Activation functions

flax.experimental.nnx.celu(x, alpha=1.0)

Continuously-differentiable exponential linear unit activation.

Computes the element-wise function:

\[\begin{split}\mathrm{celu}(x) = \begin{cases} x, & x > 0\\ \alpha \left(\exp(\frac{x}{\alpha}) - 1\right), & x \le 0 \end{cases}\end{split}\]

For more information, see Continuously Differentiable Exponential Linear Units.

Parameters

• x – input array

• alpha – array or scalar (default: 1.0)

Returns

An array.

flax.experimental.nnx.elu(x, alpha=1.0)

Exponential linear unit activation function.

Computes the element-wise function:

\[\begin{split}\mathrm{elu}(x) = \begin{cases} x, & x > 0\\ \alpha \left(\exp(x) - 1\right), & x \le 0 \end{cases}\end{split}\]

Parameters

• x – input array

• alpha – scalar or array of alpha values (default: 1.0)

Returns

An array.

See also

selu()

flax.experimental.nnx.gelu(x, approximate=True)

Gaussian error linear unit activation function.

If approximate=False, computes the element-wise function:

\[\mathrm{gelu}(x) = \frac{x}{2} \left(1 + \mathrm{erf} \left( \frac{x}{\sqrt{2}} \right) \right)\]

If approximate=True, uses the approximate formulation of GELU:

\[\mathrm{gelu}(x) = \frac{x}{2} \left(1 + \mathrm{tanh} \left( \sqrt{\frac{2}{\pi}} \left(x + 0.044715 x^3 \right) \right) \right)\]

For more information, see Gaussian Error Linear Units (GELUs), section 2.

Parameters

• x – input array

• approximate – whether to use the approximate or exact formulation.

flax.experimental.nnx.glu(x, axis=-1)

Gated linear unit activation function.

Computes the function:

\[\mathrm{glu}(x) = x\left[\ldots, 0:\frac{n}{2}, \ldots\right] \cdot \mathrm{sigmoid} \left( x\left[\ldots, \frac{n}{2}:n, \ldots\right] \right)\]

where the array is split into two along axis. The size of the axis dimension must be divisible by two.

Parameters

• x – input array

• axis – the axis along which the split should be computed (default: -1)

Returns

An array.

See also

sigmoid()

flax.experimental.nnx.hard_sigmoid(x)

Hard Sigmoid activation function.

Computes the element-wise function

\[\mathrm{hard\_sigmoid}(x) = \frac{\mathrm{relu6}(x + 3)}{6}\]

Parameters

x – input array

Returns

An array.

See also

relu6()

flax.experimental.nnx.hard_silu(x)

Hard SiLU (swish) activation function

Computes the element-wise function

\[\mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x)\]

Both hard_silu() and hard_swish() are aliases for the same function.

Parameters

x – input array

Returns

An array.

See also

hard_sigmoid()

flax.experimental.nnx.hard_swish(x)

Hard SiLU (swish) activation function

Computes the element-wise function

\[\mathrm{hard\_silu}(x) = x \cdot \mathrm{hard\_sigmoid}(x)\]

Both hard_silu() and hard_swish() are aliases for the same function.

Parameters

x – input array

Returns

An array.

See also

hard_sigmoid()

flax.experimental.nnx.hard_tanh(x)

Hard \(\mathrm{tanh}\) activation function.

Computes the element-wise function:

\[\begin{split}\mathrm{hard\_tanh}(x) = \begin{cases} -1, & x < -1\\ x, & -1 \le x \le 1\\ 1, & 1 < x \end{cases}\end{split}\]

Parameters

x – input array

Returns

An array.

flax.experimental.nnx.leaky_relu(x, negative_slope=0.01)

Leaky rectified linear unit activation function.

Computes the element-wise function:

\[\begin{split}\mathrm{leaky\_relu}(x) = \begin{cases} x, & x \ge 0\\ \alpha x, & x < 0 \end{cases}\end{split}\]

where \(\alpha\) = negative_slope.

Parameters

• x – input array

• negative_slope – array or scalar specifying the negative slope (default: 0.01)

Returns

An array.

See also

relu()

flax.experimental.nnx.log_sigmoid(x)

Log-sigmoid activation function.

Computes the element-wise function:

\[\mathrm{log\_sigmoid}(x) = \log(\mathrm{sigmoid}(x)) = -\log(1 + e^{-x})\]

Parameters

x – input array

Returns

An array.

See also

sigmoid()

flax.experimental.nnx.log_softmax(x, axis=-1, where=None, initial=_UNSPECIFIED)

Log-Softmax function.

Computes the logarithm of the softmax function, which rescales elements to the range \([-\infty, 0)\).

\[\mathrm{log\_softmax}(x)_i = \log \left( \frac{\exp(x_i)}{\sum_j \exp(x_j)} \right)\]

Parameters

• x – input array

• axis – the axis or axes along which the log_softmax should be computed. Either an integer or a tuple of integers.

• where – Elements to include in the log_softmax.

Returns

An array.

See also

softmax()

flax.experimental.nnx.logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False, where=None)

Log-sum-exp reduction.

JAX implementation of scipy.special.logsumexp().

\[\mathrm{logsumexp}(a) = \mathrm{log} \sum_j b \cdot \mathrm{exp}(a_{ij})\]

where the \(j\) indices range over one or more dimensions to be reduced.

Parameters

• a – the input array

• axis – the axis or axes over which to reduce. May be either None, an int, or a tuple of ints.

• b – scaling factors for \(\mathrm{exp}(a)\). Must be broadcastable to the shape of a.

• keepdims – If True, the axes that are reduced are left in the output as dimensions of size 1.

• return_sign – If True, the output will be a (result, sign) pair, where sign is the sign of the sums and result contains the logarithms of their absolute values. If False only result is returned and it will contain NaN values if the sums are negative.

• where – Elements to include in the reduction.

Returns

Either an array result or a pair of arrays (result, sign), depending on the value of the return_sign argument.

flax.experimental.nnx.one_hot(x, num_classes, *, dtype=<class 'jax.numpy.float64'>, axis=-1)

One-hot encodes the given indices.

Each index in the input x is encoded as a vector of zeros of length num_classes with the element at index set to one:

>>> jax.nn.one_hot(jnp.array([0, 1, 2]), 3)
Array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]], dtype=float32)

Indices outside the range [0, num_classes) will be encoded as zeros:

>>> jax.nn.one_hot(jnp.array([-1, 3]), 3)
Array([[0., 0., 0.],
       [0., 0., 0.]], dtype=float32)

Parameters

• x – A tensor of indices.

• num_classes – Number of classes in the one-hot dimension.

• dtype – optional, a float dtype for the returned values (default jnp.float_).

• axis – the axis or axes along which the function should be computed.

flax.experimental.nnx.relu(x)

Rectified linear unit activation function.

Computes the element-wise function:

\[\mathrm{relu}(x) = \max(x, 0)\]

except under differentiation, we take:

\[\nabla \mathrm{relu}(0) = 0\]

For more information see Numerical influence of ReLU’(0) on backpropagation.

Parameters

x – input array

Returns

An array.

Example

>>> jax.nn.relu(jax.numpy.array([-2., -1., -0.5, 0, 0.5, 1., 2.]))
Array([0. , 0. , 0. , 0. , 0.5, 1. , 2. ], dtype=float32)

See also

relu6()

flax.experimental.nnx.selu(x)

Scaled exponential linear unit activation.

Computes the element-wise function:

\[\begin{split}\mathrm{selu}(x) = \lambda \begin{cases} x, & x > 0\\ \alpha e^x - \alpha, & x \le 0 \end{cases}\end{split}\]

where \(\lambda = 1.0507009873554804934193349852946\) and \(\alpha = 1.6732632423543772848170429916717\).

For more information, see Self-Normalizing Neural Networks.

Parameters

x – input array

Returns

An array.

See also

elu()

flax.experimental.nnx.sigmoid(x)

Sigmoid activation function.

Computes the element-wise function:

\[\mathrm{sigmoid}(x) = \frac{1}{1 + e^{-x}}\]

Parameters

x – input array

Returns

An array.

See also

log_sigmoid()

flax.experimental.nnx.silu(x)

SiLU (aka swish) activation function.

Computes the element-wise function:

\[\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}}\]

swish() and silu() are both aliases for the same function.

Parameters

x – input array

Returns

An array.

See also

sigmoid()

flax.experimental.nnx.soft_sign(x)

Soft-sign activation function.

Computes the element-wise function

\[\mathrm{soft\_sign}(x) = \frac{x}{|x| + 1}\]

Parameters

x – input array

flax.experimental.nnx.softmax(x, axis=-1, where=None, initial=_UNSPECIFIED)

Softmax function.

Computes the function which rescales elements to the range \([0, 1]\) such that the elements along axis sum to \(1\).

\[\mathrm{softmax}(x) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}\]

Parameters

• x – input array

• axis – the axis or axes along which the softmax should be computed. The softmax output summed across these dimensions should sum to \(1\). Either an integer or a tuple of integers.

• where – Elements to include in the softmax.

Returns

An array.

See also

log_softmax()

flax.experimental.nnx.softplus(x)

Softplus activation function.

Computes the element-wise function

\[\mathrm{softplus}(x) = \log(1 + e^x)\]

Parameters

x – input array

flax.experimental.nnx.standardize(x, axis=-1, mean=None, variance=None, epsilon=1e-05, where=None)

Normalizes an array by subtracting mean and dividing by \(\sqrt{\mathrm{variance}}\).

flax.experimental.nnx.swish(x)

SiLU (aka swish) activation function.

Computes the element-wise function:

\[\mathrm{silu}(x) = x \cdot \mathrm{sigmoid}(x) = \frac{x}{1 + e^{-x}}\]

swish() and silu() are both aliases for the same function.

Parameters

x – input array

Returns

An array.

See also

sigmoid()

flax.experimental.nnx.tanh(x, /)

Compute hyperbolic tangent element-wise.

LAX-backend implementation of numpy.tanh().

Original docstring below.

Equivalent to np.sinh(x)/np.cosh(x) or -1j * np.tan(1j*x).

Parameters

x (array_like) – Input array.

Returns

y – The corresponding hyperbolic tangent values. This is a scalar if x is a scalar.

Return type

ndarray

References

1

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY: Dover, 1972, pg. 83. https://personal.math.ubc.ca/~cbm/aands/page_83.htm

2

Wikipedia, “Hyperbolic function”, https://en.wikipedia.org/wiki/Hyperbolic_function