Python tensorflow.python.ops.math_ops.rsqrt() Examples

def LRSchedule(global_step, d_model, warmup_steps=4000):
	if global_step is None:
		raise ValueError("global_step is required for learning_rate_schedule.")

	def deal_lr(global_step, d_model, warmup_steps):
		d_model = ops.convert_to_tensor(d_model, dtype=tf.float32)
		dtype = d_model.dtype
		warmup_steps = math_ops.cast(warmup_steps, dtype)

		global_step_recomp = math_ops.cast(global_step, dtype)
		arg1 = math_ops.rsqrt(global_step_recomp)
		arg2 = math_ops.multiply(global_step_recomp, math_ops.pow(warmup_steps, -1.5))

		return math_ops.multiply(math_ops.rsqrt(d_model), math_ops.minimum(arg1, arg2))

	return functools.partial(deal_lr, global_step, d_model, warmup_steps) 

def _sample_n(self, n, seed=None):
    # The sampling method comes from the fact that if:
    #   X ~ Normal(0, 1)
    #   Z ~ Chi2(df)
    #   Y = X / sqrt(Z / df)
    # then:
    #   Y ~ StudentT(df).
    shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
    normal_sample = random_ops.random_normal(shape, dtype=self.dtype, seed=seed)
    df = self.df * array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype)
    gamma_sample = random_ops.random_gamma(
        [n],
        0.5 * df,
        beta=0.5,
        dtype=self.dtype,
        seed=distribution_util.gen_new_seed(seed, salt="student_t"))
    samples = normal_sample * math_ops.rsqrt(gamma_sample / df)
    return samples * self.scale + self.loc  # Abs(scale) not wanted. 

def _sample_n(self, n, seed=None):
    # The sampling method comes from the fact that if:
    #   X ~ Normal(0, 1)
    #   Z ~ Chi2(df)
    #   Y = X / sqrt(Z / df)
    # then:
    #   Y ~ StudentT(df).
    shape = array_ops.concat([[n], self.batch_shape_tensor()], 0)
    normal_sample = random_ops.random_normal(shape, dtype=self.dtype, seed=seed)
    df = self.df * array_ops.ones(self.batch_shape_tensor(), dtype=self.dtype)
    gamma_sample = random_ops.random_gamma(
        [n],
        0.5 * df,
        beta=0.5,
        dtype=self.dtype,
        seed=distribution_util.gen_new_seed(seed, salt="student_t"))
    samples = normal_sample * math_ops.rsqrt(gamma_sample / df)
    return samples * self.scale + self.loc  # Abs(scale) not wanted. 

def _BatchNormWithGlobalNormalizationGrad(op, grad):
  """Return the gradients for the 5 inputs of BatchNormWithGlobalNormalization.

  We do not backprop anything for the mean and var intentionally as they are
  not being trained with backprop in the operation.

  Args:
    op: The BatchNormOp for which we need to generate gradients.
    grad: Tensor.  The gradients passed to the BatchNormOp.

  Returns:
    dx: Backprop for input, which is (grad * (g * rsqrt(v + epsilon)))
    dm: Backprop for mean, which is
        sum_over_rest(grad * g) * (-1 / rsqrt(v + epsilon))
    dv: Backprop for variance, which is
        sum_over_rest(grad * g * (x - m)) * (-1/2) * (v + epsilon) ^ (-3/2)
    db: Backprop for beta, which is grad reduced in all except the
        last dimension.
    dg: Backprop for gamma, which is (grad * ((x - m) * rsqrt(v + epsilon)))
  """
  dx, dm, dv, db, dg = gen_nn_ops._batch_norm_with_global_normalization_grad(
      op.inputs[0], op.inputs[1], op.inputs[2], op.inputs[4], grad,
      op.get_attr("variance_epsilon"), op.get_attr("scale_after_normalization"))
  return dx, dm, dv, db, dg 

def poincare_normalize(x, axis=1, epsilon=1e-5, name=None):
  """Project into the Poincare ball with norm <= 1.0 - epsilon.

  https://en.wikipedia.org/wiki/Poincare_ball_model

  Used in
  Poincare Embeddings for Learning Hierarchical Representations
  Maximilian Nickel, Douwe Kiela
  https://arxiv.org/pdf/1705.08039.pdf

  For a 1-D tensor with `axis = 0`, computes

                (x * (1 - epsilon)) / ||x||     if ||x|| > 1 - epsilon
      output =
                 x                              otherwise

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `axis`.

  Args:
    x: A `Tensor`.
    axis: Axis along which to normalize.  A scalar or a vector of integers.
    epsilon: A small deviation from the edge of the unit sphere for numerical
      stability.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, 'poincare_normalize', [x]) as name:
    x = ops.convert_to_tensor(x, name='x')
    square_sum = math_ops.reduce_sum(math_ops.square(x), axis, keepdims=True)
    x_inv_norm = math_ops.rsqrt(square_sum)
    x_inv_norm = math_ops.minimum((1. - epsilon) * x_inv_norm, 1.)
    return math_ops.multiply(x, x_inv_norm, name=name) 

def _test_rsqrt(data):
    """ One iteration of rsqrt """
    return _test_unary_elemwise(math_ops.rsqrt, data)
#######################################################################
# Neg
# --- 

def clip_by_average_norm(t, clip_norm, name=None):
  """Clips tensor values to a maximum average L2-norm.

  Given a tensor `t`, and a maximum clip value `clip_norm`, this operation
  normalizes `t` so that its average L2-norm is less than or equal to
  `clip_norm`. Specifically, if the average L2-norm is already less than or
  equal to `clip_norm`, then `t` is not modified. If the average L2-norm is
  greater than `clip_norm`, then this operation returns a tensor of the same
  type and shape as `t` with its values set to:

  `t * clip_norm / l2norm_avg(t)`

  In this case, the average L2-norm of the output tensor is `clip_norm`.

  This operation is typically used to clip gradients before applying them with
  an optimizer.

  Args:
    t: A `Tensor`.
    clip_norm: A 0-D (scalar) `Tensor` > 0. A maximum clipping value.
    name: A name for the operation (optional).

  Returns:
    A clipped `Tensor`.
  """
  with ops.name_scope(name, "clip_by_average_norm", [t, clip_norm]) as name:
    t = ops.convert_to_tensor(t, name="t")

    # Calculate L2-norm per element, clip elements by ratio of clip_norm to
    # L2-norm per element
    n_element = math_ops.cast(array_ops.size(t), dtypes.float32)
    l2norm_inv = math_ops.rsqrt(
        math_ops.reduce_sum(t * t, math_ops.range(array_ops.rank(t))))
    tclip = array_ops.identity(
        t * clip_norm * math_ops.minimum(
            l2norm_inv * n_element, constant_op.constant(1.0) / clip_norm),
        name=name)

  return tclip 

def call(self, inputs):
    inputs = ops.convert_to_tensor(inputs, dtype=self.dtype)
    ndim = self._input_rank

    if self.rectify:
      inputs = nn.relu(inputs)

    # Compute normalization pool.
    if ndim == 2:
      norm_pool = math_ops.matmul(math_ops.square(inputs), self.gamma)
      norm_pool = nn.bias_add(norm_pool, self.beta)
    elif self.data_format == "channels_last" and ndim <= 5:
      shape = self.gamma.shape.as_list()
      gamma = array_ops.reshape(self.gamma, (ndim - 2) * [1] + shape)
      norm_pool = nn.convolution(math_ops.square(inputs), gamma, "VALID")
      norm_pool = nn.bias_add(norm_pool, self.beta)
    else:  # generic implementation
      # This puts channels in the last dimension regardless of input.
      norm_pool = math_ops.tensordot(
          math_ops.square(inputs), self.gamma, [[self._channel_axis()], [0]])
      norm_pool += self.beta
      if self.data_format == "channels_first":
        # Return to channels_first format if necessary.
        axes = list(range(ndim - 1))
        axes.insert(1, ndim - 1)
        norm_pool = array_ops.transpose(norm_pool, axes)

    if self.inverse:
      norm_pool = math_ops.sqrt(norm_pool)
    else:
      norm_pool = math_ops.rsqrt(norm_pool)
    outputs = inputs * norm_pool

    if not context.executing_eagerly():
      outputs.set_shape(self.compute_output_shape(inputs.shape))
    return outputs 

def per_image_standardization(image):
  """Linearly scales `image` to have zero mean and unit norm.

  This op computes `(x - mean) / adjusted_stddev`, where `mean` is the average
  of all values in image, and
  `adjusted_stddev = max(stddev, 1.0/sqrt(image.NumElements()))`.

  `stddev` is the standard deviation of all values in `image`. It is capped
  away from zero to protect against division by 0 when handling uniform images.

  Args:
    image: 3-D tensor of shape `[height, width, channels]`.

  Returns:
    The standardized image with same shape as `image`.

  Raises:
    ValueError: if the shape of 'image' is incompatible with this function.
  """
  image = ops.convert_to_tensor(image, name='image')
  image = control_flow_ops.with_dependencies(
      _Check3DImage(image, require_static=False), image)
  num_pixels = math_ops.reduce_prod(array_ops.shape(image))

  image = math_ops.cast(image, dtype=dtypes.float32)
  image_mean = math_ops.reduce_mean(image)

  variance = (math_ops.reduce_mean(math_ops.square(image)) -
              math_ops.square(image_mean))
  variance = gen_nn_ops.relu(variance)
  stddev = math_ops.sqrt(variance)

  # Apply a minimum normalization that protects us against uniform images.
  min_stddev = math_ops.rsqrt(math_ops.cast(num_pixels, dtypes.float32))
  pixel_value_scale = math_ops.maximum(stddev, min_stddev)
  pixel_value_offset = image_mean

  image = math_ops.subtract(image, pixel_value_offset)
  image = math_ops.div(image, pixel_value_scale)
  return image 

def poincare_normalize(x, axis=1, epsilon=1e-5, name=None):
  """Project into the Poincare ball with norm <= 1.0 - epsilon.

  https://en.wikipedia.org/wiki/Poincare_ball_model

  Used in
  Poincare Embeddings for Learning Hierarchical Representations
  Maximilian Nickel, Douwe Kiela
  https://arxiv.org/pdf/1705.08039.pdf

  For a 1-D tensor with `axis = 0`, computes

                (x * (1 - epsilon)) / ||x||     if ||x|| > 1 - epsilon
      output =
                 x                              otherwise

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `axis`.

  Args:
    x: A `Tensor`.
    axis: Axis along which to normalize.  A scalar or a vector of integers.
    epsilon: A small deviation from the edge of the unit sphere for numerical
      stability.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, 'poincare_normalize', [x]) as name:
    x = ops.convert_to_tensor(x, name='x')
    square_sum = math_ops.reduce_sum(math_ops.square(x), axis, keepdims=True)
    x_inv_norm = math_ops.rsqrt(square_sum)
    x_inv_norm = math_ops.minimum((1. - epsilon) * x_inv_norm, 1.)
    return math_ops.multiply(x, x_inv_norm, name=name) 

def l2_normalize(x, dim, epsilon=1e-12, name=None):
  """Normalizes along dimension `dim` using an L2 norm.

  For a 1-D tensor with `dim = 0`, computes

      output = x / sqrt(max(sum(x**2), epsilon))

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `dim`.

  Args:
    x: A `Tensor`.
    dim: Dimension along which to normalize.  A scalar or a vector of
      integers.
    epsilon: A lower bound value for the norm. Will use `sqrt(epsilon)` as the
      divisor if `norm < sqrt(epsilon)`.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, "l2_normalize", [x]) as name:
    x = ops.convert_to_tensor(x, name="x")
    square_sum = math_ops.reduce_sum(math_ops.square(x), dim, keep_dims=True)
    x_inv_norm = math_ops.rsqrt(math_ops.maximum(square_sum, epsilon))
    return math_ops.multiply(x, x_inv_norm, name=name) 

def clip_by_average_norm(t, clip_norm, name=None):
  """Clips tensor values to a maximum average L2-norm.

  Given a tensor `t`, and a maximum clip value `clip_norm`, this operation
  normalizes `t` so that its average L2-norm is less than or equal to
  `clip_norm`. Specifically, if the average L2-norm is already less than or
  equal to `clip_norm`, then `t` is not modified. If the average L2-norm is
  greater than `clip_norm`, then this operation returns a tensor of the same
  type and shape as `t` with its values set to:

  `t * clip_norm / l2norm_avg(t)`

  In this case, the average L2-norm of the output tensor is `clip_norm`.

  This operation is typically used to clip gradients before applying them with
  an optimizer.

  Args:
    t: A `Tensor`.
    clip_norm: A 0-D (scalar) `Tensor` > 0. A maximum clipping value.
    name: A name for the operation (optional).

  Returns:
    A clipped `Tensor`.
  """
  with ops.name_scope(name, "clip_by_average_norm", [t, clip_norm]) as name:
    t = ops.convert_to_tensor(t, name="t")

    # Calculate L2-norm per element, clip elements by ratio of clip_norm to
    # L2-norm per element
    n_element = math_ops.cast(array_ops.size(t), dtypes.float32)
    l2norm_inv = math_ops.rsqrt(
        math_ops.reduce_sum(t * t, math_ops.range(array_ops.rank(t))))
    tclip = array_ops.identity(
        t * clip_norm * math_ops.minimum(
            l2norm_inv * n_element, constant_op.constant(1.0) / clip_norm),
        name=name)

  return tclip 

def l2_normalize(x, dim, epsilon=1e-12, name=None):
  """Normalizes along dimension `dim` using an L2 norm.

  For a 1-D tensor with `dim = 0`, computes

      output = x / sqrt(max(sum(x**2), epsilon))

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `dim`.

  Args:
    x: A `Tensor`.
    dim: Dimension along which to normalize.  A scalar or a vector of
      integers.
    epsilon: A lower bound value for the norm. Will use `sqrt(epsilon)` as the
      divisor if `norm < sqrt(epsilon)`.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, "l2_normalize", [x]) as name:
    x = ops.convert_to_tensor(x, name="x")
    square_sum = math_ops.reduce_sum(math_ops.square(x), dim, keep_dims=True)
    x_inv_norm = math_ops.rsqrt(math_ops.maximum(square_sum, epsilon))
    return math_ops.mul(x, x_inv_norm, name=name) 

def per_image_standardization(image):
  """Linearly scales `image` to have zero mean and unit norm.

  This op computes `(x - mean) / adjusted_stddev`, where `mean` is the average
  of all values in image, and
  `adjusted_stddev = max(stddev, 1.0/sqrt(image.NumElements()))`.

  `stddev` is the standard deviation of all values in `image`. It is capped
  away from zero to protect against division by 0 when handling uniform images.

  Args:
    image: 3-D tensor of shape `[height, width, channels]`.

  Returns:
    The standardized image with same shape as `image`.

  Raises:
    ValueError: if the shape of 'image' is incompatible with this function.
  """
  image = ops.convert_to_tensor(image, name='image')
  _Check3DImage(image, require_static=False)
  num_pixels = math_ops.reduce_prod(array_ops.shape(image))

  image = math_ops.cast(image, dtype=dtypes.float32)
  image_mean = math_ops.reduce_mean(image)

  variance = (math_ops.reduce_mean(math_ops.square(image)) -
              math_ops.square(image_mean))
  variance = gen_nn_ops.relu(variance)
  stddev = math_ops.sqrt(variance)

  # Apply a minimum normalization that protects us against uniform images.
  min_stddev = math_ops.rsqrt(math_ops.cast(num_pixels, dtypes.float32))
  pixel_value_scale = math_ops.maximum(stddev, min_stddev)
  pixel_value_offset = image_mean

  image = math_ops.subtract(image, pixel_value_offset)
  image = math_ops.div(image, pixel_value_scale)
  return image 

def per_image_standardization(image):
  """Linearly scales `image` to have zero mean and unit norm.

  This op computes `(x - mean) / adjusted_stddev`, where `mean` is the average
  of all values in image, and
  `adjusted_stddev = max(stddev, 1.0/sqrt(image.NumElements()))`.

  `stddev` is the standard deviation of all values in `image`. It is capped
  away from zero to protect against division by 0 when handling uniform images.

  Args:
    image: 3-D tensor of shape `[height, width, channels]`.

  Returns:
    The standardized image with same shape as `image`.

  Raises:
    ValueError: if the shape of 'image' is incompatible with this function.
  """
  image = ops.convert_to_tensor(image, name='image')
  image = control_flow_ops.with_dependencies(
      _Check3DImage(image, require_static=False), image)
  num_pixels = math_ops.reduce_prod(array_ops.shape(image))

  image = math_ops.cast(image, dtype=dtypes.float32)
  image_mean = math_ops.reduce_mean(image)

  variance = (math_ops.reduce_mean(math_ops.square(image)) -
              math_ops.square(image_mean))
  variance = gen_nn_ops.relu(variance)
  stddev = math_ops.sqrt(variance)

  # Apply a minimum normalization that protects us against uniform images.
  min_stddev = math_ops.rsqrt(math_ops.cast(num_pixels, dtypes.float32))
  pixel_value_scale = math_ops.maximum(stddev, min_stddev)
  pixel_value_offset = image_mean

  image = math_ops.subtract(image, pixel_value_offset)
  image = math_ops.div(image, pixel_value_scale)
  return image 

def _apply_dense(self, grad, var):
    # Calculates the preconditioner statistics for each tensor.
    partitioned_grads = TensorPartitioner.partition_tensor(
        grad, self._partition_info)
    shape = var.get_shape()
    fallback_to_diagonal = self._fallback_to_diagonal_for_shape(shape)

    precond_statistics_update = []
    if not fallback_to_diagonal:
      precond_statistics_update = self._updated_statistics(
          var, partitioned_grads)

    accumulator = self.get_slot(var, "accumulator")
    accumulator_updated = state_ops.assign_add(accumulator, grad * grad)
    accumulator_inv_sqrt = math_ops.rsqrt(accumulator_updated + 1e-30)
    if self._momentum > 0.0:
      scaled_g = (1.0 - self._momentum_tensor) * (grad * accumulator_inv_sqrt)
      gbar = self.get_slot(var, "momentum")
      gbar_updated = state_ops.assign_add(
          gbar,
          gbar * (self._momentum_tensor - 1.0) + scaled_g)
    else:
      gbar_updated = (grad * accumulator_inv_sqrt)

    if not fallback_to_diagonal:
      # Update the preconditioner statistics followed by computing the
      # preconditioned gradient.
      with ops.control_dependencies(precond_statistics_update):
        s = tf.cast(self._run_nondiagonal_update, tf.float32)
        preconditioned_grad = self._preconditioned_update(
            var, partitioned_grads, gbar_updated)
        # slowly adapt from diagonal to preconditioned gradient.
        w = self._run_nondiagonal_update_warmup
        warmup_update = s * self._learning_rate_tensor * (
            w * preconditioned_grad + (1.0 - w) * gbar_updated)
        fallback_update = (1 - s) * (self._learning_rate_tensor * gbar_updated)
        return state_ops.assign_sub(var, warmup_update + fallback_update)
    else:
      return state_ops.assign_sub(var,
                                  self._learning_rate_tensor * gbar_updated) 

def clip_by_average_norm(t, clip_norm, name=None):
  """Clips tensor values to a maximum average L2-norm.

  Given a tensor `t`, and a maximum clip value `clip_norm`, this operation
  normalizes `t` so that its average L2-norm is less than or equal to
  `clip_norm`. Specifically, if the average L2-norm is already less than or
  equal to `clip_norm`, then `t` is not modified. If the average L2-norm is
  greater than `clip_norm`, then this operation returns a tensor of the same
  type and shape as `t` with its values set to:

  `t * clip_norm / l2norm_avg(t)`

  In this case, the average L2-norm of the output tensor is `clip_norm`.

  This operation is typically used to clip gradients before applying them with
  an optimizer.

  Args:
    t: A `Tensor`.
    clip_norm: A 0-D (scalar) `Tensor` > 0. A maximum clipping value.
    name: A name for the operation (optional).

  Returns:
    A clipped `Tensor`.
  """
  with ops.name_scope(name, "clip_by_average_norm", [t, clip_norm]) as name:
    t = ops.convert_to_tensor(t, name="t")

    # Calculate L2-norm per element, clip elements by ratio of clip_norm to
    # L2-norm per element
    n_element = math_ops.cast(array_ops.size(t), dtypes.float32)
    l2norm_inv = math_ops.rsqrt(
        math_ops.reduce_sum(t * t, math_ops.range(array_ops.rank(t))))
    tclip = array_ops.identity(
        t * clip_norm * math_ops.minimum(
            l2norm_inv * n_element, constant_op.constant(1.0) / clip_norm),
        name=name)

  return tclip 

def per_image_standardization(image):
  """Linearly scales `image` to have zero mean and unit norm.

  This op computes `(x - mean) / adjusted_stddev`, where `mean` is the average
  of all values in image, and
  `adjusted_stddev = max(stddev, 1.0/sqrt(image.NumElements()))`.

  `stddev` is the standard deviation of all values in `image`. It is capped
  away from zero to protect against division by 0 when handling uniform images.

  Args:
    image: 3-D tensor of shape `[height, width, channels]`.

  Returns:
    The standardized image with same shape as `image`.

  Raises:
    ValueError: if the shape of 'image' is incompatible with this function.
  """
  image = ops.convert_to_tensor(image, name='image')
  image = control_flow_ops.with_dependencies(
      _Check3DImage(image, require_static=False), image)
  num_pixels = math_ops.reduce_prod(array_ops.shape(image))

  image = math_ops.cast(image, dtype=dtypes.float32)
  image_mean = math_ops.reduce_mean(image)

  variance = (math_ops.reduce_mean(math_ops.square(image)) -
              math_ops.square(image_mean))
  variance = gen_nn_ops.relu(variance)
  stddev = math_ops.sqrt(variance)

  # Apply a minimum normalization that protects us against uniform images.
  min_stddev = math_ops.rsqrt(math_ops.cast(num_pixels, dtypes.float32))
  pixel_value_scale = math_ops.maximum(stddev, min_stddev)
  pixel_value_offset = image_mean

  image = math_ops.subtract(image, pixel_value_offset)
  image = math_ops.div(image, pixel_value_scale)
  return image 

def _variance_scale_term(self):
    """Helper to `_covariance` and `_variance` which computes a shared scale."""
    return math_ops.rsqrt(1. + self.total_concentration[..., array_ops.newaxis]) 

def l2_normalize(x, dim, epsilon=1e-12, name=None):
  """Normalizes along dimension `dim` using an L2 norm.

  For a 1-D tensor with `dim = 0`, computes

      output = x / sqrt(max(sum(x**2), epsilon))

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `dim`.

  Args:
    x: A `Tensor`.
    dim: Dimension along which to normalize.  A scalar or a vector of
      integers.
    epsilon: A lower bound value for the norm. Will use `sqrt(epsilon)` as the
      divisor if `norm < sqrt(epsilon)`.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, "l2_normalize", [x]) as name:
    x = ops.convert_to_tensor(x, name="x")
    square_sum = math_ops.reduce_sum(math_ops.square(x), dim, keep_dims=True)
    x_inv_norm = math_ops.rsqrt(math_ops.maximum(square_sum, epsilon))
    return math_ops.multiply(x, x_inv_norm, name=name) 

def l2_normalize(x, dim, epsilon=1e-12, name=None):
  """Normalizes along dimension `dim` using an L2 norm.

  For a 1-D tensor with `dim = 0`, computes

      output = x / sqrt(max(sum(x**2), epsilon))

  For `x` with more dimensions, independently normalizes each 1-D slice along
  dimension `dim`.

  Args:
    x: A `Tensor`.
    dim: Dimension along which to normalize.  A scalar or a vector of
      integers.
    epsilon: A lower bound value for the norm. Will use `sqrt(epsilon)` as the
      divisor if `norm < sqrt(epsilon)`.
    name: A name for this operation (optional).

  Returns:
    A `Tensor` with the same shape as `x`.
  """
  with ops.name_scope(name, "l2_normalize", [x]) as name:
    x = ops.convert_to_tensor(x, name="x")
    square_sum = math_ops.reduce_sum(math_ops.square(x), dim, keep_dims=True)
    x_inv_norm = math_ops.rsqrt(math_ops.maximum(square_sum, epsilon))
    return math_ops.multiply(x, x_inv_norm, name=name) 

def clip_by_average_norm(t, clip_norm, name=None):
  """Clips tensor values to a maximum average L2-norm.

  Given a tensor `t`, and a maximum clip value `clip_norm`, this operation
  normalizes `t` so that its average L2-norm is less than or equal to
  `clip_norm`. Specifically, if the average L2-norm is already less than or
  equal to `clip_norm`, then `t` is not modified. If the average L2-norm is
  greater than `clip_norm`, then this operation returns a tensor of the same
  type and shape as `t` with its values set to:

  `t * clip_norm / l2norm_avg(t)`

  In this case, the average L2-norm of the output tensor is `clip_norm`.

  This operation is typically used to clip gradients before applying them with
  an optimizer.

  Args:
    t: A `Tensor`.
    clip_norm: A 0-D (scalar) `Tensor` > 0. A maximum clipping value.
    name: A name for the operation (optional).

  Returns:
    A clipped `Tensor`.
  """
  with ops.name_scope(name, "clip_by_average_norm", [t, clip_norm]) as name:
    t = ops.convert_to_tensor(t, name="t")

    # Calculate L2-norm per element, clip elements by ratio of clip_norm to
    # L2-norm per element
    n_element = math_ops.cast(array_ops.size(t), dtypes.float32)
    l2norm_inv = math_ops.rsqrt(
        math_ops.reduce_sum(t * t, math_ops.range(array_ops.rank(t))))
    tclip = array_ops.identity(
        t * clip_norm * math_ops.minimum(
            l2norm_inv * n_element, constant_op.constant(1.0) / clip_norm),
        name=name)

  return tclip 

def clip_by_average_norm(t, clip_norm, name=None):
  """Clips tensor values to a maximum average L2-norm.

  Given a tensor `t`, and a maximum clip value `clip_norm`, this operation
  normalizes `t` so that its average L2-norm is less than or equal to
  `clip_norm`. Specifically, if the average L2-norm is already less than or
  equal to `clip_norm`, then `t` is not modified. If the average L2-norm is
  greater than `clip_norm`, then this operation returns a tensor of the same
  type and shape as `t` with its values set to:

  `t * clip_norm / l2norm_avg(t)`

  In this case, the average L2-norm of the output tensor is `clip_norm`.

  This operation is typically used to clip gradients before applying them with
  an optimizer.

  Args:
    t: A `Tensor`.
    clip_norm: A 0-D (scalar) `Tensor` > 0. A maximum clipping value.
    name: A name for the operation (optional).

  Returns:
    A clipped `Tensor`.
  """
  with ops.name_scope(name, "clip_by_average_norm", [t, clip_norm]) as name:
    t = ops.convert_to_tensor(t, name="t")

    # Calculate L2-norm per element, clip elements by ratio of clip_norm to
    # L2-norm per element
    n_element = math_ops.cast(array_ops.size(t), dtypes.float32)
    l2norm_inv = math_ops.rsqrt(
        math_ops.reduce_sum(t * t, math_ops.range(array_ops.rank(t))))
    tclip = array_ops.identity(
        t * clip_norm * math_ops.minimum(
            l2norm_inv * n_element, constant_op.constant(1.0) / clip_norm),
        name=name)

  return tclip 

def per_image_standardization(image):
  """Linearly scales `image` to have zero mean and unit norm.

  This op computes `(x - mean) / adjusted_stddev`, where `mean` is the average
  of all values in image, and
  `adjusted_stddev = max(stddev, 1.0/sqrt(image.NumElements()))`.

  `stddev` is the standard deviation of all values in `image`. It is capped
  away from zero to protect against division by 0 when handling uniform images.

  Args:
    image: 3-D tensor of shape `[height, width, channels]`.

  Returns:
    The standardized image with same shape as `image`.

  Raises:
    ValueError: if the shape of 'image' is incompatible with this function.
  """
  image = ops.convert_to_tensor(image, name='image')
  _Check3DImage(image, require_static=False)
  num_pixels = math_ops.reduce_prod(array_ops.shape(image))

  image = math_ops.cast(image, dtype=dtypes.float32)
  image_mean = math_ops.reduce_mean(image)

  variance = (math_ops.reduce_mean(math_ops.square(image)) -
              math_ops.square(image_mean))
  variance = gen_nn_ops.relu(variance)
  stddev = math_ops.sqrt(variance)

  # Apply a minimum normalization that protects us against uniform images.
  min_stddev = math_ops.rsqrt(math_ops.cast(num_pixels, dtypes.float32))
  pixel_value_scale = math_ops.maximum(stddev, min_stddev)
  pixel_value_offset = image_mean

  image = math_ops.subtract(image, pixel_value_offset)
  image = math_ops.div(image, pixel_value_scale)
  return image 

def _BaseFusedBatchNormGrad(op, use_v2, *grad):
  """Return the gradients for the 3 inputs of BatchNorm.

  Args:
    op: The BatchNormOp for which we need to compute gradients.
    use_v2: Boolean indicating whether to use the V2 version of the fused batch
            norm gradient.
    *grad: An argument list for tensors of gradients wrt the outputs
          with grad[0] as grad_y.

  Returns:
    grad_x: gradient for x, which is scale * rsqrt(variance + epsilon) *
            [grad_y - mean(grad_y) - (x - mean(x)) *
            mean(grad_y * (x - mean(x))) / (variance + epsilon)]
            in training mode; grad_y * scale * rsqrt(pop_variance + epsilon)
            in freeze mode.

    grad_scale: gradient for scale, which is sum(grad_y * (x - mean(x)) *
                rsqrt(variance + epsilon)) in training mode;
                sum(grad_y * (x - pop_mean) * rsqrt(pop_variance + epsilon))
                in freeze mode.

    grad_offset: gradient for offset, which is sum(grad_y) in training mode;
                 sum(grad_y) in freeze mode.
  """
  x = op.inputs[0]
  grad_y = grad[0]
  scale = op.inputs[1]
  epsilon = op.get_attr("epsilon")
  data_format = op.get_attr("data_format")
  is_training = op.get_attr("is_training")
  grad_fun = (gen_nn_ops.fused_batch_norm_grad_v2 if use_v2
              else gen_nn_ops.fused_batch_norm_grad)
  if is_training:
    return grad_fun(
        grad_y,
        x,
        scale,
        op.outputs[3],
        op.outputs[4],
        epsilon=epsilon,
        data_format=data_format,
        is_training=is_training)
  else:
    pop_mean = op.inputs[3]
    pop_var = op.inputs[4]
    if data_format == b"NCHW":
      x = array_ops.transpose(x, [0, 2, 3, 1])
      grad_y = array_ops.transpose(grad_y, [0, 2, 3, 1])
    dx, dscale, doffset, _, _ = grad_fun(
        grad_y,
        x,
        scale,
        pop_mean,
        pop_var,
        epsilon=epsilon,
        data_format='NHWC',
        is_training=is_training)
    if data_format == b"NCHW":
      dx = array_ops.transpose(dx, [0, 3, 1, 2])
    return dx, dscale, doffset, None, None 

def batch_normalization(x,
                        mean,
                        variance,
                        offset,
                        scale,
                        variance_epsilon,
                        name=None):
  r"""Batch normalization.

  As described in http://arxiv.org/abs/1502.03167.
  Normalizes a tensor by `mean` and `variance`, and applies (optionally) a
  `scale` \\(\gamma\\) to it, as well as an `offset` \\(\beta\\):

  \\(\frac{\gamma(x-\mu)}{\sigma}+\beta\\)

  `mean`, `variance`, `offset` and `scale` are all expected to be of one of two
  shapes:

    * In all generality, they can have the same number of dimensions as the
      input `x`, with identical sizes as `x` for the dimensions that are not
      normalized over (the 'depth' dimension(s)), and dimension 1 for the
      others which are being normalized over.
      `mean` and `variance` in this case would typically be the outputs of
      `tf.nn.moments(..., keep_dims=True)` during training, or running averages
      thereof during inference.
    * In the common case where the 'depth' dimension is the last dimension in
      the input tensor `x`, they may be one dimensional tensors of the same
      size as the 'depth' dimension.
      This is the case for example for the common `[batch, depth]` layout of
      fully-connected layers, and `[batch, height, width, depth]` for
      convolutions.
      `mean` and `variance` in this case would typically be the outputs of
      `tf.nn.moments(..., keep_dims=False)` during training, or running averages
      thereof during inference.

  Args:
    x: Input `Tensor` of arbitrary dimensionality.
    mean: A mean `Tensor`.
    variance: A variance `Tensor`.
    offset: An offset `Tensor`, often denoted \\(\beta\\) in equations, or
      None. If present, will be added to the normalized tensor.
    scale: A scale `Tensor`, often denoted \\(\gamma\\) in equations, or
      `None`. If present, the scale is applied to the normalized tensor.
    variance_epsilon: A small float number to avoid dividing by 0.
    name: A name for this operation (optional).

  Returns:
    the normalized, scaled, offset tensor.
  """
  with ops.name_scope(name, "batchnorm", [x, mean, variance, scale, offset]):
    inv = math_ops.rsqrt(variance + variance_epsilon)
    if scale is not None:
      inv *= scale
    return x * inv + (offset - mean * inv
                      if offset is not None else -mean * inv) 

def _bahdanau_score(processed_query, keys, normalize):
  """Implements Bahdanau-style (additive) scoring function.

  This attention has two forms.  The first is Bhandanau attention,
  as described in:

  Dzmitry Bahdanau, Kyunghyun Cho, Yoshua Bengio.
  "Neural Machine Translation by Jointly Learning to Align and Translate."
  ICLR 2015. https://arxiv.org/abs/1409.0473

  The second is the normalized form.  This form is inspired by the
  weight normalization article:

  Tim Salimans, Diederik P. Kingma.
  "Weight Normalization: A Simple Reparameterization to Accelerate
   Training of Deep Neural Networks."
  https://arxiv.org/abs/1602.07868

  To enable the second form, set `normalize=True`.

  Args:
    processed_query: Tensor, shape `[batch_size, num_units]` to compare to keys.
    keys: Processed memory, shape `[batch_size, max_time, num_units]`.
    normalize: Whether to normalize the score function.

  Returns:
    A `[batch_size, max_time]` tensor of unnormalized score values.
  """
  dtype = processed_query.dtype
  # Get the number of hidden units from the trailing dimension of keys
  num_units = keys.shape[2].value or array_ops.shape(keys)[2]
  # Reshape from [batch_size, ...] to [batch_size, 1, ...] for broadcasting.
  processed_query = array_ops.expand_dims(processed_query, 1)
  v = variable_scope.get_variable(
      "attention_v", [num_units], dtype=dtype)
  if normalize:
    # Scalar used in weight normalization
    g = variable_scope.get_variable(
        "attention_g", dtype=dtype,
        initializer=math.sqrt((1. / num_units)))
    # Bias added prior to the nonlinearity
    b = variable_scope.get_variable(
        "attention_b", [num_units], dtype=dtype,
        initializer=init_ops.zeros_initializer())
    # normed_v = g * v / ||v||
    normed_v = g * v * math_ops.rsqrt(
        math_ops.reduce_sum(math_ops.square(v)))
    return math_ops.reduce_sum(
        normed_v * math_ops.tanh(keys + processed_query + b), [2])
  else:
    return math_ops.reduce_sum(v * math_ops.tanh(keys + processed_query), [2]) 

def batch_normalization(x,
                        mean,
                        variance,
                        offset,
                        scale,
                        variance_epsilon,
                        name=None):
  r"""Batch normalization.

  As described in http://arxiv.org/abs/1502.03167.
  Normalizes a tensor by `mean` and `variance`, and applies (optionally) a
  `scale` \\(\gamma\\) to it, as well as an `offset` \\(\beta\\):

  \\(\frac{\gamma(x-\mu)}{\sigma}+\beta\\)

  `mean`, `variance`, `offset` and `scale` are all expected to be of one of two
  shapes:

    * In all generality, they can have the same number of dimensions as the
      input `x`, with identical sizes as `x` for the dimensions that are not
      normalized over (the 'depth' dimension(s)), and dimension 1 for the
      others which are being normalized over.
      `mean` and `variance` in this case would typically be the outputs of
      `tf.nn.moments(..., keep_dims=True)` during training, or running averages
      thereof during inference.
    * In the common case where the 'depth' dimension is the last dimension in
      the input tensor `x`, they may be one dimensional tensors of the same
      size as the 'depth' dimension.
      This is the case for example for the common `[batch, depth]` layout of
      fully-connected layers, and `[batch, height, width, depth]` for
      convolutions.
      `mean` and `variance` in this case would typically be the outputs of
      `tf.nn.moments(..., keep_dims=False)` during training, or running averages
      thereof during inference.

  Args:
    x: Input `Tensor` of arbitrary dimensionality.
    mean: A mean `Tensor`.
    variance: A variance `Tensor`.
    offset: An offset `Tensor`, often denoted \\(\beta\\) in equations, or
      None. If present, will be added to the normalized tensor.
    scale: A scale `Tensor`, often denoted \\(\gamma\\) in equations, or
      `None`. If present, the scale is applied to the normalized tensor.
    variance_epsilon: A small float number to avoid dividing by 0.
    name: A name for this operation (optional).

  Returns:
    the normalized, scaled, offset tensor.
  """
  with ops.name_scope(name, "batchnorm", [x, mean, variance, scale, offset]):
    inv = math_ops.rsqrt(variance + variance_epsilon)
    if scale is not None:
      inv *= scale
    return x * inv + (offset - mean * inv
                      if offset is not None else -mean * inv) 

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 _bahdanau_score(processed_query, keys, normalize):
  """Implements Bahdanau-style (additive) scoring function.

  This attention has two forms.  The first is Bhandanau attention,
  as described in:

  Dzmitry Bahdanau, Kyunghyun Cho, Yoshua Bengio.
  "Neural Machine Translation by Jointly Learning to Align and Translate."
  ICLR 2015. https://arxiv.org/abs/1409.0473

  The second is the normalized form.  This form is inspired by the
  weight normalization article:

  Tim Salimans, Diederik P. Kingma.
  "Weight Normalization: A Simple Reparameterization to Accelerate
   Training of Deep Neural Networks."
  https://arxiv.org/abs/1602.07868

  To enable the second form, set `normalize=True`.

  Args:
    processed_query: Tensor, shape `[batch_size, num_units]` to compare to keys.
    keys: Processed memory, shape `[batch_size, max_time, num_units]`.
    normalize: Whether to normalize the score function.

  Returns:
    A `[batch_size, max_time]` tensor of unnormalized score values.
  """
  dtype = processed_query.dtype
  # Get the number of hidden units from the trailing dimension of keys
  num_units = keys.shape[2].value or array_ops.shape(keys)[2]
  # Reshape from [batch_size, ...] to [batch_size, 1, ...] for broadcasting.
  processed_query = array_ops.expand_dims(processed_query, 1)
  v = variable_scope.get_variable(
      "attention_v", [num_units], dtype=dtype)
  if normalize:
    # Scalar used in weight normalization
    g = variable_scope.get_variable(
        "attention_g", dtype=dtype,
        initializer=math.sqrt((1. / num_units)))
    # Bias added prior to the nonlinearity
    b = variable_scope.get_variable(
        "attention_b", [num_units], dtype=dtype,
        initializer=init_ops.zeros_initializer())
    # normed_v = g * v / ||v||
    normed_v = g * v * math_ops.rsqrt(
        math_ops.reduce_sum(math_ops.square(v)))
    return math_ops.reduce_sum(
        normed_v * math_ops.tanh(keys + processed_query + b), [2])
  else:
    return math_ops.reduce_sum(v * math_ops.tanh(keys + processed_query), [2])