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242 lines
8.2 KiB
Python
242 lines
8.2 KiB
Python
# Copyright 2021 The JAX Authors.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# https://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License
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"""Tests for the library of QDWH-based polar decomposition."""
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import functools
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import jax
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from jax.config import config
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import jax.numpy as jnp
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import numpy as np
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import scipy.linalg as osp_linalg
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from jax._src.lax import qdwh
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from jax._src import test_util as jtu
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from absl.testing import absltest
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config.parse_flags_with_absl()
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_JAX_ENABLE_X64_QDWH = config.x64_enabled
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# Input matrix data type for QdwhTest.
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_QDWH_TEST_DTYPE = np.float64 if _JAX_ENABLE_X64_QDWH else np.float32
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# Machine epsilon used by QdwhTest.
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_QDWH_TEST_EPS = jnp.finfo(_QDWH_TEST_DTYPE).eps
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# Largest log10 value of condition numbers used by QdwhTest.
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_MAX_LOG_CONDITION_NUM = np.log10(int(1 / _QDWH_TEST_EPS))
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def _check_symmetry(x: jax.Array) -> bool:
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"""Check if the array is symmetric."""
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m, n = x.shape
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eps = jnp.finfo(x.dtype).eps
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tol = 50.0 * eps
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is_hermitian = False
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if m == n:
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if np.linalg.norm(x - x.T.conj()) / np.linalg.norm(x) < tol:
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is_hermitian = True
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return is_hermitian
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def _compute_relative_diff(actual, expected):
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"""Computes relative difference between two matrices."""
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return np.linalg.norm(actual - expected) / np.linalg.norm(expected)
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_dot = functools.partial(jnp.dot, precision="highest")
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class QdwhTest(jtu.JaxTestCase):
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@jtu.sample_product(
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[dict(m=m, n=n) for m, n in [(8, 6), (10, 10), (20, 18)]],
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log_cond=np.linspace(1, _MAX_LOG_CONDITION_NUM, 4),
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)
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def testQdwhUnconvergedAfterMaxNumberIterations(
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self, m, n, log_cond):
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"""Tests unconvergence after maximum number of iterations."""
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a = jnp.triu(jnp.ones((m, n)))
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u, s, v = jnp.linalg.svd(a, full_matrices=False)
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cond = 10**log_cond
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s = jnp.expand_dims(jnp.linspace(cond, 1, min(m, n)), range(u.ndim - 1))
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with jax.numpy_dtype_promotion('standard'):
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a = (u * s) @ v
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is_hermitian = _check_symmetry(a)
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max_iterations = 2
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_, _, actual_num_iterations, is_converged = qdwh.qdwh(
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a, is_hermitian=is_hermitian, max_iterations=max_iterations)
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with self.subTest('Number of iterations.'):
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self.assertEqual(max_iterations, actual_num_iterations)
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with self.subTest('Converged.'):
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self.assertFalse(is_converged)
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@jtu.sample_product(
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[dict(m=m, n=n) for m, n in [(8, 6), (10, 10), (20, 18)]],
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log_cond=np.linspace(1, _MAX_LOG_CONDITION_NUM, 4),
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)
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def testQdwhWithUpperTriangularInputAllOnes(self, m, n, log_cond):
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"""Tests qdwh with upper triangular input of all ones."""
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a = jnp.triu(jnp.ones((m, n))).astype(_QDWH_TEST_DTYPE)
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u, s, v = jnp.linalg.svd(a, full_matrices=False)
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cond = 10**log_cond
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s = jnp.expand_dims(jnp.linspace(cond, 1, min(m, n)), range(u.ndim - 1))
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a = (u * s) @ v
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is_hermitian = _check_symmetry(a)
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max_iterations = 10
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actual_u, actual_h, _, _ = qdwh.qdwh(a, is_hermitian=is_hermitian,
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max_iterations=max_iterations)
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expected_u, expected_h = osp_linalg.polar(a)
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# Sets the test tolerance.
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rtol = 1E6 * _QDWH_TEST_EPS
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with self.subTest('Test u.'):
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relative_diff_u = _compute_relative_diff(actual_u, expected_u)
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np.testing.assert_almost_equal(relative_diff_u, 1E-6, decimal=5)
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with self.subTest('Test h.'):
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relative_diff_h = _compute_relative_diff(actual_h, expected_h)
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np.testing.assert_almost_equal(relative_diff_h, 1E-6, decimal=5)
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with self.subTest('Test u.dot(h).'):
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a_round_trip = _dot(actual_u, actual_h)
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relative_diff_a = _compute_relative_diff(a_round_trip, a)
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np.testing.assert_almost_equal(relative_diff_a, 1E-6, decimal=5)
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with self.subTest('Test orthogonality.'):
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actual_results = _dot(actual_u.T, actual_u)
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expected_results = np.eye(n)
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self.assertAllClose(
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actual_results, expected_results, rtol=rtol, atol=1E-5)
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@jtu.sample_product(
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[dict(m=m, n=n) for m, n in [(6, 6), (8, 4)]],
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padding=(None, (3, 2)),
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log_cond=np.linspace(1, 4, 4),
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)
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def testQdwhWithRandomMatrix(self, m, n, log_cond, padding):
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"""Tests qdwh with random input."""
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rng = jtu.rand_uniform(self.rng(), low=0.3, high=0.9)
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a = rng((m, n), _QDWH_TEST_DTYPE)
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u, s, v = jnp.linalg.svd(a, full_matrices=False)
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cond = 10**log_cond
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s = jnp.expand_dims(jnp.linspace(cond, 1, min(m, n)), range(u.ndim - 1))
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a = (u * s) @ v
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is_hermitian = _check_symmetry(a)
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max_iterations = 10
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def lsp_linalg_fn(a):
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if padding is not None:
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pm, pn = padding
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a = jnp.pad(a, [(0, pm), (0, pn)], constant_values=jnp.nan)
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u, h, _, _ = qdwh.qdwh(
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a, is_hermitian=is_hermitian, max_iterations=max_iterations,
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dynamic_shape=(m, n) if padding else None)
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if padding is not None:
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u = u[:m, :n]
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h = h[:n, :n]
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return u, h
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args_maker = lambda: [a]
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# Sets the test tolerance.
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rtol = 1E6 * _QDWH_TEST_EPS
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with self.subTest('Test JIT compatibility'):
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self._CompileAndCheck(lsp_linalg_fn, args_maker)
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with self.subTest('Test against numpy.'):
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self._CheckAgainstNumpy(osp_linalg.polar, lsp_linalg_fn, args_maker,
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rtol=rtol, atol=1E-3)
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@jtu.sample_product(
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[dict(m=m, n=n) for m, n in [(10, 10), (8, 8)]],
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log_cond=np.linspace(1, 4, 4),
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)
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def testQdwhWithOnRankDeficientInput(self, m, n, log_cond):
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"""Tests qdwh with rank-deficient input."""
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a = jnp.triu(jnp.ones((m, n))).astype(_QDWH_TEST_DTYPE)
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# Generates a rank-deficient input.
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u, s, v = jnp.linalg.svd(a, full_matrices=False)
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cond = 10**log_cond
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s = jnp.linspace(cond, 1, min(m, n))
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s = jnp.expand_dims(s.at[-1].set(0), range(u.ndim - 1))
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a = (u * s) @ v
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is_hermitian = _check_symmetry(a)
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max_iterations = 15
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actual_u, actual_h, _, _ = qdwh.qdwh(a, is_hermitian=is_hermitian,
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max_iterations=max_iterations)
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_, expected_h = osp_linalg.polar(a)
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# Sets the test tolerance.
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rtol = 1E4 * _QDWH_TEST_EPS
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# For rank-deficient matrix, `u` is not unique.
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with self.subTest('Test h.'):
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relative_diff_h = _compute_relative_diff(actual_h, expected_h)
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np.testing.assert_almost_equal(relative_diff_h, 1E-6, decimal=5)
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with self.subTest('Test u.dot(h).'):
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a_round_trip = _dot(actual_u, actual_h)
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relative_diff_a = _compute_relative_diff(a_round_trip, a)
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np.testing.assert_almost_equal(relative_diff_a, 1E-6, decimal=5)
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with self.subTest('Test orthogonality.'):
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actual_results = _dot(actual_u.T.conj(), actual_u)
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expected_results = np.eye(n)
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self.assertAllClose(
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actual_results, expected_results, rtol=rtol, atol=1E-6)
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@jtu.sample_product(
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[dict(m=m, n=n, r=r, c=c) for m, n, r, c in [(4, 3, 1, 1), (5, 2, 0, 0)]],
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dtype=jtu.dtypes.floating,
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)
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def testQdwhWithTinyElement(self, m, n, r, c, dtype):
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"""Tests qdwh on matrix with zeros and close-to-zero entries."""
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a = jnp.zeros((m, n), dtype=dtype)
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tiny_elem = jnp.finfo(a).tiny
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a = a.at[r, c].set(tiny_elem)
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is_hermitian = _check_symmetry(a)
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max_iterations = 10
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@jax.jit
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def lsp_linalg_fn(a):
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u, h, _, _ = qdwh.qdwh(
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a, is_hermitian=is_hermitian, max_iterations=max_iterations)
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return u, h
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actual_u, actual_h = lsp_linalg_fn(a)
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expected_u = jnp.zeros((m, n), dtype=dtype)
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expected_u = expected_u.at[r, c].set(1.0)
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with self.subTest('Test u.'):
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np.testing.assert_array_equal(expected_u, actual_u)
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expected_h = jnp.zeros((n, n), dtype=dtype)
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expected_h = expected_h.at[r, c].set(tiny_elem)
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with self.subTest('Test h.'):
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np.testing.assert_array_equal(expected_h, actual_h)
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if __name__ == '__main__':
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absltest.main(testLoader=jtu.JaxTestLoader())
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