torch.special

The torch.special module, modeled after SciPy’s special module.

Functions

torch.special.airy_ai(input, *, out=None) → Tensor

Airy function $\text{Ai}\left(\text{input}\right)$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.bessel_j0(input, *, out=None) → Tensor

Bessel function of the first kind of order $0$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.bessel_j1(input, *, out=None) → Tensor

Bessel function of the first kind of order $1$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.digamma(input, *, out=None) → Tensor

Computes the logarithmic derivative of the gamma function on input.

$$\digamma(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}$$

Parameters

input (Tensor) – the tensor to compute the digamma function on

Keyword Arguments

out (Tensor, optional) – the output tensor.

Note

This function is similar to SciPy’s scipy.special.digamma.

Note

From PyTorch 1.8 onwards, the digamma function returns -Inf for 0. Previously it returned NaN for 0.

Example:

>>> a = torch.tensor([1, 0.5])
>>> torch.special.digamma(a)
tensor([-0.5772, -1.9635])

torch.special.entr(input, *, out=None) → Tensor

Computes the entropy on input (as defined below), elementwise.

$$\begin{align} \text{entr(x)} = \begin{cases} -x * \ln(x) & x > 0 \\ 0 & x = 0.0 \\ -\infty & x < 0 \end{cases} \end{align}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> a = torch.arange(-0.5, 1, 0.5)
>>> a
tensor([-0.5000,  0.0000,  0.5000])
>>> torch.special.entr(a)
tensor([  -inf, 0.0000, 0.3466])

torch.special.erf(input, *, out=None) → Tensor

Computes the error function of input. The error function is defined as follows:

$$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.erf(torch.tensor([0, -1., 10.]))
tensor([ 0.0000, -0.8427,  1.0000])

torch.special.erfc(input, *, out=None) → Tensor

Computes the complementary error function of input. The complementary error function is defined as follows:

$$\mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.erfc(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 1.8427,  0.0000])

torch.special.erfcx(input, *, out=None) → Tensor

Computes the scaled complementary error function for each element of input. The scaled complementary error function is defined as follows:

$$\mathrm{erfcx}(x) = e^{x^2} \mathrm{erfc}(x)$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.erfcx(torch.tensor([0, -1., 10.]))
tensor([ 1.0000, 5.0090, 0.0561])

torch.special.erfinv(input, *, out=None) → Tensor

Computes the inverse error function of input. The inverse error function is defined in the range $(-1, 1)$ as:

$$\mathrm{erfinv}(\mathrm{erf}(x)) = x$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.erfinv(torch.tensor([0, 0.5, -1.]))
tensor([ 0.0000,  0.4769,    -inf])

torch.special.exp2(input, *, out=None) → Tensor

Computes the base two exponential function of input.

$$y_{i} = 2^{x_{i}}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
tensor([ 1.,  2.,  8., 16.])

torch.special.expit(input, *, out=None) → Tensor

Computes the expit (also known as the logistic sigmoid function) of the elements of input.

$$\text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> t = torch.randn(4)
>>> t
tensor([ 0.9213,  1.0887, -0.8858, -1.7683])
>>> torch.special.expit(t)
tensor([ 0.7153,  0.7481,  0.2920,  0.1458])

torch.special.expm1(input, *, out=None) → Tensor

Computes the exponential of the elements minus 1 of input.

$$y_{i} = e^{x_{i}} - 1$$

Note

This function provides greater precision than exp(x) - 1 for small values of x.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.special.expm1(torch.tensor([0, math.log(2.)]))
tensor([ 0.,  1.])

torch.special.gammainc(input, other, *, out=None) → Tensor

Computes the regularized lower incomplete gamma function:

$$\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_0^{\text{other}_i} t^{\text{input}_i-1} e^{-t} dt$$

where both $\text{input}_i$ and $\text{other}_i$ are weakly positive and at least one is strictly positive. If both are zero or either is negative then $\text{out}_i=\text{nan}$. $\Gamma(\cdot)$ in the equation above is the gamma function,

$$\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.$$

See torch.special.gammaincc() and torch.special.gammaln() for related functions.

Supports broadcasting to a common shape and float inputs.

Note

The backward pass with respect to input is not yet supported. Please open an issue on PyTorch’s Github to request it.

Parameters

• input (Tensor) – the first non-negative input tensor

• other (Tensor) – the second non-negative input tensor

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> a1 = torch.tensor([4.0])
>>> a2 = torch.tensor([3.0, 4.0, 5.0])
>>> a = torch.special.gammaincc(a1, a2)
tensor([0.3528, 0.5665, 0.7350])
tensor([0.3528, 0.5665, 0.7350])
>>> b = torch.special.gammainc(a1, a2) + torch.special.gammaincc(a1, a2)
tensor([1., 1., 1.])

torch.special.gammaincc(input, other, *, out=None) → Tensor

Computes the regularized upper incomplete gamma function:

$$\text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_{\text{other}_i}^{\infty} t^{\text{input}_i-1} e^{-t} dt$$

where both $\text{input}_i$ and $\text{other}_i$ are weakly positive and at least one is strictly positive. If both are zero or either is negative then $\text{out}_i=\text{nan}$. $\Gamma(\cdot)$ in the equation above is the gamma function,

$$\Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.$$

See torch.special.gammainc() and torch.special.gammaln() for related functions.

Supports broadcasting to a common shape and float inputs.

Note

The backward pass with respect to input is not yet supported. Please open an issue on PyTorch’s Github to request it.

Parameters

• input (Tensor) – the first non-negative input tensor

• other (Tensor) – the second non-negative input tensor

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> a1 = torch.tensor([4.0])
>>> a2 = torch.tensor([3.0, 4.0, 5.0])
>>> a = torch.special.gammaincc(a1, a2)
tensor([0.6472, 0.4335, 0.2650])
>>> b = torch.special.gammainc(a1, a2) + torch.special.gammaincc(a1, a2)
tensor([1., 1., 1.])

torch.special.gammaln(input, *, out=None) → Tensor

Computes the natural logarithm of the absolute value of the gamma function on input.

$$\text{out}_{i} = \ln \Gamma(|\text{input}_{i}|)$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> a = torch.arange(0.5, 2, 0.5)
>>> torch.special.gammaln(a)
tensor([ 0.5724,  0.0000, -0.1208])

torch.special.i0(input, *, out=None) → Tensor

Computes the zeroth order modified Bessel function of the first kind for each element of input.

$$\text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}$$

Parameters

input (Tensor) – the input tensor

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> torch.i0(torch.arange(5, dtype=torch.float32))
tensor([ 1.0000,  1.2661,  2.2796,  4.8808, 11.3019])

torch.special.i0e(input, *, out=None) → Tensor

Computes the exponentially scaled zeroth order modified Bessel function of the first kind (as defined below) for each element of input.

$$\text{out}_{i} = \exp(-|x|) * i0(x) = \exp(-|x|) * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.i0e(torch.arange(5, dtype=torch.float32))
tensor([1.0000, 0.4658, 0.3085, 0.2430, 0.2070])

torch.special.i1(input, *, out=None) → Tensor

Computes the first order modified Bessel function of the first kind (as defined below) for each element of input.

$$\text{out}_{i} = \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.i1(torch.arange(5, dtype=torch.float32))
tensor([0.0000, 0.5652, 1.5906, 3.9534, 9.7595])

torch.special.i1e(input, *, out=None) → Tensor

Computes the exponentially scaled first order modified Bessel function of the first kind (as defined below) for each element of input.

$$\text{out}_{i} = \exp(-|x|) * i1(x) = \exp(-|x|) * \frac{(\text{input}_{i})}{2} * \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!) * (k+1)!}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.i1e(torch.arange(5, dtype=torch.float32))
tensor([0.0000, 0.2079, 0.2153, 0.1968, 0.1788])

torch.special.log1p(input, *, out=None) → Tensor

Alias for torch.log1p().

torch.special.log_ndtr(input, *, out=None) → Tensor

Computes the log of the area under the standard Gaussian probability density function, integrated from minus infinity to input, elementwise.

$$\text{log\_ndtr}(x) = \log\left(\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt \right)$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.log_ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
tensor([-6.6077 -3.7832 -1.841  -0.6931 -0.1728 -0.023  -0.0014])

torch.special.log_softmax(input, dim, *, dtype=None) → Tensor

Computes softmax followed by a logarithm.

While mathematically equivalent to log(softmax(x)), doing these two operations separately is slower and numerically unstable. This function is computed as:

$$\text{log\_softmax}(x_{i}) = \log\left(\frac{\exp(x_i) }{ \sum_j \exp(x_j)} \right)$$

Parameters

• input (Tensor) – input

• dim (int) – A dimension along which log_softmax will be computed.

• dtype (torch.dtype, optional) – the desired data type of returned tensor. If specified, the input tensor is cast to dtype before the operation is performed. This is useful for preventing data type overflows. Default: None.

Example::

>>> t = torch.ones(2, 2)
>>> torch.special.log_softmax(t, 0)
tensor([[-0.6931, -0.6931],
        [-0.6931, -0.6931]])

torch.special.logit(input, eps=None, *, out=None) → Tensor

Returns a new tensor with the logit of the elements of input. input is clamped to [eps, 1 - eps] when eps is not None. When eps is None and input < 0 or input > 1, the function will yields NaN.

$$\begin{align} y_{i} &= \ln(\frac{z_{i}}{1 - z_{i}}) \\ z_{i} &= \begin{cases} x_{i} & \text{if eps is None} \\ \text{eps} & \text{if } x_{i} < \text{eps} \\ x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\ 1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps} \end{cases} \end{align}$$

Parameters

• input (Tensor) – the input tensor.

• eps (float, optional) – the epsilon for input clamp bound. Default: None

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> a = torch.rand(5)
>>> a
tensor([0.2796, 0.9331, 0.6486, 0.1523, 0.6516])
>>> torch.special.logit(a, eps=1e-6)
tensor([-0.9466,  2.6352,  0.6131, -1.7169,  0.6261])

torch.special.logsumexp(input, dim, keepdim=False, *, out=None)

Alias for torch.logsumexp().

torch.special.multigammaln(input, p, *, out=None) → Tensor

Computes the multivariate log-gamma function with dimension $p$ element-wise, given by

$$\log(\Gamma_{p}(a)) = C + \displaystyle \sum_{i=1}^{p} \log\left(\Gamma\left(a - \frac{i - 1}{2}\right)\right)$$

where $C = \log(\pi) \cdot \frac{p (p - 1)}{4}$ and $\Gamma(-)$ is the Gamma function.

All elements must be greater than $\frac{p - 1}{2}$, otherwise the behavior is undefiend.

Parameters

• input (Tensor) – the tensor to compute the multivariate log-gamma function

• p (int) – the number of dimensions

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> a = torch.empty(2, 3).uniform_(1, 2)
>>> a
tensor([[1.6835, 1.8474, 1.1929],
        [1.0475, 1.7162, 1.4180]])
>>> torch.special.multigammaln(a, 2)
tensor([[0.3928, 0.4007, 0.7586],
        [1.0311, 0.3901, 0.5049]])

torch.special.ndtr(input, *, out=None) → Tensor

Computes the area under the standard Gaussian probability density function, integrated from minus infinity to input, elementwise.

$$\text{ndtr}(x) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x} e^{-\frac{1}{2}t^2} dt$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.ndtr(torch.tensor([-3., -2, -1, 0, 1, 2, 3]))
tensor([0.0013, 0.0228, 0.1587, 0.5000, 0.8413, 0.9772, 0.9987])

torch.special.ndtri(input, *, out=None) → Tensor

Computes the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to input, elementwise.

$$\text{ndtri}(p) = \sqrt{2}\text{erf}^{-1}(2p - 1)$$

Note

Also known as quantile function for Normal Distribution.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> torch.special.ndtri(torch.tensor([0, 0.25, 0.5, 0.75, 1]))
tensor([   -inf, -0.6745,  0.0000,  0.6745,     inf])

torch.special.polygamma(n, input, *, out=None) → Tensor

Computes the $n^{th}$ derivative of the digamma function on input. $n \geq 0$ is called the order of the polygamma function.

$$\psi^{(n)}(x) = \frac{d^{(n)}}{dx^{(n)}} \psi(x)$$

Note

This function is implemented only for nonnegative integers $n \geq 0$.

Parameters

• n (int) – the order of the polygamma function

• input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> a = torch.tensor([1, 0.5])
>>> torch.special.polygamma(1, a)
tensor([1.64493, 4.9348])
>>> torch.special.polygamma(2, a)
tensor([ -2.4041, -16.8288])
>>> torch.special.polygamma(3, a)
tensor([ 6.4939, 97.4091])
>>> torch.special.polygamma(4, a)
tensor([ -24.8863, -771.4742])

torch.special.psi(input, *, out=None) → Tensor

Alias for torch.special.digamma().

torch.special.round(input, *, out=None) → Tensor

Alias for torch.round().

torch.special.scaled_modified_bessel_k0(input, *, out=None) → Tensor

Scaled modified Bessel function of the second kind of order $0$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.scaled_modified_bessel_k1(input, *, out=None) → Tensor

Scaled modified Bessel function of the second kind of order $1$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.sinc(input, *, out=None) → Tensor

Computes the normalized sinc of input.

$$\text{out}_{i} = \begin{cases} 1, & \text{if}\ \text{input}_{i}=0 \\ \sin(\pi \text{input}_{i}) / (\pi \text{input}_{i}), & \text{otherwise} \end{cases}$$

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> t = torch.randn(4)
>>> t
tensor([ 0.2252, -0.2948,  1.0267, -1.1566])
>>> torch.special.sinc(t)
tensor([ 0.9186,  0.8631, -0.0259, -0.1300])

torch.special.softmax(input, dim, *, dtype=None) → Tensor

Computes the softmax function.

Softmax is defined as:

$\text{Softmax}(x_{i}) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}$

It is applied to all slices along dim, and will re-scale them so that the elements lie in the range [0, 1] and sum to 1.

Parameters

• input (Tensor) – input

• dim (int) – A dimension along which softmax will be computed.

• dtype (torch.dtype, optional) – the desired data type of returned tensor. If specified, the input tensor is cast to dtype before the operation is performed. This is useful for preventing data type overflows. Default: None.

Examples::

>>> t = torch.ones(2, 2)
>>> torch.special.softmax(t, 0)
tensor([[0.5000, 0.5000],
        [0.5000, 0.5000]])

torch.special.spherical_bessel_j0(input, *, out=None) → Tensor

Spherical Bessel function of the first kind of order $0$.

Parameters

input (Tensor) – the input tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

torch.special.xlog1py(input, other, *, out=None) → Tensor

Computes input * log1p(other) with the following cases.

$$\text{out}_{i} = \begin{cases} \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\ 0 & \text{if } \text{input}_{i} = 0.0 \text{ and } \text{other}_{i} != \text{NaN} \\ \text{input}_{i} * \text{log1p}(\text{other}_{i})& \text{otherwise} \end{cases}$$

Similar to SciPy’s scipy.special.xlog1py.

Parameters

• input (Number or Tensor) – Multiplier

• other (Number or Tensor) – Argument

Note

At least one of input or other must be a tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> x = torch.zeros(5,)
>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
>>> torch.special.xlog1py(x, y)
tensor([0., 0., 0., 0., nan])
>>> x = torch.tensor([1, 2, 3])
>>> y = torch.tensor([3, 2, 1])
>>> torch.special.xlog1py(x, y)
tensor([1.3863, 2.1972, 2.0794])
>>> torch.special.xlog1py(x, 4)
tensor([1.6094, 3.2189, 4.8283])
>>> torch.special.xlog1py(2, y)
tensor([2.7726, 2.1972, 1.3863])

torch.special.xlogy(input, other, *, out=None) → Tensor

Computes input * log(other) with the following cases.

$$\text{out}_{i} = \begin{cases} \text{NaN} & \text{if } \text{other}_{i} = \text{NaN} \\ 0 & \text{if } \text{input}_{i} = 0.0 \\ \text{input}_{i} * \log{(\text{other}_{i})} & \text{otherwise} \end{cases}$$

Similar to SciPy’s scipy.special.xlogy.

Parameters

• input (Number or Tensor) – Multiplier

• other (Number or Tensor) – Argument

Note

At least one of input or other must be a tensor.

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example:

>>> x = torch.zeros(5,)
>>> y = torch.tensor([-1, 0, 1, float('inf'), float('nan')])
>>> torch.special.xlogy(x, y)
tensor([0., 0., 0., 0., nan])
>>> x = torch.tensor([1, 2, 3])
>>> y = torch.tensor([3, 2, 1])
>>> torch.special.xlogy(x, y)
tensor([1.0986, 1.3863, 0.0000])
>>> torch.special.xlogy(x, 4)
tensor([1.3863, 2.7726, 4.1589])
>>> torch.special.xlogy(2, y)
tensor([2.1972, 1.3863, 0.0000])

torch.special.zeta(input, other, *, out=None) → Tensor

Computes the Hurwitz zeta function, elementwise.

$$\zeta(x, q) = \sum_{k=0}^{\infty} \frac{1}{(k + q)^x}$$

Parameters

• input (Tensor) – the input tensor corresponding to x.

• other (Tensor) – the input tensor corresponding to q.

Note

The Riemann zeta function corresponds to the case when q = 1

Keyword Arguments

out (Tensor, optional) – the output tensor.

Example::

>>> x = torch.tensor([2., 4.])
>>> torch.special.zeta(x, 1)
tensor([1.6449, 1.0823])
>>> torch.special.zeta(x, torch.tensor([1., 2.]))
tensor([1.6449, 0.0823])
>>> torch.special.zeta(2, torch.tensor([1., 2.]))
tensor([1.6449, 0.6449])