torch.special¶
The torch.special module, modeled after SciPy’s special module.
Functions¶
- torch.special.airy_ai(input, *, out=None) Tensor¶
Airy function \(\text{Ai}\left(\text{input}\right)\).
- torch.special.bessel_j0(input, *, out=None) Tensor¶
Bessel function of the first kind of order \(0\).
- torch.special.bessel_j1(input, *, out=None) Tensor¶
Bessel function of the first kind of order \(1\).
- 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()andtorch.special.gammaln()for related functions.Supports broadcasting to a common shape and float inputs.
Note
The backward pass with respect to
inputis not yet supported. Please open an issue on PyTorch’s Github to request it.- Parameters:
- 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()andtorch.special.gammaln()for related functions.Supports broadcasting to a common shape and float inputs.
Note
The backward pass with respect to
inputis not yet supported. Please open an issue on PyTorch’s Github to request it.- Parameters:
- 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 todtypebefore 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.inputis clamped to [eps, 1 - eps] when eps is not None. When eps is None andinput< 0 orinput> 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:
- 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:
- 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:
- 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 \(0\).
- torch.special.scaled_modified_bessel_k1(input, *, out=None) Tensor¶
Scaled modified Bessel function of the second kind of order \(1\).
- 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 todtypebefore 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 \(0\).
- 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.
Note
At least one of
inputorothermust 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.
Note
At least one of
inputorothermust 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:
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])
