def svd(tensor, full_matrices=False, compute_uv=True, name=None):
"""Computes the singular value decompositions of one or more matrices.
Computes the SVD of each inner matrix in `tensor` such that
`tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(v[..., :,
:])`
```python
# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
# v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)
```
Args:
tensor: `Tensor` of shape `[..., M, N]`. Let `P` be the minimum of `M` and
`N`.
full_matrices: If true, compute full-sized `u` and `v`. If false
(the default), compute only the leading `P` singular vectors.
Ignored if `compute_uv` is `False`.
compute_uv: If `True` then left and right singular vectors will be
computed and returned in `u` and `v`, respectively. Otherwise, only the
singular values will be computed, which can be significantly faster.
name: string, optional name of the operation.
Returns:
s: Singular values. Shape is `[..., P]`. The values are sorted in reverse
order of magnitude, so s[..., 0] is the largest value, s[..., 1] is the
second largest, etc.
u: Left singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
`[..., M, M]`. Not returned if `compute_uv` is `False`.
v: Right singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., N, P]`. If `full_matrices` is `True` then shape is
`[..., N, N]`. Not returned if `compute_uv` is `False`.
@compatibility(numpy)
Mostly equivalent to numpy.linalg.svd, except that the order of output
arguments here is `s`, `u`, `v` when `compute_uv` is `True`, as opposed to
`u`, `s`, `v` for numpy.linalg.svd.
@end_compatibility
"""
# pylint: disable=protected-access
s, u, v = gen_linalg_ops._svd(tensor,
compute_uv=compute_uv,
full_matrices=full_matrices,
name=name)
# pylint: enable=protected-access
if compute_uv:
return math_ops.real(s), u, v
else:
return math_ops.real(s)
def svd(tensor, full_matrices=False, compute_uv=True, name=None):
"""Computes the singular value decompositions of one or more matrices.
Computes the SVD of each inner matrix in `tensor` such that
`tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(v[..., :,
:])`
```prettyprint
# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
# v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)
```
Args:
tensor: `Tensor` of shape `[..., M, N]`. Let `P` be the minimum of `M` and
`N`.
full_matrices: If true, compute full-sized `u` and `v`. If false
(the default), compute only the leading `P` singular vectors.
Ignored if `compute_uv` is `False`.
compute_uv: If `True` then left and right singular vectors will be
computed and returned in `u` and `v`, respectively. Otherwise, only the
singular values will be computed, which can be significantly faster.
name: string, optional name of the operation.
Returns:
s: Singular values. Shape is `[..., P]`. The values are sorted in reverse
order of magnitude, so s[..., 0] is the largest value, s[..., 1] is the
second largest, etc.
u: Left singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
`[..., M, M]`. Not returned if `compute_uv` is `False`.
v: Right singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., N, P]`. If `full_matrices` is `True` then shape is
`[..., N, N]`. Not returned if `compute_uv` is `False`.
@compatibility(numpy)
Mostly equivalent to numpy.linalg.svd, except that the order of output
arguments here is `s`, `u`, `v` when `compute_uv` is `True`, as opposed to
`u`, `s`, `v` for numpy.linalg.svd.
@end_compatibility
"""
# pylint: disable=protected-access
s, u, v = gen_linalg_ops._svd(
tensor, compute_uv=compute_uv, full_matrices=full_matrices)
# pylint: enable=protected-access
if compute_uv:
return math_ops.real(s), u, v
else:
return math_ops.real(s)
def _compareRealImag(self, cplx, use_gpu):
np_real, np_imag = np.real(cplx), np.imag(cplx)
np_zeros = np_real * 0
with test_util.device(use_gpu=use_gpu):
inx = ops.convert_to_tensor(cplx)
tf_real = math_ops.real(inx)
tf_imag = math_ops.imag(inx)
tf_real_real = math_ops.real(tf_real)
tf_imag_real = math_ops.imag(tf_real)
self.assertAllEqual(np_real, self.evaluate(tf_real))
self.assertAllEqual(np_imag, self.evaluate(tf_imag))
self.assertAllEqual(np_real, self.evaluate(tf_real_real))
self.assertAllEqual(np_zeros, self.evaluate(tf_imag_real))
def _compareRealImag(self, cplx, use_gpu):
np_real, np_imag = np.real(cplx), np.imag(cplx)
np_zeros = np_real * 0
with self.test_session(use_gpu=use_gpu,
force_gpu=use_gpu and test_util.is_gpu_available()):
inx = ops.convert_to_tensor(cplx)
tf_real = math_ops.real(inx)
tf_imag = math_ops.imag(inx)
tf_real_real = math_ops.real(tf_real)
tf_imag_real = math_ops.imag(tf_real)
self.assertAllEqual(np_real, self.evaluate(tf_real))
self.assertAllEqual(np_imag, self.evaluate(tf_imag))
self.assertAllEqual(np_real, self.evaluate(tf_real_real))
self.assertAllEqual(np_zeros, self.evaluate(tf_imag_real))
def _resource_apply_sparse(self, grad, var, indices, apply_state=None):
var_device, var_dtype = var.device, var.dtype.base_dtype
coefficients = ((apply_state or {}).get((var_device, var_dtype))
or self._fallback_apply_state(var_device, var_dtype))
# m_t = beta1 * m + (1 - beta1) * g_t
m = self.get_slot(var, 'm')
m_scaled_g_values = grad * coefficients['one_minus_beta_1_t']
m_t = state_ops.assign(m,
m * coefficients['beta_1_t'],
use_locking=self._use_locking)
with ops.control_dependencies([m_t]):
m_t = self._resource_scatter_add(m, indices, m_scaled_g_values)
# v_t = beta2 * v + (1 - beta2) * (g_t * conj(g_t))
v = self.get_slot(var, 'v')
v_scaled_g_values = (
grad * math_ops.conj(grad)) * coefficients['one_minus_beta_2_t']
v_t = state_ops.assign(v,
v * coefficients['beta_2_t'],
use_locking=self._use_locking)
with ops.control_dependencies([v_t]):
v_t = self._resource_scatter_add(v, indices, v_scaled_g_values)
if not self.amsgrad:
v_sqrt = math_ops.sqrt(v_t)
var_update = state_ops.assign_sub(
var,
coefficients['lr'] * m_t / (v_sqrt + coefficients['epsilon']),
use_locking=self._use_locking)
return control_flow_ops.group(*[var_update, m_t, v_t])
else:
cxmax = lambda a, b: math_ops.cast(
math_ops.maximum(math_ops.real(a), math_ops.real(b)), var_dtype
)
v_hat = self.get_slot(var, 'vhat')
v_hat_t = cxmax(v_hat, v_t)
with ops.control_dependencies([v_hat_t]):
v_hat_t = state_ops.assign(v_hat,
v_hat_t,
use_locking=self._use_locking)
v_hat_sqrt = math_ops.sqrt(v_hat_t)
var_update = state_ops.assign_sub(
var,
coefficients['lr'] * m_t /
(v_hat_sqrt + coefficients['epsilon']),
use_locking=self._use_locking)
return control_flow_ops.group(*[var_update, m_t, v_t, v_hat_t])
def _operator_and_matrix(self, build_info, dtype, use_placeholder):
shape = build_info.shape
# For this test class, we are creating Hermitian spectrums.
# We also want the spectrum to have eigenvalues bounded away from zero.
#
# pre_spectrum is bounded away from zero.
pre_spectrum = linear_operator_test_util.random_uniform(
shape=self._shape_to_spectrum_shape(shape), minval=1., maxval=2.)
pre_spectrum_c = _to_complex(pre_spectrum)
# Real{IFFT[pre_spectrum]}
# = IFFT[EvenPartOf[pre_spectrum]]
# is the IFFT of something that is also bounded away from zero.
# Therefore, FFT[pre_h] would be a well-conditioned spectrum.
pre_h = math_ops.ifft2d(pre_spectrum_c)
# A spectrum is Hermitian iff it is the DFT of a real convolution kernel.
# So we will make spectrum = FFT[h], for real valued h.
h = math_ops.real(pre_h)
h_c = _to_complex(h)
spectrum = math_ops.fft2d(h_c)
lin_op_spectrum = spectrum
if use_placeholder:
lin_op_spectrum = array_ops.placeholder_with_default(spectrum, shape=None)
operator = linalg.LinearOperatorCirculant2D(
lin_op_spectrum, input_output_dtype=dtype)
mat = self._spectrum_to_circulant_2d(spectrum, shape, dtype=dtype)
return operator, mat
def f(inx): inx.set_shape(x.shape) # func is a forward RFFT function (batched or unbatched). z = func(inx) # loss = sum(|z|^2) loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z))) return loss
def logdet(matrix, name=None):
"""Computes log of the determinant of a hermitian positive definite matrix.
```python
# Compute the determinant of a matrix while reducing the chance of over- or
underflow:
A = ... # shape 10 x 10
det = tf.exp(tf.linalg.logdet(A)) # scalar
```
Args:
matrix: A `Tensor`. Must be `float16`, `float32`, `float64`, `complex64`,
or `complex128` with shape `[..., M, M]`.
name: A name to give this `Op`. Defaults to `logdet`.
Returns:
The natural log of the determinant of `matrix`.
@compatibility(numpy)
Equivalent to numpy.linalg.slogdet, although no sign is returned since only
hermitian positive definite matrices are supported.
@end_compatibility
"""
# This uses the property that the log det(A) = 2*sum(log(real(diag(C))))
# where C is the cholesky decomposition of A.
with ops.name_scope(name, 'logdet', [matrix]):
chol = gen_linalg_ops.cholesky(matrix)
return 2.0 * math_ops.reduce_sum(
math_ops.log(math_ops.real(array_ops.matrix_diag_part(chol))),
axis=[-1])
def logdet(matrix, name=None):
"""Computes log of the determinant of a hermitian positive definite matrix.
```python
# Compute the determinant of a matrix while reducing the chance of over- or
underflow:
A = ... # shape 10 x 10
det = tf.exp(tf.logdet(A)) # scalar
```
Args:
matrix: A `Tensor`. Must be `float16`, `float32`, `float64`, `complex64`,
or `complex128` with shape `[..., M, M]`.
name: A name to give this `Op`. Defaults to `logdet`.
Returns:
The natural log of the determinant of `matrix`.
@compatibility(numpy)
Equivalent to numpy.linalg.slogdet, although no sign is returned since only
hermitian positive definite matrices are supported.
@end_compatibility
"""
# This uses the property that the log det(A) = 2*sum(log(real(diag(C))))
# where C is the cholesky decomposition of A.
with ops.name_scope(name, 'logdet', [matrix]):
chol = gen_linalg_ops.cholesky(matrix)
return 2.0 * math_ops.reduce_sum(
math_ops.log(math_ops.real(array_ops.matrix_diag_part(chol))),
reduction_indices=[-1])
def _dct2_1d(signals, name=None):
"""Computes the type II 1D Discrete Cosine Transform (DCT) of `signals`.
Args:
signals: A `[..., samples]` `float32` `Tensor` containing the signals to
take the DCT of.
name: An optional name for the operation.
Returns:
A `[..., samples]` `float32` `Tensor` containing the DCT of `signals`.
"""
with ops.name_scope(name, 'dct', [signals]):
# We use the FFT to compute the DCT and TensorFlow only supports float32 for
# FFTs at the moment.
signals = ops.convert_to_tensor(signals, dtype=dtypes.float32)
axis_dim = signals.shape[-1].value or array_ops.shape(signals)[-1]
axis_dim_float = math_ops.to_float(axis_dim)
scale = 2.0 * math_ops.exp(math_ops.complex(
0.0, -math.pi * math_ops.range(axis_dim_float) /
(2.0 * axis_dim_float)))
rfft = spectral_ops.rfft(signals, fft_length=[2 * axis_dim])[..., :axis_dim]
dct2 = math_ops.real(rfft * scale)
return dct2
def test_defining_spd_operator_by_taking_real_part(self):
with self.cached_session() as sess:
# S is real and positive.
s = linear_operator_test_util.random_uniform(
shape=(10, 2, 3, 4), dtype=dtypes.float32, minval=1., maxval=2.)
# Let S = S1 + S2, the Hermitian and anti-hermitian parts.
# S1 = 0.5 * (S + S^H), S2 = 0.5 * (S - S^H),
# where ^H is the Hermitian transpose of the function:
# f(n0, n1, n2)^H := ComplexConjugate[f(N0-n0, N1-n1, N2-n2)].
# We want to isolate S1, since
# S1 is Hermitian by construction
# S1 is real since S is
# S1 is positive since it is the sum of two positive kernels
# IDFT[S] = IDFT[S1] + IDFT[S2]
# = H1 + H2
# where H1 is real since it is Hermitian,
# and H2 is imaginary since it is anti-Hermitian.
ifft_s = fft_ops.ifft3d(math_ops.cast(s, dtypes.complex64))
# Throw away H2, keep H1.
real_ifft_s = math_ops.real(ifft_s)
# This is the perfect spectrum!
# spectrum = DFT[H1]
# = S1,
fft_real_ifft_s = fft_ops.fft3d(
math_ops.cast(real_ifft_s, dtypes.complex64))
# S1 is Hermitian ==> operator is real.
# S1 is real ==> operator is self-adjoint.
# S1 is positive ==> operator is positive-definite.
operator = linalg.LinearOperatorCirculant3D(fft_real_ifft_s)
# Allow for complex output so we can check operator has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix, matrix_t = sess.run([
operator.to_dense(),
array_ops.matrix_transpose(operator.to_dense())
])
operator.assert_positive_definite().run() # Should not fail.
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
self.assertAllClose(matrix, matrix_t)
# Just to test the theory, get S2 as well.
# This should create an imaginary operator.
# S2 is anti-Hermitian ==> operator is imaginary.
# S2 is real ==> operator is self-adjoint.
imag_ifft_s = math_ops.imag(ifft_s)
fft_imag_ifft_s = fft_ops.fft3d(
1j * math_ops.cast(imag_ifft_s, dtypes.complex64))
operator_imag = linalg.LinearOperatorCirculant3D(fft_imag_ifft_s)
matrix, matrix_h = sess.run([
operator_imag.to_dense(),
array_ops.matrix_transpose(math_ops.conj(operator_imag.to_dense()))
])
self.assertAllClose(matrix, matrix_h)
np.testing.assert_allclose(0, np.real(matrix), atol=1e-7)
def _checkGradComplex(self,
func,
x,
y,
result_is_complex=True,
rtol=1e-2,
atol=1e-2):
with self.cached_session(use_gpu=True):
inx = ops.convert_to_tensor(x)
iny = ops.convert_to_tensor(y)
# func is a forward or inverse, real or complex, batched or unbatched FFT
# function with a complex input.
z = func(math_ops.complex(inx, iny))
# loss = sum(|z|^2)
loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z)))
((x_jacob_t, x_jacob_n),
(y_jacob_t, y_jacob_n)) = gradient_checker.compute_gradient(
[inx, iny], [list(x.shape), list(y.shape)],
loss, [1],
x_init_value=[x, y],
delta=1e-2)
self.assertAllClose(x_jacob_t, x_jacob_n, rtol=rtol, atol=atol)
self.assertAllClose(y_jacob_t, y_jacob_n, rtol=rtol, atol=atol)
def testRealReal(self):
for dtype in (dtypes_lib.int32, dtypes_lib.int64, dtypes_lib.float32,
dtypes_lib.float64):
with self.subTest(dtype=dtype):
x = array_ops.placeholder(dtype)
y = math_ops.real(x)
self.assertEqual(x, y)
def _trace(self):
# The diagonal of the [[nested] block] circulant operator is the mean of
# the spectrum.
# Proof: For the [0,...,0] element, this follows from the IDFT formula.
# Then the result follows since all diagonal elements are the same.
# Therefore, the trace is the sum of the spectrum.
# Get shape of diag along with the axis over which to reduce the spectrum.
# We will reduce the spectrum over all block indices.
if self.spectrum.get_shape().is_fully_defined():
spec_rank = self.spectrum.get_shape().ndims
axis = np.arange(spec_rank - self.block_depth, spec_rank, dtype=np.int32)
else:
spec_rank = array_ops.rank(self.spectrum)
axis = math_ops.range(spec_rank - self.block_depth, spec_rank)
# Real diag part "re_d".
# Suppose spectrum.shape = [B1,...,Bb, N1, N2]
# self.shape = [B1,...,Bb, N, N], with N1 * N2 = N.
# re_d_value.shape = [B1,...,Bb]
re_d_value = math_ops.reduce_sum(math_ops.real(self.spectrum), axis=axis)
if not self.dtype.is_complex:
return math_ops.cast(re_d_value, self.dtype)
# Imaginary part, "im_d".
if self.is_self_adjoint:
im_d_value = 0.
else:
im_d_value = math_ops.reduce_sum(math_ops.imag(self.spectrum), axis=axis)
return math_ops.cast(math_ops.complex(re_d_value, im_d_value), self.dtype)
def _compareMulGradient(self, data):
# data is a float matrix of shape [n, 4]. data[:, 0], data[:, 1],
# data[:, 2], data[:, 3] are real parts of x, imaginary parts of
# x, real parts of y and imaginary parts of y.
with self.cached_session():
inp = ops.convert_to_tensor(data)
xr, xi, yr, yi = array_ops.split(value=inp, num_or_size_splits=4, axis=1)
def vec(x): # Reshape to a vector
return array_ops.reshape(x, [-1])
xr, xi, yr, yi = vec(xr), vec(xi), vec(yr), vec(yi)
def cplx(r, i): # Combine to a complex vector
return math_ops.complex(r, i)
x, y = cplx(xr, xi), cplx(yr, yi)
# z is x times y in complex plane.
z = x * y
# Defines the loss function as the sum of all coefficients of z.
loss = math_ops.reduce_sum(math_ops.real(z) + math_ops.imag(z))
epsilon = 0.005
jacob_t, jacob_n = gradient_checker.compute_gradient(
inp, list(data.shape), loss, [1], x_init_value=data, delta=epsilon)
self.assertAllClose(jacob_t, jacob_n, rtol=epsilon, atol=epsilon)
def _trace(self):
# The diagonal of the [[nested] block] circulant operator is the mean of
# the spectrum.
# Proof: For the [0,...,0] element, this follows from the IDFT formula.
# Then the result follows since all diagonal elements are the same.
# Therefore, the trace is the sum of the spectrum.
# Get shape of diag along with the axis over which to reduce the spectrum.
# We will reduce the spectrum over all block indices.
if self.spectrum.get_shape().is_fully_defined():
spec_rank = self.spectrum.get_shape().ndims
axis = np.arange(spec_rank - self.block_depth, spec_rank, dtype=np.int32)
else:
spec_rank = array_ops.rank(self.spectrum)
axis = math_ops.range(spec_rank - self.block_depth, spec_rank)
# Real diag part "re_d".
# Suppose spectrum.shape = [B1,...,Bb, N1, N2]
# self.shape = [B1,...,Bb, N, N], with N1 * N2 = N.
# re_d_value.shape = [B1,...,Bb]
re_d_value = math_ops.reduce_sum(math_ops.real(self.spectrum), axis=axis)
if not self.dtype.is_complex:
return math_ops.cast(re_d_value, self.dtype)
# Imaginary part, "im_d".
if self.is_self_adjoint:
im_d_value = 0.
else:
im_d_value = math_ops.reduce_sum(math_ops.imag(self.spectrum), axis=axis)
return math_ops.cast(math_ops.complex(re_d_value, im_d_value), self.dtype)
def _operator_and_matrix(self, build_info, dtype, use_placeholder):
shape = build_info.shape
# For this test class, we are creating Hermitian spectrums.
# We also want the spectrum to have eigenvalues bounded away from zero.
#
# pre_spectrum is bounded away from zero.
pre_spectrum = linear_operator_test_util.random_uniform(
shape=self._shape_to_spectrum_shape(shape), minval=1., maxval=2.)
pre_spectrum_c = _to_complex(pre_spectrum)
# Real{IFFT[pre_spectrum]}
# = IFFT[EvenPartOf[pre_spectrum]]
# is the IFFT of something that is also bounded away from zero.
# Therefore, FFT[pre_h] would be a well-conditioned spectrum.
pre_h = math_ops.ifft2d(pre_spectrum_c)
# A spectrum is Hermitian iff it is the DFT of a real convolution kernel.
# So we will make spectrum = FFT[h], for real valued h.
h = math_ops.real(pre_h)
h_c = _to_complex(h)
spectrum = math_ops.fft2d(h_c)
lin_op_spectrum = spectrum
if use_placeholder:
lin_op_spectrum = array_ops.placeholder_with_default(spectrum,
shape=None)
operator = linalg.LinearOperatorCirculant2D(lin_op_spectrum,
input_output_dtype=dtype)
mat = self._spectrum_to_circulant_2d(spectrum, shape, dtype=dtype)
return operator, mat
def test_defining_spd_operator_by_taking_real_part(self):
with self.cached_session() as sess:
# S is real and positive.
s = linear_operator_test_util.random_uniform(
shape=(10, 2, 3, 4), dtype=dtypes.float32, minval=1., maxval=2.)
# Let S = S1 + S2, the Hermitian and anti-hermitian parts.
# S1 = 0.5 * (S + S^H), S2 = 0.5 * (S - S^H),
# where ^H is the Hermitian transpose of the function:
# f(n0, n1, n2)^H := ComplexConjugate[f(N0-n0, N1-n1, N2-n2)].
# We want to isolate S1, since
# S1 is Hermitian by construction
# S1 is real since S is
# S1 is positive since it is the sum of two positive kernels
# IDFT[S] = IDFT[S1] + IDFT[S2]
# = H1 + H2
# where H1 is real since it is Hermitian,
# and H2 is imaginary since it is anti-Hermitian.
ifft_s = fft_ops.ifft3d(math_ops.cast(s, dtypes.complex64))
# Throw away H2, keep H1.
real_ifft_s = math_ops.real(ifft_s)
# This is the perfect spectrum!
# spectrum = DFT[H1]
# = S1,
fft_real_ifft_s = fft_ops.fft3d(
math_ops.cast(real_ifft_s, dtypes.complex64))
# S1 is Hermitian ==> operator is real.
# S1 is real ==> operator is self-adjoint.
# S1 is positive ==> operator is positive-definite.
operator = linalg.LinearOperatorCirculant3D(fft_real_ifft_s)
# Allow for complex output so we can check operator has zero imag part.
self.assertEqual(operator.dtype, dtypes.complex64)
matrix, matrix_t = sess.run([
operator.to_dense(),
array_ops.matrix_transpose(operator.to_dense())
])
operator.assert_positive_definite().run() # Should not fail.
np.testing.assert_allclose(0, np.imag(matrix), atol=1e-6)
self.assertAllClose(matrix, matrix_t)
# Just to test the theory, get S2 as well.
# This should create an imaginary operator.
# S2 is anti-Hermitian ==> operator is imaginary.
# S2 is real ==> operator is self-adjoint.
imag_ifft_s = math_ops.imag(ifft_s)
fft_imag_ifft_s = fft_ops.fft3d(
1j * math_ops.cast(imag_ifft_s, dtypes.complex64))
operator_imag = linalg.LinearOperatorCirculant3D(fft_imag_ifft_s)
matrix, matrix_h = sess.run([
operator_imag.to_dense(),
array_ops.matrix_transpose(math_ops.conj(operator_imag.to_dense()))
])
self.assertAllClose(matrix, matrix_h)
np.testing.assert_allclose(0, np.real(matrix), atol=1e-7)
def operator_and_matrix(self,
shape_info,
dtype,
use_placeholder,
ensure_self_adjoint_and_pd=False):
shape = shape_info.shape
# For this test class, we are creating Hermitian spectrums.
# We also want the spectrum to have eigenvalues bounded away from zero.
#
# pre_spectrum is bounded away from zero.
pre_spectrum = linear_operator_test_util.random_uniform(
shape=self._shape_to_spectrum_shape(shape),
dtype=dtype,
minval=1.,
maxval=2.)
pre_spectrum_c = _to_complex(pre_spectrum)
# Real{IFFT[pre_spectrum]}
# = IFFT[EvenPartOf[pre_spectrum]]
# is the IFFT of something that is also bounded away from zero.
# Therefore, FFT[pre_h] would be a well-conditioned spectrum.
pre_h = fft_ops.ifft2d(pre_spectrum_c)
# A spectrum is Hermitian iff it is the DFT of a real convolution kernel.
# So we will make spectrum = FFT[h], for real valued h.
h = math_ops.real(pre_h)
h_c = _to_complex(h)
spectrum = fft_ops.fft2d(h_c)
lin_op_spectrum = spectrum
if use_placeholder:
lin_op_spectrum = array_ops.placeholder_with_default(spectrum, shape=None)
operator = linalg.LinearOperatorCirculant2D(
lin_op_spectrum,
is_positive_definite=True if ensure_self_adjoint_and_pd else None,
is_self_adjoint=True if ensure_self_adjoint_and_pd else None,
input_output_dtype=dtype)
self.assertEqual(
operator.parameters,
{
"input_output_dtype": dtype,
"is_non_singular": None,
"is_positive_definite": (
True if ensure_self_adjoint_and_pd else None),
"is_self_adjoint": (
True if ensure_self_adjoint_and_pd else None),
"is_square": True,
"name": "LinearOperatorCirculant2D",
"spectrum": lin_op_spectrum,
})
mat = self._spectrum_to_circulant_2d(spectrum, shape, dtype=dtype)
return operator, mat
def dct(input, type=2, n=None, axis=-1, norm=None, name=None): # pylint: disable=redefined-builtin
"""Computes the 1D [Discrete Cosine Transform (DCT)][dct] of `input`.
Currently only Type II is supported. Implemented using a length `2N` padded
@{tf.spectral.rfft}, as described here: https://dsp.stackexchange.com/a/10606
@compatibility(scipy)
Equivalent to scipy.fftpack.dct for the Type-II DCT.
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.fftpack.dct.html
@end_compatibility
Args:
input: A `[..., samples]` `float32` `Tensor` containing the signals to
take the DCT of.
type: The DCT type to perform. Must be 2.
n: For future expansion. The length of the transform. Must be `None`.
axis: For future expansion. The axis to compute the DCT along. Must be `-1`.
norm: The normalization to apply. `None` for no normalization or `'ortho'`
for orthonormal normalization.
name: An optional name for the operation.
Returns:
A `[..., samples]` `float32` `Tensor` containing the DCT of `input`.
Raises:
ValueError: If `type` is not `2`, `n` is not `None, `axis` is not `-1`, or
`norm` is not `None` or `'ortho'`.
[dct]: https://en.wikipedia.org/wiki/Discrete_cosine_transform
"""
_validate_dct_arguments(type, n, axis, norm)
with _ops.name_scope(name, "dct", [input]):
# We use the RFFT to compute the DCT and TensorFlow only supports float32
# for FFTs at the moment.
input = _ops.convert_to_tensor(input, dtype=_dtypes.float32)
axis_dim = input.shape[-1].value or _array_ops.shape(input)[-1]
axis_dim_float = _math_ops.to_float(axis_dim)
scale = 2.0 * _math_ops.exp(
_math_ops.complex(
0.0, -_math.pi * _math_ops.range(axis_dim_float) /
(2.0 * axis_dim_float)))
# TODO(rjryan): Benchmark performance and memory usage of the various
# approaches to computing a DCT via the RFFT.
dct2 = _math_ops.real(
rfft(input, fft_length=[2 * axis_dim])[..., :axis_dim] * scale)
if norm == "ortho":
n1 = 0.5 * _math_ops.rsqrt(axis_dim_float)
n2 = n1 * _math_ops.sqrt(2.0)
# Use tf.pad to make a vector of [n1, n2, n2, n2, ...].
weights = _array_ops.pad(_array_ops.expand_dims(n1, 0),
[[0, axis_dim - 1]],
constant_values=n2)
dct2 *= weights
return dct2
def svd(tensor, full_matrices=False, compute_uv=True, name=None):
"""Computes the singular value decompositions of one or more matrices.
Computes the SVD of each inner matrix in `tensor` such that
`tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(v[..., :,
:])`
```prettyprint
# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
#v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)
```
Args:
tensor: `Tensor` of shape `[..., M, N]`. Let `P` be the minimum of `M` and
`N`.
full_matrices: If true, compute full-sized `u` and `v`. If false
(the default), compute only the leading `P` singular vectors.
Ignored if `compute_uv` is `False`.
compute_uv: If `True` then left and right singular vectors will be
computed and returned in `u` and `v`, respectively. Otherwise, only the
singular values will be computed, which can be significantly faster.
name: string, optional name of the operation.
Returns:
s: Singular values. Shape is `[..., P]`.
u: Right singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
`[..., M, M]`. Not returned if `compute_uv` is `False`.
v: Left singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., N, P]`. If `full_matrices` is `True` then shape is
`[..., N, N]`. Not returned if `compute_uv` is `False`.
"""
# pylint: disable=protected-access
s, u, v = gen_linalg_ops._svd(tensor,
compute_uv=compute_uv,
full_matrices=full_matrices)
# pylint: enable=protected-access
if compute_uv:
return math_ops.real(s), u, v
else:
return math_ops.real(s)
def dct(input, type=2, n=None, axis=-1, norm=None, name=None): # pylint: disable=redefined-builtin
"""Computes the 1D [Discrete Cosine Transform (DCT)][dct] of `input`.
Currently only Type II is supported. Implemented using a length `2N` padded
@{tf.spectral.rfft}, as described here: https://dsp.stackexchange.com/a/10606
@compatibility(scipy)
Equivalent to scipy.fftpack.dct for the Type-II DCT.
https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.fftpack.dct.html
@end_compatibility
Args:
input: A `[..., samples]` `float32` `Tensor` containing the signals to
take the DCT of.
type: The DCT type to perform. Must be 2.
n: For future expansion. The length of the transform. Must be `None`.
axis: For future expansion. The axis to compute the DCT along. Must be `-1`.
norm: The normalization to apply. `None` for no normalization or `'ortho'`
for orthonormal normalization.
name: An optional name for the operation.
Returns:
A `[..., samples]` `float32` `Tensor` containing the DCT of `input`.
Raises:
ValueError: If `type` is not `2`, `n` is not `None, `axis` is not `-1`, or
`norm` is not `None` or `'ortho'`.
[dct]: https://en.wikipedia.org/wiki/Discrete_cosine_transform
"""
_validate_dct_arguments(type, n, axis, norm)
with _ops.name_scope(name, "dct", [input]):
# We use the RFFT to compute the DCT and TensorFlow only supports float32
# for FFTs at the moment.
input = _ops.convert_to_tensor(input, dtype=_dtypes.float32)
axis_dim = input.shape[-1].value or _array_ops.shape(input)[-1]
axis_dim_float = _math_ops.to_float(axis_dim)
scale = 2.0 * _math_ops.exp(_math_ops.complex(
0.0, -_math.pi * _math_ops.range(axis_dim_float) /
(2.0 * axis_dim_float)))
# TODO(rjryan): Benchmark performance and memory usage of the various
# approaches to computing a DCT via the RFFT.
dct2 = _math_ops.real(
rfft(input, fft_length=[2 * axis_dim])[..., :axis_dim] * scale)
if norm == "ortho":
n1 = 0.5 * _math_ops.rsqrt(axis_dim_float)
n2 = n1 * _math_ops.sqrt(2.0)
# Use tf.pad to make a vector of [n1, n2, n2, n2, ...].
weights = _array_ops.pad(
_array_ops.expand_dims(n1, 0), [[0, axis_dim - 1]],
constant_values=n2)
dct2 *= weights
return dct2
def _ComplexGrad(op, grad):
"""Returns the real and imaginary components of 'grad', respectively."""
x = op.inputs[0]
y = op.inputs[1]
sx = array_ops.shape(x)
sy = array_ops.shape(y)
rx, ry = gen_array_ops._broadcast_gradient_args(sx, sy)
return (array_ops.reshape(math_ops.reduce_sum(math_ops.real(grad), rx), sx),
array_ops.reshape(math_ops.reduce_sum(math_ops.imag(grad), ry), sy))
def f(inx, iny): inx.set_shape(x.shape) iny.set_shape(y.shape) # func is a forward or inverse, real or complex, batched or unbatched # FFT function with a complex input. z = func(math_ops.complex(inx, iny)) # loss = sum(|z|^2) loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z))) return loss
def testRealImagNumericType(self):
for use_gpu in [True, False]:
for value in [1., 1j, 1. + 1j]:
np_real, np_imag = np.real(value), np.imag(value)
with test_util.device(use_gpu=use_gpu):
tf_real = math_ops.real(value)
tf_imag = math_ops.imag(value)
self.assertAllEqual(np_real, self.evaluate(tf_real))
self.assertAllEqual(np_imag, self.evaluate(tf_imag))
def _ComplexGrad(op, grad):
"""Returns the real and imaginary components of 'grad', respectively."""
x = op.inputs[0]
y = op.inputs[1]
sx = array_ops.shape(x)
sy = array_ops.shape(y)
rx, ry = gen_array_ops._broadcast_gradient_args(sx, sy)
return (array_ops.reshape(math_ops.reduce_sum(math_ops.real(grad), rx), sx),
array_ops.reshape(math_ops.reduce_sum(math_ops.imag(grad), ry), sy))
def _assert_positive_definite(self):
# This operator has the action Ax = F^H D F x,
# where D is the diagonal matrix with self.spectrum on the diag. Therefore,
# <x, Ax> = <Fx, DFx>,
# Since F is bijective, the condition for positive definite is the same as
# for a diagonal matrix, i.e. real part of spectrum is positive.
message = (
"Not positive definite: Real part of spectrum was not all positive.")
return check_ops.assert_positive(
math_ops.real(self.spectrum), message=message)
def _AngleGrad(op, grad):
"""Returns -grad / (Im(x) + iRe(x))"""
x = op.inputs[0]
with ops.control_dependencies([grad]):
re = math_ops.real(x)
im = math_ops.imag(x)
z = math_ops.reciprocal(math_ops.complex(im, re))
zero = constant_op.constant(0, dtype=grad.dtype)
complex_grad = math_ops.complex(grad, zero)
return -complex_grad * z
def svd(tensor, full_matrices=False, compute_uv=True, name=None):
"""Computes the singular value decompositions of one or more matrices.
Computes the SVD of each inner matrix in `tensor` such that
`tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) * transpose(v[..., :,
:])`
```prettyprint
# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
# v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)
```
Args:
matrix: `Tensor` of shape `[..., M, N]`. Let `P` be the minimum of `M` and
`N`.
full_matrices: If true, compute full-sized `u` and `v`. If false
(the default), compute only the leading `P` singular vectors.
Ignored if `compute_uv` is `False`.
compute_uv: If `True` then left and right singular vectors will be
computed and returned in `u` and `v`, respectively. Otherwise, only the
singular values will be computed, which can be significantly faster.
name: string, optional name of the operation.
Returns:
s: Singular values. Shape is `[..., P]`.
u: Right singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
`[..., M, M]`. Not returned if `compute_uv` is `False`.
v: Left singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., N, P]`. If `full_matrices` is `True` then shape is
`[..., N, N]`. Not returned if `compute_uv` is `False`.
"""
# pylint: disable=protected-access
s, u, v = gen_linalg_ops._svd(
tensor, compute_uv=compute_uv, full_matrices=full_matrices)
if compute_uv:
return math_ops.real(s), u, v
else:
return math_ops.real(s)
def _AngleGrad(op, grad):
"""Returns -grad / (Im(x) + iRe(x))"""
x = op.inputs[0]
with ops.control_dependencies([grad]):
re = math_ops.real(x)
im = math_ops.imag(x)
z = math_ops.reciprocal(math_ops.complex(im, re))
zero = constant_op.constant(0, dtype=grad.dtype)
complex_grad = math_ops.complex(grad, zero)
return -complex_grad * z
def _assert_positive_definite(self):
# This operator has the action Ax = F^H D F x,
# where D is the diagonal matrix with self.spectrum on the diag. Therefore,
# <x, Ax> = <Fx, DFx>,
# Since F is bijective, the condition for positive definite is the same as
# for a diagonal matrix, i.e. real part of spectrum is positive.
message = (
"Not positive definite: Real part of spectrum was not all positive.")
return check_ops.assert_positive(
math_ops.real(self.spectrum), message=message)
def _checkGradReal(self, func, x, use_gpu=False):
with self.test_session(use_gpu=use_gpu):
inx = ops.convert_to_tensor(x)
# func is a forward RFFT function (batched or unbatched).
z = func(inx)
# loss = sum(|z|^2)
loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z)))
x_jacob_t, x_jacob_n = test.compute_gradient(
inx, list(x.shape), loss, [1], x_init_value=x, delta=1e-2)
self.assertAllClose(x_jacob_t, x_jacob_n, rtol=1e-2, atol=1e-2)
def _assert_positive_definite(self):
if self.dtype.is_complex:
message = (
"Diagonal operator had diagonal entries with non-positive real part, "
"thus was not positive definite.")
else:
message = (
"Real diagonal operator had non-positive diagonal entries, "
"thus was not positive definite.")
return check_ops.assert_positive(math_ops.real(self._diag),
message=message)
def modrelu(z, b, comp):
if comp:
z_norm = math_ops.sqrt(math_ops.square(math_ops.real(z)) + math_ops.square(math_ops.imag(z))) + 0.00001
step1 = nn_ops.bias_add(z_norm, b)
step2 = math_ops.complex(nn_ops.relu(step1), array_ops.zeros_like(z_norm))
step3 = z/math_ops.complex(z_norm, array_ops.zeros_like(z_norm))
else:
z_norm = math_ops.abs(z) + 0.00001
step1 = nn_ops.bias_add(z_norm, b)
step2 = nn_ops.relu(step1)
step3 = math_ops.sign(z)
return math_ops.multiply(step3, step2)
def _EigGrad(op, grad_e, grad_v):
"""Gradient for Eig.
Based on eq. 4.77 from paper by
Christoph Boeddeker et al.
https://arxiv.org/abs/1701.00392
See also
"Computation of eigenvalue and eigenvector derivatives
for a general complex-valued eigensystem" by Nico van der Aa.
As for now only distinct eigenvalue case is considered.
"""
e = op.outputs[0]
compute_v = op.get_attr("compute_v")
# a = op.inputs[0], which satisfies
# a[...,:,:] * v[...,:,i] = e[...,i] * v[...,i]
with ops.control_dependencies([grad_e, grad_v]):
if compute_v:
v = op.outputs[1]
vt = _linalg.adjoint(v)
# Construct the matrix f(i,j) = (i != j ? 1 / (e_i - e_j) : 0).
# Notice that because of the term involving f, the gradient becomes
# infinite (or NaN in practice) when eigenvalues are not unique.
# Mathematically this should not be surprising, since for (k-fold)
# degenerate eigenvalues, the corresponding eigenvectors are only defined
# up to arbitrary rotation in a (k-dimensional) subspace.
f = array_ops.matrix_set_diag(
_SafeReciprocal(
array_ops.expand_dims(e, -2) -
array_ops.expand_dims(e, -1)), array_ops.zeros_like(e))
f = math_ops.conj(f)
vgv = math_ops.matmul(vt, grad_v)
mid = array_ops.matrix_diag(grad_e)
diag_grad_part = array_ops.matrix_diag(
array_ops.matrix_diag_part(
math_ops.cast(math_ops.real(vgv), vgv.dtype)))
mid += f * (
vgv - math_ops.matmul(math_ops.matmul(vt, v), diag_grad_part))
# vt is formally invertible as long as the original matrix is
# diagonalizable. However, in practice, vt may
# be ill-conditioned when matrix original matrix is close to
# non-diagonalizable one
grad_a = linalg_ops.matrix_solve(vt, math_ops.matmul(mid, vt))
else:
_, v = linalg_ops.eig(op.inputs[0])
vt = _linalg.adjoint(v)
# vt is formally invertible as long as the original matrix is
# diagonalizable. However, in practice, vt may
# be ill-conditioned when matrix original matrix is close to
# non-diagonalizable one
grad_a = linalg_ops.matrix_solve(
vt, math_ops.matmul(array_ops.matrix_diag(grad_e), vt))
return math_ops.cast(grad_a, op.inputs[0].dtype)
def _assert_positive_definite(self):
if self.dtype.is_complex:
message = (
"Diagonal operator had diagonal entries with non-positive real part, "
"thus was not positive definite.")
else:
message = (
"Real diagonal operator had non-positive diagonal entries, "
"thus was not positive definite.")
return check_ops.assert_positive(
math_ops.real(self._diag),
message=message)
def _checkGrad(self, func, x, y, use_gpu=False):
with self.test_session(use_gpu=use_gpu):
inx = ops.convert_to_tensor(x)
iny = ops.convert_to_tensor(y)
# func is a forward or inverse FFT function (batched or unbatched)
z = func(math_ops.complex(inx, iny))
# loss = sum(|z|^2)
loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z)))
((x_jacob_t, x_jacob_n),
(y_jacob_t, y_jacob_n)) = gradient_checker.compute_gradient(
[inx, iny], [list(x.shape), list(y.shape)],
loss, [1],
x_init_value=[x, y],
delta=1e-2)
self.assertAllClose(x_jacob_t, x_jacob_n, rtol=1e-2, atol=1e-2)
self.assertAllClose(y_jacob_t, y_jacob_n, rtol=1e-2, atol=1e-2)
def _checkGrad(self, func, x, y, use_gpu=False):
with self.test_session(use_gpu=use_gpu):
inx = ops.convert_to_tensor(x)
iny = ops.convert_to_tensor(y)
# func is a forward or inverse FFT function (batched or unbatched)
z = func(math_ops.complex(inx, iny))
# loss = sum(|z|^2)
loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z)))
((x_jacob_t, x_jacob_n),
(y_jacob_t, y_jacob_n)) = gradient_checker.compute_gradient(
[inx, iny], [list(x.shape), list(y.shape)],
loss, [1],
x_init_value=[x, y],
delta=1e-2)
self.assertAllClose(x_jacob_t, x_jacob_n, rtol=1e-2, atol=1e-2)
self.assertAllClose(y_jacob_t, y_jacob_n, rtol=1e-2, atol=1e-2)
def real(val):
"""Returns real parts of all elements in `a`.
Uses `tf.real`.
Args:
val: array_like. Could be an ndarray, a Tensor or any object that can be
converted to a Tensor using `tf.convert_to_tensor`.
Returns:
An ndarray with the same shape as `a`.
"""
val = asarray(val)
# TODO(srbs): np.real returns a scalar if val is a scalar, whereas we always
# return an ndarray.
return np_utils.tensor_to_ndarray(math_ops.real(val.data))
def Compute(x):
e, v = linalg_ops.eig(x)
# We sort eigenvalues by e.real+e.imag to have consistent
# order between runs
b_dims = len(e.shape) - 1
idx = sort_ops.argsort(math_ops.real(e) + math_ops.imag(e), axis=-1)
e = array_ops.gather(e, idx, batch_dims=b_dims)
v = array_ops.gather(v, idx, batch_dims=b_dims)
# (complex) Eigenvectors are only unique up to an arbitrary phase
# We normalize the vectors such that the first component has phase 0.
top_rows = v[..., 0:1, :]
angle = -math_ops.angle(top_rows)
phase = math_ops.complex(math_ops.cos(angle), math_ops.sin(angle))
v *= phase
return e, v
def _compareGradient(self, x):
# x[:, 0] is real, x[:, 1] is imag. We combine real and imag into
# complex numbers. Then, we extract real and imag parts and
# computes the squared sum. This is obviously the same as sum(real
# * real) + sum(imag * imag). We just want to make sure the
# gradient function is checked.
with self.cached_session():
inx = ops.convert_to_tensor(x)
real, imag = array_ops.split(value=inx, num_or_size_splits=2, axis=1)
real, imag = array_ops.reshape(real, [-1]), array_ops.reshape(imag, [-1])
cplx = math_ops.complex(real, imag)
cplx = math_ops.conj(cplx)
loss = math_ops.reduce_sum(math_ops.square(
math_ops.real(cplx))) + math_ops.reduce_sum(
math_ops.square(math_ops.imag(cplx)))
epsilon = 1e-3
jacob_t, jacob_n = gradient_checker.compute_gradient(
inx, list(x.shape), loss, [1], x_init_value=x, delta=epsilon)
self.assertAllClose(jacob_t, jacob_n, rtol=epsilon, atol=epsilon)
def _checkGradComplex(self, func, x, y, result_is_complex=True,
rtol=1e-2, atol=1e-2):
with self.cached_session(use_gpu=True):
inx = ops.convert_to_tensor(x)
iny = ops.convert_to_tensor(y)
# func is a forward or inverse, real or complex, batched or unbatched FFT
# function with a complex input.
z = func(math_ops.complex(inx, iny))
# loss = sum(|z|^2)
loss = math_ops.reduce_sum(math_ops.real(z * math_ops.conj(z)))
((x_jacob_t, x_jacob_n),
(y_jacob_t, y_jacob_n)) = gradient_checker.compute_gradient(
[inx, iny], [list(x.shape), list(y.shape)],
loss, [1],
x_init_value=[x, y],
delta=1e-2)
self.assertAllClose(x_jacob_t, x_jacob_n, rtol=rtol, atol=atol)
self.assertAllClose(y_jacob_t, y_jacob_n, rtol=rtol, atol=atol)
def _operator_and_mat_and_feed_dict(self, build_info, dtype,
use_placeholder):
shape = build_info.shape
# For this test class, we are creating Hermitian spectrums.
# We also want the spectrum to have eigenvalues bounded away from zero.
#
# pre_spectrum is bounded away from zero.
pre_spectrum = linear_operator_test_util.random_uniform(
shape=self._shape_to_spectrum_shape(shape), minval=1., maxval=2.)
pre_spectrum_c = _to_complex(pre_spectrum)
# Real{IFFT[pre_spectrum]}
# = IFFT[EvenPartOf[pre_spectrum]]
# is the IFFT of something that is also bounded away from zero.
# Therefore, FFT[pre_h] would be a well-conditioned spectrum.
pre_h = math_ops.ifft2d(pre_spectrum_c)
# A spectrum is Hermitian iff it is the DFT of a real convolution kernel.
# So we will make spectrum = FFT[h], for real valued h.
h = math_ops.real(pre_h)
h_c = _to_complex(h)
spectrum = math_ops.fft2d(h_c)
if use_placeholder:
spectrum_ph = array_ops.placeholder(dtypes.complex64)
# Evaluate here because (i) you cannot feed a tensor, and (ii)
# it is random and we want the same value used for both mat and feed_dict.
spectrum = spectrum.eval()
operator = linalg.LinearOperatorCirculant2D(
spectrum_ph, input_output_dtype=dtype)
feed_dict = {spectrum_ph: spectrum}
else:
operator = linalg.LinearOperatorCirculant2D(
spectrum, input_output_dtype=dtype)
feed_dict = None
mat = self._spectrum_to_circulant_2d(spectrum, shape, dtype=dtype)
return operator, mat, feed_dict
def _operator_and_mat_and_feed_dict(self, build_info, dtype, use_placeholder):
shape = build_info.shape
# For this test class, we are creating Hermitian spectrums.
# We also want the spectrum to have eigenvalues bounded away from zero.
#
# pre_spectrum is bounded away from zero.
pre_spectrum = linear_operator_test_util.random_uniform(
shape=self._shape_to_spectrum_shape(shape), minval=1., maxval=2.)
pre_spectrum_c = _to_complex(pre_spectrum)
# Real{IFFT[pre_spectrum]}
# = IFFT[EvenPartOf[pre_spectrum]]
# is the IFFT of something that is also bounded away from zero.
# Therefore, FFT[pre_h] would be a well-conditioned spectrum.
pre_h = math_ops.ifft2d(pre_spectrum_c)
# A spectrum is Hermitian iff it is the DFT of a real convolution kernel.
# So we will make spectrum = FFT[h], for real valued h.
h = math_ops.real(pre_h)
h_c = _to_complex(h)
spectrum = math_ops.fft2d(h_c)
if use_placeholder:
spectrum_ph = array_ops.placeholder(dtypes.complex64)
# Evaluate here because (i) you cannot feed a tensor, and (ii)
# it is random and we want the same value used for both mat and feed_dict.
spectrum = spectrum.eval()
operator = linalg.LinearOperatorCirculant2D(
spectrum_ph, input_output_dtype=dtype)
feed_dict = {spectrum_ph: spectrum}
else:
operator = linalg.LinearOperatorCirculant2D(
spectrum, input_output_dtype=dtype)
feed_dict = None
mat = self._spectrum_to_circulant_2d(spectrum, shape, dtype=dtype)
return operator, mat, feed_dict
def testRealReal(self):
for dtype in (dtypes_lib.int32, dtypes_lib.int64, dtypes_lib.float32,
dtypes_lib.float64):
x = array_ops.placeholder(dtype)
y = math_ops.real(x)
self.assertEqual(x, y)
def _Grad(op, grad):
"""A gradient function for RFFT with the provided `rank` and `irfft_fn`."""
fft_length = op.inputs[1]
input_shape = array_ops.shape(op.inputs[0])
is_even = math_ops.cast(1 - (fft_length[-1] % 2), dtypes.complex64)
def _TileForBroadcasting(matrix, t):
expanded = array_ops.reshape(
matrix,
array_ops.concat([
array_ops.ones([array_ops.rank(t) - 2], dtypes.int32),
array_ops.shape(matrix)
], 0))
return array_ops.tile(
expanded, array_ops.concat([array_ops.shape(t)[:-2], [1, 1]], 0))
def _MaskMatrix(length):
# TODO(rjryan): Speed up computation of twiddle factors using the
# following recurrence relation and cache them across invocations of RFFT.
#
# t_n = exp(sqrt(-1) * pi * n^2 / line_len)
# for n = 0, 1,..., line_len-1.
# For n > 2, use t_n = t_{n-1}^2 / t_{n-2} * t_1^2
a = array_ops.tile(
array_ops.expand_dims(math_ops.range(length), 0), (length, 1))
b = array_ops.transpose(a, [1, 0])
return math_ops.exp(-2j * np.pi * math_ops.cast(a * b, dtypes.complex64) /
math_ops.cast(length, dtypes.complex64))
def _YMMask(length):
"""A sequence of [1+0j, -1+0j, 1+0j, -1+0j, ...] with length `length`."""
return math_ops.cast(1 - 2 * (math_ops.range(length) % 2),
dtypes.complex64)
y0 = grad[..., 0:1]
if rank == 1:
ym = grad[..., -1:]
extra_terms = y0 + is_even * ym * _YMMask(input_shape[-1])
elif rank == 2:
# Create a mask matrix for y0 and ym.
base_mask = _MaskMatrix(input_shape[-2])
# Tile base_mask to match y0 in shape so that we can batch-matmul the
# inner 2 dimensions.
tiled_mask = _TileForBroadcasting(base_mask, y0)
y0_term = math_ops.matmul(tiled_mask, math_ops.conj(y0))
extra_terms = y0_term
ym = grad[..., -1:]
ym_term = math_ops.matmul(tiled_mask, math_ops.conj(ym))
inner_dim = input_shape[-1]
ym_term = array_ops.tile(
ym_term,
array_ops.concat([
array_ops.ones([array_ops.rank(grad) - 1], dtypes.int32),
[inner_dim]
], 0)) * _YMMask(inner_dim)
extra_terms += is_even * ym_term
# The gradient of RFFT is the IRFFT of the incoming gradient times a scaling
# factor, plus some additional terms to make up for the components dropped
# due to Hermitian symmetry.
input_size = math_ops.to_float(_FFTSizeForGrad(op.inputs[0], rank))
irfft = irfft_fn(grad, fft_length)
return 0.5 * (irfft * input_size + math_ops.real(extra_terms)), None
def svd(tensor, full_matrices=False, compute_uv=True, name=None):
r"""Computes the singular value decompositions of one or more matrices.
Computes the SVD of each inner matrix in `tensor` such that
`tensor[..., :, :] = u[..., :, :] * diag(s[..., :, :]) *
transpose(conj(v[..., :, :]))`
```python
# a is a tensor.
# s is a tensor of singular values.
# u is a tensor of left singular vectors.
# v is a tensor of right singular vectors.
s, u, v = svd(a)
s = svd(a, compute_uv=False)
```
Args:
tensor: `Tensor` of shape `[..., M, N]`. Let `P` be the minimum of `M` and
`N`.
full_matrices: If true, compute full-sized `u` and `v`. If false
(the default), compute only the leading `P` singular vectors.
Ignored if `compute_uv` is `False`.
compute_uv: If `True` then left and right singular vectors will be
computed and returned in `u` and `v`, respectively. Otherwise, only the
singular values will be computed, which can be significantly faster.
name: string, optional name of the operation.
Returns:
s: Singular values. Shape is `[..., P]`. The values are sorted in reverse
order of magnitude, so s[..., 0] is the largest value, s[..., 1] is the
second largest, etc.
u: Left singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., M, P]`; if `full_matrices` is `True` then shape is
`[..., M, M]`. Not returned if `compute_uv` is `False`.
v: Right singular vectors. If `full_matrices` is `False` (default) then
shape is `[..., N, P]`. If `full_matrices` is `True` then shape is
`[..., N, N]`. Not returned if `compute_uv` is `False`.
@compatibility(numpy)
Mostly equivalent to numpy.linalg.svd, except that
* The order of output arguments here is `s`, `u`, `v` when `compute_uv` is
`True`, as opposed to `u`, `s`, `v` for numpy.linalg.svd.
* full_matrices is `False` by default as opposed to `True` for
numpy.linalg.svd.
* tf.linalg.svd uses the standard definition of the SVD
\\(A = U \Sigma V^H\\), such that the left singular vectors of `a` are
the columns of `u`, while the right singular vectors of `a` are the
columns of `v`. On the other hand, numpy.linalg.svd returns the adjoint
\\(V^H\\) as the third output argument.
```python
import tensorflow as tf
import numpy as np
s, u, v = tf.linalg.svd(a)
tf_a_approx = tf.matmul(u, tf.matmul(tf.linalg.diag(s), v, adjoint_b=True))
u, s, v_adj = np.linalg.svd(a, full_matrices=False)
np_a_approx = np.dot(u, np.dot(np.diag(s), v_adj))
# tf_a_approx and np_a_approx should be numerically close.
```
@end_compatibility
"""
s, u, v = gen_linalg_ops.svd(
tensor, compute_uv=compute_uv, full_matrices=full_matrices, name=name)
if compute_uv:
return math_ops.real(s), u, v
else:
return math_ops.real(s)
def _abs_square(x):
if x.dtype.is_complex:
return math_ops.square(math_ops.real(x)) + math_ops.square(math_ops.imag(x))
else:
return math_ops.square(x)
def _assert_positive_definite(self):
return check_ops.assert_positive(
math_ops.real(self.multiplier),
message="LinearOperator was not positive definite.")
def _ComplexGrad(_, grad): """Returns the real and imaginary components of 'grad', respectively.""" return math_ops.real(grad), math_ops.imag(grad)
