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309 lines
8.4 KiB
Python
309 lines
8.4 KiB
Python
# Copyright 2019 Google LLC
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# https://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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"""Shared neural network activations and other functions."""
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import numpy as np
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from jax import custom_jvp
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from jax import dtypes
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from jax import lax
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from jax import core
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from jax.scipy.special import expit
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import jax.numpy as jnp
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# activations
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@custom_jvp
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def relu(x):
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r"""Rectified linear unit activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{relu}(x) = \max(x, 0)
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"""
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return jnp.maximum(x, 0)
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relu.defjvps(lambda g, ans, x: lax.select(x > 0, g, lax.full_like(g, 0)))
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def softplus(x):
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r"""Softplus activation function.
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Computes the element-wise function
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.. math::
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\mathrm{softplus}(x) = \log(1 + e^x)
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"""
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return jnp.logaddexp(x, 0)
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def soft_sign(x):
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r"""Soft-sign activation function.
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Computes the element-wise function
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.. math::
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\mathrm{soft\_sign}(x) = \frac{x}{|x| + 1}
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"""
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return x / (jnp.abs(x) + 1)
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def sigmoid(x):
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r"""Sigmoid activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{sigmoid}(x) = \frac{1}{1 + e^{-x}}
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"""
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return expit(x)
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def silu(x):
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r"""SiLU activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}}
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"""
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return x * sigmoid(x)
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swish = silu
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def log_sigmoid(x):
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r"""Log-sigmoid activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{log\_sigmoid}(x) = \log(\mathrm{sigmoid}(x)) = -\log(1 + e^{-x})
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"""
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return -softplus(-x)
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def elu(x, alpha=1.0):
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r"""Exponential linear unit activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{elu}(x) = \begin{cases}
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x, & x > 0\\
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\alpha \left(\exp(x) - 1\right), & x \le 0
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\end{cases}
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"""
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safe_x = jnp.where(x > 0, 0., x)
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return jnp.where(x > 0, x, alpha * jnp.expm1(safe_x))
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def leaky_relu(x, negative_slope=1e-2):
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r"""Leaky rectified linear unit activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{leaky\_relu}(x) = \begin{cases}
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x, & x \ge 0\\
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\alpha x, & x < 0
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\end{cases}
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where :math:`\alpha` = :code:`negative_slope`.
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"""
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return jnp.where(x >= 0, x, negative_slope * x)
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def hard_tanh(x):
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r"""Hard :math:`\mathrm{tanh}` activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{hard\_tanh}(x) = \begin{cases}
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-1, & x < -1\\
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x, & 0 \le x \le 1\\
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1, & 1 < x
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\end{cases}
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"""
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return jnp.where(x > 1, 1, jnp.where(x < -1, -1, x))
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def celu(x, alpha=1.0):
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r"""Continuously-differentiable exponential linear unit activation.
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Computes the element-wise function:
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.. math::
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\mathrm{celu}(x) = \begin{cases}
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x, & x > 0\\
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\alpha \left(\exp(\frac{x}{\alpha}) - 1\right), & x \le 0
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\end{cases}
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For more information, see
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`Continuously Differentiable Exponential Linear Units
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<https://arxiv.org/pdf/1704.07483.pdf>`_."""
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return jnp.where(x > 0, x, alpha * jnp.expm1(x / alpha))
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def selu(x):
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r"""Scaled exponential linear unit activation.
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Computes the element-wise function:
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.. math::
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\mathrm{selu}(x) = \lambda \begin{cases}
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x, & x > 0\\
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\alpha e^x - \alpha, & x \le 0
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\end{cases}
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where :math:`\lambda = 1.0507009873554804934193349852946` and
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:math:`\alpha = 1.6732632423543772848170429916717`.
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For more information, see
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`Self-Normalizing Neural Networks
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<https://papers.nips.cc/paper/6698-self-normalizing-neural-networks.pdf>`_.
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"""
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alpha = 1.6732632423543772848170429916717
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scale = 1.0507009873554804934193349852946
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return scale * elu(x, alpha)
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def gelu(x):
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r"""Gaussian error linear unit activation function.
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Computes the element-wise function:
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.. math::
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\mathrm{gelu}(x) = \frac{x}{2} \left(1 + \mathrm{tanh} \left(
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\sqrt{\frac{2}{\pi}} \left(x + 0.044715 x^3 \right) \right) \right)
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We explicitly use the approximation rather than the exact formulation for
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speed. For more information, see `Gaussian Error Linear Units (GELUs)
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<https://arxiv.org/abs/1606.08415>`_, section 2.
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"""
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sqrt_2_over_pi = np.sqrt(2 / np.pi).astype(x.dtype)
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cdf = 0.5 * (1.0 + jnp.tanh(sqrt_2_over_pi * (x + 0.044715 * (x ** 3))))
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return x * cdf
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def glu(x, axis=-1):
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"""Gated linear unit activation function."""
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size = x.shape[axis]
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assert size % 2 == 0, "axis size must be divisible by 2"
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x1, x2 = jnp.split(x, 2, axis)
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return x1 * sigmoid(x2)
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# other functions
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def log_softmax(x, axis=-1):
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r"""Log-Softmax function.
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Computes the logarithm of the :code:`softmax` function, which rescales
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elements to the range :math:`[-\infty, 0)`.
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.. math ::
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\mathrm{log\_softmax}(x) = \log \left( \frac{\exp(x_i)}{\sum_j \exp(x_j)}
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\right)
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Args:
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axis: the axis or axes along which the :code:`log_softmax` should be
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computed. Either an integer or a tuple of integers.
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"""
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shifted = x - lax.stop_gradient(x.max(axis, keepdims=True))
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return shifted - jnp.log(jnp.sum(jnp.exp(shifted), axis, keepdims=True))
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def softmax(x, axis=-1):
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r"""Softmax function.
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Computes the function which rescales elements to the range :math:`[0, 1]`
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such that the elements along :code:`axis` sum to :math:`1`.
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.. math ::
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\mathrm{softmax}(x) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}
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Args:
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axis: the axis or axes along which the softmax should be computed. The
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softmax output summed across these dimensions should sum to :math:`1`.
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Either an integer or a tuple of integers.
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"""
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unnormalized = jnp.exp(x - lax.stop_gradient(x.max(axis, keepdims=True)))
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return unnormalized / unnormalized.sum(axis, keepdims=True)
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def normalize(x, axis=-1, mean=None, variance=None, epsilon=1e-5):
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"""Normalizes an array by subtracting mean and dividing by sqrt(var)."""
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if mean is None:
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mean = jnp.mean(x, axis, keepdims=True)
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if variance is None:
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# this definition is traditionally seen as less accurate than jnp.var's
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# mean((x - mean(x))**2) but may be faster and even, given typical
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# activation distributions and low-precision arithmetic, more accurate
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# when used in neural network normalization layers
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variance = jnp.mean(jnp.square(x), axis, keepdims=True) - jnp.square(mean)
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return (x - mean) * lax.rsqrt(variance + epsilon)
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def one_hot(x, num_classes, *, dtype=jnp.float64):
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"""One-hot encodes the given indicies.
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Each index in the input ``x`` is encoded as a vector of zeros of length
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``num_classes`` with the element at ``index`` set to one::
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>>> jax.nn.one_hot(jnp.array([0, 1, 2]), 3)
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DeviceArray([[1., 0., 0.],
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[0., 1., 0.],
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[0., 0., 1.]], dtype=float32)
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Indicies outside the range [0, num_classes) will be encoded as zeros::
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>>> jax.nn.one_hot(jnp.array([-1, 3]), 3)
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DeviceArray([[0., 0., 0.],
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[0., 0., 0.]], dtype=float32)
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Args:
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x: A tensor of indices.
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num_classes: Number of classes in the one-hot dimension.
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dtype: optional, a float dtype for the returned values (default float64 if
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jax_enable_x64 is true, otherwise float32).
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"""
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num_classes = core.concrete_or_error(
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int, num_classes,
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"The error arose in jax.nn.one_hot argument `num_classes`.")
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dtype = dtypes.canonicalize_dtype(dtype)
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x = jnp.asarray(x)
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lhs = x[..., jnp.newaxis]
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rhs = lax.broadcast_to_rank(jnp.arange(num_classes, dtype=x.dtype), lhs.ndim)
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return jnp.array(lhs == rhs, dtype=dtype)
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def relu6(x):
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r"""Rectified Linear Unit 6 activation function.
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Computes the element-wise function
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.. math::
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\mathrm{relu6}(x) = \min(\max(x, 0), 6)
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"""
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return jnp.minimum(jnp.maximum(x, 0), 6.)
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def hard_sigmoid(x):
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r"""Hard Sigmoid activation function.
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Computes the element-wise function
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.. math::
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\mathrm{hard\_sigmoid}(x) = \frac{\mathrm{relu6}(x + 3)}{6}
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"""
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return relu6(x + 3.) / 6.
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def hard_silu(x):
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r"""Hard SiLU activation function
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Computes the element-wise function
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.. math::
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\mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x)
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"""
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return x * hard_sigmoid(x)
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hard_swish = hard_silu
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