def sign_magnitude_positive_definite(
raw, off_diagonal_scale=0., overall_scale=0.):
"""Constructs a positive definite matrix from an unconstrained input matrix.
We want to keep the whole matrix on a log scale, but also allow off-diagonal
elements to be negative, so the sign of off-diagonal elements is modeled
separately from their magnitude (using the lower and upper triangles
respectively). Specifically:
for i < j, we have:
output_cholesky[i, j] = raw[j, i] / (abs(raw[j, i]) + 1) *
exp((off_diagonal_scale + overall_scale + raw[i, j]) / 2)
output_cholesky[i, i] = exp((raw[i, i] + overall_scale) / 2)
output = output_cholesky^T * output_cholesky
where raw, off_diagonal_scale, and overall_scale are
un-constrained real-valued variables. The resulting values are stable
around zero due to the exponential (and the softsign keeps the function
smooth).
Args:
raw: A [..., M, M] Tensor.
off_diagonal_scale: A scalar or [...] shaped Tensor controlling the relative
scale of off-diagonal values in the output matrix.
overall_scale: A scalar or [...] shaped Tensor controlling the overall scale
of the output matrix.
Returns:
The `output` matrix described above, a [..., M, M] positive definite matrix.
"""
raw = ops.convert_to_tensor(raw)
diagonal = array_ops.matrix_diag_part(raw)
def _right_pad_with_ones(tensor, target_rank):
# Allow broadcasting even if overall_scale and off_diagonal_scale have batch
# dimensions
tensor = ops.convert_to_tensor(tensor, dtype=raw.dtype.base_dtype)
return array_ops.reshape(tensor,
array_ops.concat(
[
array_ops.shape(tensor), array_ops.ones(
[target_rank - array_ops.rank(tensor)],
dtype=target_rank.dtype)
],
axis=0))
# We divide the log values by 2 to compensate for the squaring that happens
# when transforming Cholesky factors into positive definite matrices.
sign_magnitude = (gen_math_ops.exp(
(raw + _right_pad_with_ones(off_diagonal_scale, array_ops.rank(raw)) +
_right_pad_with_ones(overall_scale, array_ops.rank(raw))) / 2.) *
nn.softsign(array_ops.matrix_transpose(raw)))
sign_magnitude.set_shape(raw.get_shape())
cholesky_factor = array_ops.matrix_set_diag(
input=array_ops.matrix_band_part(sign_magnitude, 0, -1),
diagonal=gen_math_ops.exp((diagonal + _right_pad_with_ones(
overall_scale, array_ops.rank(diagonal))) / 2.))
return math_ops.matmul(cholesky_factor, cholesky_factor, transpose_a=True)
def sign_magnitude_positive_definite(
raw, off_diagonal_scale=0., overall_scale=0.):
"""Constructs a positive definite matrix from an unconstrained input matrix.
We want to keep the whole matrix on a log scale, but also allow off-diagonal
elements to be negative, so the sign of off-diagonal elements is modeled
separately from their magnitude (using the lower and upper triangles
respectively). Specifically:
for i < j, we have:
output_cholesky[i, j] = raw[j, i] / (abs(raw[j, i]) + 1) *
exp((off_diagonal_scale + overall_scale + raw[i, j]) / 2)
output_cholesky[i, i] = exp((raw[i, i] + overall_scale) / 2)
output = output_cholesky^T * output_cholesky
where raw, off_diagonal_scale, and overall_scale are
un-constrained real-valued variables. The resulting values are stable
around zero due to the exponential (and the softsign keeps the function
smooth).
Args:
raw: A [..., M, M] Tensor.
off_diagonal_scale: A scalar or [...] shaped Tensor controlling the relative
scale of off-diagonal values in the output matrix.
overall_scale: A scalar or [...] shaped Tensor controlling the overall scale
of the output matrix.
Returns:
The `output` matrix described above, a [..., M, M] positive definite matrix.
"""
raw = ops.convert_to_tensor(raw)
diagonal = array_ops.matrix_diag_part(raw)
def _right_pad_with_ones(tensor, target_rank):
# Allow broadcasting even if overall_scale and off_diagonal_scale have batch
# dimensions
tensor = ops.convert_to_tensor(tensor, dtype=raw.dtype.base_dtype)
return array_ops.reshape(tensor,
array_ops.concat(
[
array_ops.shape(tensor), array_ops.ones(
[target_rank - array_ops.rank(tensor)],
dtype=target_rank.dtype)
],
axis=0))
# We divide the log values by 2 to compensate for the squaring that happens
# when transforming Cholesky factors into positive definite matrices.
sign_magnitude = (gen_math_ops.exp(
(raw + _right_pad_with_ones(off_diagonal_scale, array_ops.rank(raw)) +
_right_pad_with_ones(overall_scale, array_ops.rank(raw))) / 2.) *
nn.softsign(array_ops.matrix_transpose(raw)))
sign_magnitude.set_shape(raw.get_shape())
cholesky_factor = array_ops.matrix_set_diag(
input=array_ops.matrix_band_part(sign_magnitude, 0, -1),
diagonal=gen_math_ops.exp((diagonal + _right_pad_with_ones(
overall_scale, array_ops.rank(diagonal))) / 2.))
return math_ops.matmul(cholesky_factor, cholesky_factor, transpose_a=True)
def softsign(x):
"""Softsign activation function.
Arguments:
x: Input tensor.
Returns:
The softplus activation: `x / (abs(x) + 1)`.
"""
return nn.softsign(x)
def softsign(x):
"""Softsign activation function.
Arguments:
x: Input tensor.
Returns:
The softplus activation: `x / (abs(x) + 1)`.
"""
return nn.softsign(x)
def template(x_shape=[2, 3, 4, 5], description: str = ""):
from tensorflow.python.ops import nn
x = tf.placeholder(np.float32, x_shape, "x")
y = nn.softsign(x)
vx = np.random.rand(*x_shape).astype(np.float32) - 0.5
with tf.Session() as sess:
vy, = sess.run([y], {x: vx})
graph = TensorFlowConverter(sess, batch_size=2).convert([x], [y])
generate_kernel_test_case(
description=f"[TensorFlow] Softsign {description}",
graph=graph,
inputs={graph.inputs[0]: vx},
expected={graph.outputs[0]: vy}
)
def softsign(x):
"""Softsign activation function, `softsign(x) = x / (abs(x) + 1)`.
Example Usage:
>>> a = tf.constant([-1.0, 0.0, 1.0], dtype = tf.float32)
>>> b = tf.keras.activations.softsign(a)
>>> b.numpy()
array([-0.5, 0. , 0.5], dtype=float32)
Args:
x: Input tensor.
Returns:
The softsign activation: `x / (abs(x) + 1)`.
"""
return nn.softsign(x)
def softsign(x): return nn.softsign(x)
def softsign(x):
return nn.softsign(x)
