.assertTrue(np.allclose(result, expected))
# Type promotion
a = mx.array([1.0, 2.0, 3.0])
b = mx.array([4, 5, 6])
result = mx.linalg.cross(a, b)
expected = np.cross(a, b)
self.assertTrue(np.allclose(result, expected))
# Test with incorrect vector size (should raise an exception)
a = mx.array([1.0])
b = mx.array([4.0])
with self.assertRaises(ValueError):
mx.linalg.cross(a, b)
def test_eig(self):
tols = {"atol": 1e-5, "rtol": 1e-5}
def check_eigs_and_vecs(A_np, kwargs={}):
A = mx.array(A_np)
eig_vals, eig_vecs = mx.linalg.eig(A, stream=mx.cpu, **kwargs)
self.assertTrue(
mx.allclose(A @ eig_vecs, eig_vals[..., None, :] * eig_vecs, **tols)
)
eig_vals_only = mx.linalg.eigvals(A, stream=mx.cpu, **kwargs)
self.assertTrue(mx.allclose(eig_vals, eig_vals_only, **tols))
# Test a simple 2x2 matrix
A_np = np.array([[1.0, 1.0], [3.0, 4.0]], dtype=np.float32)
check_eigs_and_vecs(A_np)
# Test complex eigenvalues
A_np = np.array([[1.0, -1.0], [1.0, 1.0]], dtype=np.float32)
check_eigs_and_vecs(A_np)
# Test a larger random symmetric matrix
n = 5
np.random.seed(1)
A_np = np.random.randn(n, n).astype(np.float32)
check_eigs_and_vecs(A_np)
# Test with batched input
A_np = np.random.randn(3, n, n).astype(np.float32)
check_eigs_and_vecs(A_np)
# Test error cases
with self.assertRaises(ValueError):
mx.linalg.eig(mx.array([1.0, 2.0])) # 1D array
with self.assertRaises(ValueError):
mx.linalg.eig(
mx.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
) # Non-square matrix
with self.assertRaises(ValueError):
mx.linalg.eigvals(mx.array([1.0, 2.0])) # 1D array
with self.assertRaises(ValueError):
mx.linalg.eigvals(
mx.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
) # Non-square matrix
def test_eigh(self):
tols = {"atol": 1e-5, "rtol": 1e-5}
def check_eigs_and_vecs(A_np, kwargs={}):
A = mx.array(A_np)
eig_vals, eig_vecs = mx.linalg.eigh(A, stream=mx.cpu, **kwargs)
eig_vals_np, _ = np.linalg.eigh(A_np, **kwargs)
self.assertTrue(np.allclose(eig_vals, eig_vals_np, **tols))
self.assertTrue(
mx.allclose(A @ eig_vecs, eig_vals[..., None, :] * eig_vecs, **tols)
)
eig_vals_only = mx.linalg.eigvalsh(A, stream=mx.cpu, **kwargs)
self.assertTrue(mx.allclose(eig_vals, eig_vals_only, **tols))
# Test a simple 2x2 symmetric matrix
A_np = np.array([[1.0, 2.0], [2.0, 4.0]], dtype=np.float32)
check_eigs_and_vecs(A_np)
# Test a larger random symmetric matrix
n = 5
np.random.seed(1)
A_np = np.random.randn(n, n).astype(np.float32)
A_np = (A_np + A_np.T) / 2
check_eigs_and_vecs(A_np)
# Test with upper triangle
check_eigs_and_vecs(A_np, {"UPLO": "U"})
# Test with batched input
A_np = np.random.randn(3, n, n).astype(np.float32)
A_np = (A_np + np.transpose(A_np, (0, 2, 1))) / 2
check_eigs_and_vecs(A_np)
# Test with complex inputs
A_np = (
np.random.randn(8, 8, 2).astype(np.float32).view(np.complex64).squeeze(-1)
)
A_np = A_np + A_np.T.conj()
check_eigs_and_vecs(A_np)
# Test error cases
with self.assertRaises(ValueError):
mx.linalg.eigh(mx.array([1.0, 2.0])) # 1D array
with self.assertRaises(ValueError):
mx.linalg.eigh(
mx.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
) # Non-square matrix
with self.assertRaises(ValueError):
mx.linalg.eigvalsh(mx.array([1.0, 2.0])) # 1D array
with self.assertRaises(ValueError):
mx.linalg.eigvalsh(
mx.array([[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]])
) # Non-square matrix
def test_lu(self):
with self.assertRaises(ValueError):
mx.linalg.lu(mx.array(0.0), stream=mx.cpu)
with self.assertRaises(ValueError):
mx.linalg.lu(mx.array([0.0, 1.0]), stream=mx.cpu)
with self.assertRaises(ValueError):
mx.linalg.lu(mx.array([[0, 1], [1, 0]]), stream=mx.cpu)
# Test 3x3 matrix
a = mx.array([[3.0, 1.0, 2.0], [1.0, 8.0, 6.0], [9.0, 2.0, 5.0]])
P, L, U = mx.linalg.lu(a, stream=mx.cpu)
self.assertTrue(mx.allclose(L[P, :] @ U, a))
# Test batch dimension
a = mx.broadcast_to(a, (5, 5, 3, 3))
P, L, U = mx.linalg.lu(a, stream=mx.cpu)
L = mx.take_along_axis(L, P[..., None], axis=-2)
self.assertTrue(mx.allclose(L @ U, a))
# Test non-square matrix
a = mx.array([[3.0, 1.0, 2.0], [1.0, 8.0, 6.0]])
P, L, U = mx.linalg.lu(a, stream=mx.cpu)
self.assertTrue(mx.allclose(L[P, :] @ U, a))
a = mx.array([[3.0, 1.0], [1.0, 8.0], [9.0, 2.0]])
P, L, U = mx.linalg.lu(a, stream=mx.cpu)
self.assertTrue(mx.allclose(L[P, :] @ U, a))
def test_lu_factor(self):
mx.random.seed(7)
# Test 3x3 matrix
a = mx.random.uniform(shape=(5, 5))
LU, pivots = mx.linalg.lu_factor(a, stream=mx.cpu)
n = a.shape[-1]
pivots = pivots.tolist()
perm = list(range(n))
for i in range(len(pivots)):
perm[i], perm[pivots[i]] = perm[pivots[i]], perm[i]
L = mx.add(mx.tril(LU, k=-1), mx.eye(n))
U = mx.triu(LU)
self.assertTrue(mx.allclose(L @ U, a[perm, :]))
def test_solve(self):
mx.random.seed(7)
# Test 3x3 matrix with 1D rhs
a = mx.array([[3.0, 1.0, 2.0], [1.0, 8.0, 6.0], [9.0, 2.0, 5.0]])
b = mx.array([11.0, 35.0, 28.0])
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
# Test symmetric positive-definite matrix
N = 5
a = mx.random.uniform(shape=(N, N))
a = mx.matmul(a, a.T) + N * mx.eye(N)
b = mx.random.uniform(shape=(N, 1))
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
# Test batch dimension
a = mx.random.uniform(shape=(5, 5, 4, 4))
b = mx.random.uniform(shape=(5, 5, 4, 1))
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected, atol=1e-5))
# Test large matrix
N = 1000
a = mx.random.uniform(shape=(N, N))
b = mx.random.uniform(shape=(N, 1))
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected, atol=1e-3))
# Test multi-column rhs
a = mx.random.uniform(shape=(5, 5))
b = mx.random.uniform(shape=(5, 8))
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
# Test batched multi-column rhs
a = mx.broadcast_to(a, (3, 2, 5, 5))
b = mx.broadcast_to(b, (3, 1, 5, 8))
result = mx.linalg.solve(a, b, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected, rtol=1e-5, atol=1e-5))
def test_solve_triangular(self):
# Test lower triangular matrix
a = mx.array([[4.0, 0.0, 0.0], [2.0, 3.0, 0.0], [1.0, -2.0, 5.0]])
b = mx.array([8.0, 14.0, 3.0])
result = mx.linalg.solve_triangular(a, b, upper=False, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
# Test upper triangular matrix
a = mx.array([[3.0, 2.0, 1.0], [0.0, 5.0, 4.0], [0.0, 0.0, 6.0]])
b = mx.array([13.0, 33.0, 18.0])
result = mx.linalg.solve_triangular(a, b, upper=True, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
# Test batch multi-column rhs
a = mx.broadcast_to(a, (3, 4, 3, 3))
b = mx.broadcast_to(mx.expand_dims(b, -1), (3, 4, 3, 8))
result = mx.linalg.solve_triangular(a, b, upper=True, stream=mx.cpu)
expected = np.linalg.solve(a, b)
self.assertTrue(np.allclose(result, expected))
if __name__ == "__main__":
mlx_tests.MLXTestRunner()
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