"""Functional interface"""

import warnings
import math
from operator import mul
from functools import reduce

import torch
from torch._C import _infer_size, _add_docstr
from . import _functions
from .modules import utils
from ._functions.padding import ConstantPadNd
from ._functions import vision
from ._functions.thnn.fold import Col2Im, Im2Col
from torch.autograd import Variable
from .modules.utils import _single, _pair, _triple
from . import grad


conv1d = _add_docstr(torch.conv1d, r"""
conv1d(input, weight, bias=None, stride=1, padding=0, dilation=1, groups=1) -> Tensor

Applies a 1D convolution over an input signal composed of several input
planes.

See :class:`~torch.nn.Conv1d` for details and output shape.

Args:
    input: input tensor of shape :math:`minibatch \times in\_channels \times iW`
    weight: filters of shape :math:`out\_channels \times \frac{in\_channels}{groups} \times kW`
    bias: optional bias of shape (:math:`out\_channels`). Default: ``None``
    stride: the stride of the convolving kernel. Can be a single number or
      a one-element tuple `(sW,)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a one-element tuple `(padW,)`. Default: 0
    dilation: the spacing between kernel elements. Can be a single number or
      a one-element tuple `(dW,)`. Default: 1
    groups: split input into groups, :math:`in\_channels` should be divisible by
      the number of groups. Default: 1

Examples::

    >>> filters = torch.randn(33, 16, 3)
    >>> inputs = torch.randn(20, 16, 50)
    >>> F.conv1d(inputs, filters)
""")

conv2d = _add_docstr(torch.conv2d, r"""
conv2d(input, weight, bias=None, stride=1, padding=0, dilation=1, groups=1) -> Tensor

Applies a 2D convolution over an input image composed of several input
planes.

See :class:`~torch.nn.Conv2d` for details and output shape.

Args:
    input: input tensor of shape (:math:`minibatch \times in\_channels \times iH \times iW`)
    weight: filters of shape (:math:`out\_channels \times \frac{in\_channels}{groups} \times kH \times kW`)
    bias: optional bias tensor of shape (:math:`out\_channels`). Default: ``None``
    stride: the stride of the convolving kernel. Can be a single number or a
      tuple `(sH, sW)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padH, padW)`. Default: 0
    dilation: the spacing between kernel elements. Can be a single number or
      a tuple `(dH, dW)`. Default: 1
    groups: split input into groups, :math:`in\_channels` should be divisible by the
      number of groups. Default: 1

Examples::

    >>> # With square kernels and equal stride
    >>> filters = torch.randn(8,4,3,3)
    >>> inputs = torch.randn(1,4,5,5)
    >>> F.conv2d(inputs, filters, padding=1)
""")

conv3d = _add_docstr(torch.conv3d, r"""
conv3d(input, weight, bias=None, stride=1, padding=0, dilation=1, groups=1) -> Tensor

Applies a 3D convolution over an input image composed of several input
planes.

See :class:`~torch.nn.Conv3d` for details and output shape.

Args:
    input: input tensor of shape (:math:`minibatch \times in\_channels \times iT \times iH \times iW`)
    weight: filters of shape (:math:`out\_channels \times \frac{in\_channels}{groups} \times kT \times kH \times kW`)
    bias: optional bias tensor of shape (:math:`out\_channels`). Default: None
    stride: the stride of the convolving kernel. Can be a single number or a
      tuple `(sT, sH, sW)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padT, padH, padW)`. Default: 0
    dilation: the spacing between kernel elements. Can be a single number or
      a tuple `(dT, dH, dW)`. Default: 1
    groups: split input into groups, :math:`in\_channels` should be divisible by
      the number of groups. Default: 1

Examples::

    >>> filters = torch.randn(33, 16, 3, 3, 3)
    >>> inputs = torch.randn(20, 16, 50, 10, 20)
    >>> F.conv3d(inputs, filters)
""")

conv_transpose1d = _add_docstr(torch.conv_transpose1d, r"""
conv_transpose1d(input, weight, bias=None, stride=1, padding=0, output_padding=0, groups=1, dilation=1) -> Tensor

Applies a 1D transposed convolution operator over an input signal
composed of several input planes, sometimes also called "deconvolution".

See :class:`~torch.nn.ConvTranspose1d` for details and output shape.

Args:
    input: input tensor of shape (:math:`minibatch \times in\_channels \times iW`)
    weight: filters of shape (:math:`in\_channels \times \frac{out\_channels}{groups} \times kW`)
    bias: optional bias of shape (:math:`out\_channels`). Default: None
    stride: the stride of the convolving kernel. Can be a single number or a
      tuple `(sW,)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padW,)`. Default: 0
    output_padding: implicit zero-paddings of :math:`0 \leq padding < stride` on both
      sides of the output. Can be a single number or a tuple `(out_padW,)`.
      Default: 0
    groups: split input into groups, :math:`in\_channels` should be divisible by the
      number of groups. Default: 1
    dilation: the spacing between kernel elements. Can be a single number or
      a tuple `(dW,)`. Default: 1

Examples::

    >>> inputs = torch.randn(20, 16, 50)
    >>> weights = torch.randn(16, 33, 5)
    >>> F.conv_transpose1d(inputs, weights)
""")

conv_transpose2d = _add_docstr(torch.conv_transpose2d, r"""
conv_transpose2d(input, weight, bias=None, stride=1, padding=0, output_padding=0, groups=1, dilation=1) -> Tensor

Applies a 2D transposed convolution operator over an input image
composed of several input planes, sometimes also called "deconvolution".

See :class:`~torch.nn.ConvTranspose2d` for details and output shape.

Args:
    input: input tensor of shape (:math:`minibatch \times in\_channels \times iH \times iW`)
    weight: filters of shape (:math:`in\_channels \times \frac{out\_channels}{groups} \times kH \times kW`)
    bias: optional bias of shape (:math:`out\_channels`). Default: None
    stride: the stride of the convolving kernel. Can be a single number or a
      tuple `(sH, sW)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padH, padW)`. Default: 0
    output_padding: implicit zero-paddings of :math:`0 \leq padding < stride` on both
      sides of the output. Can be a single number or a tuple
      `(out_padH, out_padW)`. Default: 0
    groups: split input into groups, :math:`in\_channels` should be divisible by the
      number of groups. Default: 1
    dilation: the spacing between kernel elements. Can be a single number or
      a tuple `(dH, dW)`. Default: 1

Examples::

    >>> # With square kernels and equal stride
    >>> inputs = torch.randn(1, 4, 5, 5)
    >>> weights = torch.randn(4, 8, 3, 3)
    >>> F.conv_transpose2d(inputs, weights, padding=1)
""")

conv_transpose3d = _add_docstr(torch.conv_transpose3d, r"""
conv_transpose3d(input, weight, bias=None, stride=1, padding=0, output_padding=0, groups=1, dilation=1) -> Tensor

Applies a 3D transposed convolution operator over an input image
composed of several input planes, sometimes also called "deconvolution"

See :class:`~torch.nn.ConvTranspose3d` for details and output shape.

Args:
    input: input tensor of shape (:math:`minibatch \times in\_channels \times iT \times iH \times iW`)
    weight: filters of shape (:math:`in\_channels \times \frac{out\_channels}{groups} \times kT \times kH \times kW`)
    bias: optional bias of shape (:math:`out\_channels`). Default: None
    stride: the stride of the convolving kernel. Can be a single number or a
      tuple `(sT, sH, sW)`. Default: 1
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padT, padH, padW)`. Default: 0
    output_padding: implicit zero-paddings of `0 \leq padding < stride` on both
      sides of the output. Can be a single number or a tuple
      `(out_padT, out_padH, out_padW)`. Default: 0
    groups: split input into groups, :math:`in\_channels` should be divisible by the
      number of groups. Default: 1
    dilation: the spacing between kernel elements. Can be a single number or
      a tuple `(dT, dH, dW)`. Default: 1

Examples::

    >>> inputs = torch.randn(20, 16, 50, 10, 20)
    >>> weights = torch.randn(16, 33, 3, 3, 3)
    >>> F.conv_transpose3d(inputs, weights)
""")


def conv_tbc(input, weight, bias, pad=0):
    r"""Applies a 1-dimensional sequence convolution over an input sequence.
    Input and output dimensions are (Time, Batch, Channels) - hence TBC.

    Args:
        input: input tensor of shape (:math:`\text{sequence length} \times batch \times in\_channels`)
        weight: filter of shape (:math:`\text{kernel width} \times in\_channels \times out\_channels`)
        bias: bias of shape (:math:`out\_channels`)
        pad: number of timesteps to pad
    """
    return input.conv_tbc(weight, bias, pad)


# Pooling
def avg_pool1d(input, kernel_size, stride=None, padding=0,
               ceil_mode=False, count_include_pad=True):
    r"""Applies a 1D average pooling over an input signal composed of several
    input planes.

    See :class:`~torch.nn.AvgPool1d` for details and output shape.

    Args:
        input: input tensor of shape (:math:`minibatch \times in\_channels \times iW`)
        kernel_size: the size of the window. Can be a single number or a
          tuple `(kW,)`
        stride: the stride of the window. Can be a single number or a tuple
          `(sW,)`. Default: :attr:`kernel_size`
        padding: implicit zero paddings on both sides of the input. Can be a
          single number or a tuple `(padW,)`. Default: 0
        ceil_mode: when True, will use `ceil` instead of `floor` to compute the
            output shape. Default: ``False``
        count_include_pad: when True, will include the zero-padding in the
            averaging calculation. Default: ``True``

    Example::
        >>> # pool of square window of size=3, stride=2
        >>> input = torch.Tensor([[[1,2,3,4,5,6,7]]])
        >>> F.avg_pool1d(input, kernel_size=3, stride=2)
        (0 ,.,.) =
          2  4  6
        [torch.FloatTensor of size (1,1,3)]
    """
    if input.dim() != 3:
        raise ValueError('expected 3D input (got {} dimensions)'
                         .format(input.dim()))
    kernel_size = _single(kernel_size) + (1,)
    stride = _single(stride) + (1,) if stride is not None else kernel_size
    padding = _single(padding) + (0,)
    return avg_pool2d(input.unsqueeze(3), kernel_size, stride, padding,
                      ceil_mode, count_include_pad).squeeze(3)


avg_pool2d = _add_docstr(torch._C._nn.avg_pool2d, r"""
avg_pool2d(input, kernel_size, stride=None, padding=0, ceil_mode=False, count_include_pad=True) -> Tensor

Applies 2D average-pooling operation in :math:`kH \times kW` regions by step size
:math:`sH \times sW` steps. The number of output features is equal to the number of
input planes.

See :class:`~torch.nn.AvgPool2d` for details and output shape.

Args:
    input: input tensor (:math:`minibatch \times in\_channels \times iH \times iW`)
    kernel_size: size of the pooling region. Can be a single number or a
      tuple (:math:`kH \times kW`)
    stride: stride of the pooling operation. Can be a single number or a
      tuple `(sH, sW)`. Default: :attr:`kernel_size`
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padH, padW)`. Default: 0
    ceil_mode: when True, will use `ceil` instead of `floor` in the formula
        to compute the output shape. Default: ``False``
    count_include_pad: when True, will include the zero-padding in the
        averaging calculation. Default: ``True``
""")

avg_pool3d = _add_docstr(torch._C._nn.avg_pool3d, r"""
avg_pool3d(input, kernel_size, stride=None, padding=0, ceil_mode=False, count_include_pad=True) -> Tensor

Applies 3D average-pooling operation in :math:`kT \times kH \times kW` regions by step
size :math:`sT \times sH \times sW` steps. The number of output features is equal to
:math:`\lfloor\frac{\text{input planes}}{sT}\rfloor`.

See :class:`~torch.nn.AvgPool3d` for details and output shape.

Args:
    input: input tensor (:math:`minibatch \times in\_channels \times iT \times iH \times iW`)
    kernel_size: size of the pooling region. Can be a single number or a
      tuple (:math:`kT \times kH \times kW`)
    stride: stride of the pooling operation. Can be a single number or a
      tuple `(sT, sH, sW)`. Default: :attr:`kernel_size`
    padding: implicit zero paddings on both sides of the input. Can be a
      single number or a tuple `(padT, padH, padW)`, Default: 0
    ceil_mode: when True, will use `ceil` instead of `floor` in the formula
        to compute the output shape
    count_include_pad: when True, will include the zero-padding in the
        averaging calculation
""")


def fractional_max_pool2d(input, kernel_size, output_size=None,
                          output_ratio=None, return_indices=False,
                          _random_samples=None):
    r"""Applies 2D fractional max pooling over an input signal composed of several input planes.

    Fractional MaxPooling is described in detail in the paper `Fractional MaxPooling`_ by Ben Graham

    The max-pooling operation is applied in :math:`kH \times kW` regions by a stochastic
    step size determined by the target output size.
    The number of output features is equal to the number of input planes.

    Args:
        kernel_size: the size of the window to take a max over.
                     Can be a single number :math:`k` (for a square kernel of :math:`k \times k`)
                     or a tuple (:math:`kH \times kW`)
        output_size: the target output size of the image of the form :math:`oH \times oW`.
                     Can be a tuple `(oH, oW)` or a single number :math:`oH` for a square image :math:`oH \times oH`
        output_ratio: If one wants to have an output size as a ratio of the input size, this option can be given.
                      This has to be a number or tuple in the range (0, 1)
        return_indices: if ``True``, will return the indices along with the outputs.
                        Useful to pass to `max_unpool2d`.

    Examples::
        >>> input = torch.randn(20, 16, 50, 32)
        >>> # pool of square window of size=3, and target output size 13x12
        >>> F.fractional_max_pool2d(input, 3, output_size=(13, 12))
        >>> # pool of square window and target output size being half of input image size
        >>> F.fractional_max_pool2d(input, 3, output_ratio=(0.5, 0.5))

    .. _Fractional MaxPooling:
        http://arxiv.org/abs/1412.6071
    """
    if output_size is None and output_ratio is None:
        raise ValueError("fractional_max_pool2d requires specifying either "
                         "an output_size, or a output_ratio")
    if output_size is None:
        output_ratio = _pair(output_ratio)
        output_size = (int(input.size(2) * output_ratio[0]),
                       int(input.size(3) * output_ratio[1]))

    if _random_samples is None:
        _random_samples = input.new(input.size(0), input.size(1), 2).uniform_()
    ret = torch._C._nn.fractional_max_pool2d(input, kernel_size, output_size, _random_samples)
    return ret if return_indices else ret[0]


def max_pool1d(input, kernel_size, stride=None, padding=0, dilation=1,
               ceil_mode=False, return_indices=False):
    r"""Applies a 1D max pooling over an input signal composed of several input
    planes.

    See :class:`~torch.nn.MaxPool1d` for details.
    """
    ret = torch.max_pool1d(input, kernel_size, stride, padding, dilation, ceil_mode)
    return ret if return_indices else ret[0]


def max_pool2d(input, kernel_size, stride=None, padding=0, dilation=1,
               ceil_mode=False, return_indices=False):
    r"""Applies a 2D max pooling over an input signal composed of several input
    planes.

    See :class:`~torch.nn.MaxPool2d` for details.
    """
    ret = torch._C._nn.max_pool2d(input, kernel_size, stride, padding, dilation, ceil_mode)
    return ret if return_indices else ret[0]


def max_pool3d(input, kernel_size, stride=None, padding=0, dilation=1,
               ceil_mode=False, return_indices=False):
    r"""Applies a 3D max pooling over an input signal composed of several input
    planes.

    See :class:`~torch.nn.MaxPool3d` for details.
    """
    ret = torch._C._nn.max_pool3d(input, kernel_size, stride, padding, dilation, ceil_mode)
    return ret if return_indices else ret[0]


def _unpool_output_size(input, kernel_size, stride, padding, output_size):
    input_size = input.size()
    default_size = []
    for d in range(len(kernel_size)):
        default_size.append((input_size[d + 2] - 1) * stride[d] +
                            kernel_size[d] - 2 * padding[d])
    if output_size is None:
        return default_size

    output_size = list(output_size)
    if len(output_size) == len(kernel_size) + 2:
        output_size = output_size[2:]
    if len(output_size) != len(kernel_size):
        raise ValueError("output_size should be a sequence containing "
                         "{} or {} elements, but it has a length of '{}'"
                         .format(len(kernel_size), len(kernel_size) + 2,
                                 len(output_size)))
    for d in range(len(kernel_size)):
        min_size = default_size[d] - stride[d]
        max_size = default_size[d] + stride[d]
        if not (min_size < output_size[d] < max_size):
            raise ValueError(
                'invalid output_size "{}" (dim {} must be between {} and {})'
                .format(output_size, d, min_size, max_size))

    return output_size


def max_unpool1d(input, indices, kernel_size, stride=None, padding=0,
                 output_size=None):
    r"""Computes a partial inverse of :class:`MaxPool1d`.

    See :class:`~torch.nn.MaxUnpool1d` for details.
    """
    kernel_size = _single(kernel_size)
    stride = _single(stride)
    padding = _single(padding)
    output_size = _unpool_output_size(input, kernel_size, stride, padding,
                                      output_size)
    return torch._C._nn.max_unpool2d(input.unsqueeze(3), indices.unsqueeze(3), output_size + [1]).squeeze(3)


def max_unpool2d(input, indices, kernel_size, stride=None, padding=0,
                 output_size=None):
    r"""Computes a partial inverse of :class:`MaxPool2d`.

    See :class:`~torch.nn.MaxUnpool2d` for details.
    """
    kernel_size = _pair(kernel_size)
    stride = _pair(stride)
    padding = _pair(padding)
    output_size = _unpool_output_size(input, kernel_size, stride, padding,
                                      output_size)
    return torch._C._nn.max_unpool2d(input, indices, output_size)


def max_unpool3d(input, indices, kernel_size, stride=None, padding=0,
                 output_size=None):
    r"""Computes a partial inverse of :class:`MaxPool3d`.

    See :class:`~torch.nn.MaxUnpool3d` for details.
    """
    kernel_size = _triple(kernel_size)
    stride = _triple(stride)
    padding = _triple(padding)
    output_size = _unpool_output_size(input, kernel_size, stride, padding,
                                      output_size)
    return torch._C._nn.max_unpool3d(input, indices, output_size, stride, padding)


def lp_pool2d(input, norm_type, kernel_size, stride=None, ceil_mode=False):
    r"""Applies a 2D power-average pooling over an input signal composed of
    several input planes.

    See :class:`~torch.nn.LPPool2d` for details.
    """
    kw, kh = utils._pair(kernel_size)
    out = avg_pool2d(input.pow(norm_type), kernel_size, stride, 0, ceil_mode)
    return out.mul(kw * kh).pow(1. / norm_type)


def lp_pool1d(input, norm_type, kernel_size, stride=None, ceil_mode=False):
    r"""Applies a 1D power-average pooling over an input signal composed of
    several input planes.

    See :class:`~torch.nn.LPPool1d` for details.
    """
    out = avg_pool1d(input.pow(norm_type), kernel_size, stride, 0, ceil_mode)
    return out.mul(kernel_size).pow(1. / norm_type)


def adaptive_max_pool1d(input, output_size, return_indices=False):
    r"""Applies a 1D adaptive max pooling over an input signal composed of
    several input planes.

    See :class:`~torch.nn.AdaptiveMaxPool1d` for details and output shape.

    Args:
        output_size: the target output size (single integer)
        return_indices: whether to return pooling indices. Default: ``False``
    """
    ret = torch.adaptive_max_pool1d(input, output_size)
    return ret if return_indices else ret[0]


def adaptive_max_pool2d(input, output_size, return_indices=False):
    r"""Applies a 2D adaptive max pooling over an input signal composed of
    several input planes.

    See :class:`~torch.nn.AdaptiveMaxPool2d` for details and output shape.

    Args:
        output_size: the target output size (single integer or
            double-integer tuple)
        return_indices: whether to return pooling indices. Default: ``False``
    """
    ret = torch._C._nn.adaptive_max_pool2d(input, output_size)
    return ret if return_indices else ret[0]


def adaptive_max_pool3d(input, output_size, return_indices=False):
    r"""Applies a 3D adaptive max pooling over an input signal composed of
    several input planes.

    See :class:`~torch.nn.AdaptiveMaxPool3d` for details and output shape.

    Args:
        output_size: the target output size (single integer or
            triple-integer tuple)
        return_indices: whether to return pooling indices. Default: ``False``
    """
    ret = torch._C._nn.adaptive_max_pool3d(input, output_size)
    return ret if return_indices else ret[0]


adaptive_avg_pool1d = _add_docstr(torch.adaptive_avg_pool1d, r"""
adaptive_avg_pool1d(input, output_size) -> Tensor

Applies a 1D adaptive average pooling over an input signal composed of
several input planes.

See :class:`~torch.nn.AdaptiveAvgPool1d` for details and output shape.

Args:
    output_size: the target output size (single integer)
""")

adaptive_avg_pool2d = _add_docstr(torch._C._nn.adaptive_avg_pool2d, r"""
adaptive_avg_pool2d(input, output_size) -> Tensor

Applies a 2D adaptive average pooling over an input signal composed of
several input planes.

See :class:`~torch.nn.AdaptiveAvgPool2d` for details and output shape.

Args:
    output_size: the target output size (single integer or
        double-integer tuple)
""")

adaptive_avg_pool3d = _add_docstr(torch._C._nn.adaptive_avg_pool3d, r"""
adaptive_avg_pool3d(input, output_size) -> Tensor

Applies a 3D adaptive average pooling over an input signal composed of
several input planes.

See :class:`~torch.nn.AdaptiveAvgPool3d` for details and output shape.

Args:
    output_size: the target output size (single integer or
        triple-integer tuple)
""")


# Activation functions

def dropout(input, p=0.5, training=False, inplace=False):
    return _functions.dropout.Dropout.apply(input, p, training, inplace)


def alpha_dropout(input, p=0.5, training=False):
    r"""Applies alpha dropout to the input.

    See :class:`~torch.nn.AlphaDropout` for details.

    Args:
        p (float, optional): the drop probability. Default: 0.5
        training (bool, optional): switch between training and evaluation mode. Default: ``False``
    """
    if p < 0 or p > 1:
        raise ValueError("dropout probability has to be between 0 and 1, "
                         "but got {}".format(p))

    if p == 0 or not training:
        return input

    alpha = -1.7580993408473766
    keep_prob = 1 - p
    # TODO avoid casting to byte after resize
    noise = input.data.new().resize_(input.size())
    noise.bernoulli_(p)
    noise = noise.byte()

    output = input.masked_fill(noise, alpha)

    a = (keep_prob + alpha ** 2 * keep_prob * (1 - keep_prob)) ** (-0.5)
    b = -a * alpha * (1 - keep_prob)

    return output.mul_(a).add_(b)


def dropout2d(input, p=0.5, training=False, inplace=False):
    return _functions.dropout.FeatureDropout.apply(input, p, training, inplace)


def dropout3d(input, p=0.5, training=False, inplace=False):
    return _functions.dropout.FeatureDropout.apply(input, p, training, inplace)


def threshold(input, threshold, value, inplace=False):
    r"""Thresholds each element of the input Tensor.

    See :class:`~torch.nn.Threshold` for more details.
    """
    if inplace:
        return torch._C._nn.threshold_(input, threshold, value)
    return torch._C._nn.threshold(input, threshold, value)


threshold_ = _add_docstr(torch._C._nn.threshold_, r"""
threshold_(input, threshold, value) -> Tensor

In-place version of :func:`~threshold`.
""")


def relu(input, inplace=False):
    r"""relu(input, threshold, value, inplace=False) -> Tensor

    Applies the rectified linear unit function element-wise. See
    :class:`~torch.nn.ReLU` for more details.
    """
    if inplace:
        return torch.relu_(input)
    return torch.relu(input)


relu_ = _add_docstr(torch.relu_, r"""
relu_(input) -> Tensor

In-place verison of :func:`~relu`.
""")


def glu(input, dim=-1):
    r"""
    glu(input, dim=-1) -> Tensor

    The gated linear unit. Computes:

    .. math ::

        H = A \times \sigma(B)

    where `input` is split in half along `dim` to form `A` and `B`.

    See `Language Modeling with Gated Convolutional Networks <https://arxiv.org/abs/1612.08083>`_.

    Args:
        input (Variable): input variable
        dim (int): dimension on which to split the input
    """
    if input.dim() == 0:
        raise RuntimeError("glu does not suppport scalars because halving size must be even")
    return torch._C._nn.glu(input, dim)


def hardtanh(input, min_val=-1., max_val=1., inplace=False):
    r"""
    hardtanh(input, min_val=-1., max_val=1., inplace=False) -> Tensor

    Applies the HardTanh function element-wise. See :class:`~torch.nn.Hardtanh` for more
    details.
    """
    if inplace:
        return torch._C._nn.hardtanh_(input, min_val, max_val)
    return torch._C._nn.hardtanh(input, min_val, max_val)


hardtanh_ = _add_docstr(torch._C._nn.hardtanh_, r"""
hardtanh_(input, min_val=-1., max_val=1.) -> Tensor

In-place version of :func:`~hardtanh`.
""")


def relu6(input, inplace=False):
    r"""relu6(input, inplace=False) -> Tensor

    Applies the element-wise function :math:`\text{ReLU6}(x) = \min(\max(0,x), 6)`.

    See :class:`~torch.nn.ReLU6` for more details.
    """
    return hardtanh(input, 0, 6, inplace)


def elu(input, alpha=1., inplace=False):
    r"""Applies element-wise,
    :math:`\text{ELU}(x) = \max(0,x) + \min(0, \alpha * (\exp(x) - 1))`.

    See :class:`~torch.nn.ELU` for more details.
    """
    if inplace:
        return torch._C._nn.elu_(input, alpha)
    return torch._C._nn.elu(input, alpha)


elu_ = _add_docstr(torch._C._nn.elu_, r"""
elu_(input, alpha=1.) -> Tensor

In-place verison of :func:`~elu`.
""")


def selu(input, inplace=False):
    r"""selu(input, inplace=False) -> Tensor

    Applies element-wise,
    :math:`\text{SELU}(x) = scale * (\max(0,x) + \min(0, \alpha * (\exp(x) - 1)))`,
    with :math:`\alpha=1.6732632423543772848170429916717` and
    :math:`scale=1.0507009873554804934193349852946`.

    See :class:`~torch.nn.SELU` for more details.
    """
    if inplace:
        return torch.selu_(input)
    return torch.selu(input)

selu_ = _add_docstr(torch.selu_, r"""
selu_(input) -> Tensor

In-place verison of :func:`~selu`.
""")


def leaky_relu(input, negative_slope=0.01, inplace=False):
    r"""
    leaky_relu(input, negative_slope=0.01, inplace=False) -> Tensor

    Applies element-wise,
    :math:`\text{LeakyReLU}(x) = \max(0, x) + \text{negative_slope} * \min(0, x)`

    See :class:`~torch.nn.LeakyReLU` for more details.
    """
    if inplace:
        return torch._C._nn.leaky_relu_(input, negative_slope)
    return torch._C._nn.leaky_relu(input, negative_slope)


leaky_relu_ = _add_docstr(torch._C._nn.leaky_relu_, r"""
leaky_relu_(input, negative_slope=0.01) -> Tensor

In-place version of :func:`~leaky_relu`.
""")


prelu = _add_docstr(torch._C._nn.prelu, r"""
prelu(input, weight) -> Tensor

Applies element-wise the function
:math:`\text{PReLU}(x) = \max(0,x) + \text{weight} * \min(0,x)` where weight is a
learnable parameter.

See :class:`~torch.nn.PReLU` for more details.
""")


def rrelu(input, lower=1. / 8, upper=1. / 3, training=False, inplace=False):
    r"""rrelu(input, lower=1./8, upper=1./3, training=False, inplace=False) -> Tensor

    Randomized leaky ReLU.

    See :class:`~torch.nn.RReLU` for more details.
    """
    if inplace:
        return torch.rrelu_(input, lower, upper, training)
    return torch.rrelu(input, lower, upper, training)


rrelu_ = _add_docstr(torch.rrelu_, r"""
rrelu_(input, lower=1./8, upper=1./3, training=False) -> Tensor

In-place version of :func:`~rrelu`.
""")

logsigmoid = _add_docstr(torch._C._nn.log_sigmoid, r"""
logsigmoid(input) -> Tensor

Applies element-wise :math:`\text{LogSigmoid}(x) = \log \left(\frac{1}{1 + \exp(-x_i)}\right)`

See :class:`~torch.nn.LogSigmoid` for more details.
""")

hardshrink = _add_docstr(torch._C._nn.hardshrink, r"""
hardshrink(input, lambd=0.5) -> Tensor

Applies the hard shrinkage function element-wise

See :class:`~torch.nn.Hardshrink` for more details.
""")


def tanhshrink(input):
    r"""tanhshrink(input) -> Tensor

    Applies element-wise, :math:`\text{Tanhshrink}(x) = x - \text{Tanh}(x)`

    See :class:`~torch.nn.Tanhshrink` for more details.
    """
    return input - input.tanh()


def softsign(input):
    r"""softsign(input) -> Tensor

    Applies element-wise, the function :math:`\text{SoftSign}(x) = \frac{x}{1 + |x|}`

    See :class:`~torch.nn.Softsign` for more details.
    """
    return input / (input.abs() + 1)


softplus = _add_docstr(torch._C._nn.softplus, r"""
softplus(input, beta=1, threshold=20) -> Tensor
""")


def _get_softmax_dim(name, ndim, stacklevel):
    warnings.warn("Implicit dimension choice for " + name + " has been deprecated. "
                  "Change the call to include dim=X as an argument.", stacklevel=stacklevel)
    if ndim == 0 or ndim == 1 or ndim == 3:
        return 0
    else:
        return 1


def softmin(input, dim=None, _stacklevel=3):
    r"""Applies a softmin function.

    Note that :math:`\text{Softmin}(x) = \text{Softmax}(-x)`. See softmax definition for mathematical formula.

    See :class:`~torch.nn.Softmin` for more details.

    Arguments:
        input (Variable): input
        dim (int): A dimension along which softmin will be computed (so every slice
            along dim will sum to 1).
    """
    if dim is None:
        dim = _get_softmax_dim('softmin', input.dim(), _stacklevel)
    return torch._C._nn.softmax(-input, dim)


def softmax(input, dim=None, _stacklevel=3):
    r"""Applies a softmax function.

    Softmax is defined as:

    :math:`\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.

    See :class:`~torch.nn.Softmax` for more details.

    Arguments:
        input (Variable): input
        dim (int): A dimension along which softmax will be computed.

    .. note::
        This function doesn't work directly with NLLLoss,
        which expects the Log to be computed between the Softmax and itself.
        Use log_softmax instead (it's faster and has better numerical properties).

    """
    if dim is None:
        dim = _get_softmax_dim('softmax', input.dim(), _stacklevel)
    return torch._C._nn.softmax(input, dim)


def _sample_gumbel(shape, eps=1e-10, out=None):
    """
    Sample from Gumbel(0, 1)

    based on
    https://github.com/ericjang/gumbel-softmax/blob/3c8584924603869e90ca74ac20a6a03d99a91ef9/Categorical%20VAE.ipynb ,
    (MIT license)
    """
    U = out.resize_(shape).uniform_() if out is not None else torch.rand(shape)
    return - torch.log(eps - torch.log(U + eps))


def _gumbel_softmax_sample(logits, tau=1, eps=1e-10):
    """
    Draw a sample from the Gumbel-Softmax distribution

    based on
    https://github.com/ericjang/gumbel-softmax/blob/3c8584924603869e90ca74ac20a6a03d99a91ef9/Categorical%20VAE.ipynb
    (MIT license)
    """
    dims = logits.dim()
    gumbel_noise = _sample_gumbel(logits.size(), eps=eps, out=logits.data.new())
    y = logits + gumbel_noise
    return softmax(y / tau, dims - 1)


def gumbel_softmax(logits, tau=1, hard=False, eps=1e-10):
    """
    Sample from the Gumbel-Softmax distribution and optionally discretize.
    Args:
      logits: `[batch_size, n_class]` unnormalized log-probs
      tau: non-negative scalar temperature
      hard: if ``True``, take `argmax`, but differentiate w.r.t. soft sample y
    Returns:
      [batch_size, n_class] sample from the Gumbel-Softmax distribution.
      If hard=True, then the returned sample will be one-hot, otherwise it will
      be a probability distribution that sums to 1 across classes

    Constraints:
    - this implementation only works on batch_size x num_features tensor for now

    based on
    https://github.com/ericjang/gumbel-softmax/blob/3c8584924603869e90ca74ac20a6a03d99a91ef9/Categorical%20VAE.ipynb ,
    (MIT license)
    """
    shape = logits.size()
    assert len(shape) == 2
    y_soft = _gumbel_softmax_sample(logits, tau=tau, eps=eps)
    if hard:
        _, k = y_soft.data.max(-1)
        # this bit is based on
        # https://discuss.pytorch.org/t/stop-gradients-for-st-gumbel-softmax/530/5
        y_hard = logits.data.new(*shape).zero_().scatter_(-1, k.view(-1, 1), 1.0)
        # this cool bit of code achieves two things:
        # - makes the output value exactly one-hot (since we add then
        #   subtract y_soft value)
        # - makes the gradient equal to y_soft gradient (since we strip
        #   all other gradients)
        y = Variable(y_hard - y_soft.data) + y_soft
    else:
        y = y_soft
    return y


def log_softmax(input, dim=None, _stacklevel=3):
    r"""Applies a softmax followed by a logarithm.

    While mathematically equivalent to log(softmax(x)), doing these two
    operations separately is slower, and numerically unstable. This function
    uses an alternative formulation to compute the output and gradient correctly.

    See :class:`~torch.nn.LogSoftmax` for more details.

    Arguments:
        input (Variable): input
        dim (int): A dimension along which log_softmax will be computed.
    """
    if dim is None:
        dim = _get_softmax_dim('log_softmax', input.dim(), _stacklevel)
    return torch._C._nn.log_softmax(input, dim)


softshrink = _add_docstr(torch._C._nn.softshrink, r"""
softshrink(input, lambd=0.5) -> Tensor

Applies the soft shrinkage function elementwise

See :class:`~torch.nn.Softshrink` for more details.
""")


def tanh(input):
    r"""tanh(input) -> Tensor

    Applies element-wise,
    :math:`\text{Tanh}(x) = \tanh(x) = \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)}`

    See :class:`~torch.nn.Tanh` for more details.
    """
    return input.tanh()


def sigmoid(input):
    r"""sigmoid(input) -> Tensor

    Applies the element-wise function :math:`\text{Sigmoid}(x) = \frac{1}{1 + \exp(-x)}`

    See :class:`~torch.nn.Sigmoid` for more details.
    """
    return input.sigmoid()


# etc.

def linear(input, weight, bias=None):
    """
    Applies a linear transformation to the incoming data: :math:`y = xA^T + b`.

    Shape:
        - Input: :math:`(N, *, in\_features)` where `*` means any number of
          additional dimensions
        - Weight: :math:`(out\_features, in\_features)`
        - Bias: :math:`(out\_features)`
        - Output: :math:`(N, *, out\_features)`
    """
    if input.dim() == 2 and bias is not None:
        # fused op is marginally faster
        return torch.addmm(bias, input, weight.t())

    output = input.matmul(weight.t())
    if bias is not None:
        output += bias
    return output


def bilinear(input1, input2, weight, bias=None):
    return torch._C._VariableFunctions.bilinear(input1, input2, weight, bias)


def embedding(input, weight, padding_idx=None, max_norm=None, norm_type=2,
              scale_grad_by_freq=False, sparse=False):
    r"""A simple lookup table that looks up embeddings in a fixed dictionary and size.

    This module is often used to retrieve word embeddings using indices.
    The input to the module is a list of indices, and the embedding matrix,
    and the output is the corresponding word embeddings.

    Args:
        input: tensor, containing indices into the embedding matrix
        weight:
            Number of rows should correspond to the maximum possible index + 1,
            number of columns is the embedding size
        padding_idx (int, optional): Entries at the given index do not contribute to the gradient
        max_norm (float, optional): If given, will renormalize the embeddings to always have a norm lesser than this
        norm_type (float, optional): The p of the p-norm to compute for the max_norm option
        scale_grad_by_freq (boolean, optional): if given, this will scale gradients by the frequency of
                                                the words in the mini-batch.
        sparse (boolean, optional): if ``True``, gradient w.r.t. weight matrix will be a sparse tensor. See Notes for
                                    more details regarding sparse gradients.

    Shape:
        - Input: LongTensor `(N, W)`, N = mini-batch, W = number of indices to extract per mini-batch
        - Embedding_matrix: FloatTensor `(V, embedding_dim)`, V = maximum index + 1, embedding_dim = embedding size
        - Output: `(N, W, embedding_dim)`

    Notes:
        It is advised to only use `sparse=True` if `embedding_matrix` is a leaf Variable,
        since some autograd functions may not propagate sparse gradients correctly.
        Additionally, keep in mind that only a limited number of optimizers support
        sparse gradients: currently it's :class:`optim.SGD` (`CUDA` and `CPU`), and :class:`optim.Adagrad` (`CPU`)

    Examples::

        >>> # a batch of 2 samples of 4 indices each
        >>> input = torch.LongTensor([[1,2,4,5],[4,3,2,9]])
        >>> # an embedding matrix containing 10 tensors of size 3
        >>> embedding_matrix = torch.rand(10, 3)
        >>> F.embedding(input, embedding_matrix)

        (0 ,.,.) =
         -1.0822  1.2522  0.2434
          0.8393 -0.6062 -0.3348
          0.6597  0.0350  0.0837
          0.5521  0.9447  0.0498

        (1 ,.,.) =
          0.6597  0.0350  0.0837
         -0.1527  0.0877  0.4260
          0.8393 -0.6062 -0.3348
         -0.8738 -0.9054  0.4281
        [torch.FloatTensor of size (2,4,3)]

        >>> # example with padding_idx
        >>> weights = torch.rand(10, 3)
        >>> weights[0, :].zero_()
        >>> embedding_matrix = weights
        >>> input = torch.LongTensor([[0,2,0,5]])
        >>> F.embedding(input, embedding_matrix, padding_idx=0)

        (0 ,.,.) =
          0.0000  0.0000  0.0000
          0.3452  0.4937 -0.9361
          0.0000  0.0000  0.0000
          0.0706 -2.1962 -0.6276
        [torch.FloatTensor of size (1,4,3)]

    """
    input = input.contiguous()
    if padding_idx is not None:
        if padding_idx > 0:
            assert padding_idx < weight.size(0), 'Padding_idx must be within num_embeddings'
        elif padding_idx < 0:
            assert padding_idx >= -weight.size(0), 'Padding_idx must be within num_embeddings'
            padding_idx = weight.size(0) + padding_idx
    elif padding_idx is None:
            padding_idx = -1
    if max_norm is not None:
        with torch.no_grad():
            torch.embedding_renorm_(weight, input, max_norm, norm_type)
    return torch.embedding(weight, input, padding_idx, scale_grad_by_freq, sparse)


def embedding_bag(embedding_matrix, indices, offsets=None,
                  max_norm=None, norm_type=2, scale_grad_by_freq=False, mode='mean', sparse=False):
    r"""Computes sums or means of 'bags' of embeddings, without instantiating the
        intermediate embeddings.

        For bags of constant length,
            * embedding_bag with `mode=sum` is equivalent to nn.functional.embedding followed by `torch.sum(dim=1)`
            * with `mode=mean` is equivalent to nn.functional.embedding followed by `torch.mean(dim=1)`

        However, embedding_bag is much more time and memory efficient than using a chain of these
        operations.

        Args:
            embedding_matrix: FloatTensor, where number of rows should correspond to the maximum possible index + 1,
                              number of columns is the embedding size
            indices (N or BxN): LongTensor containing the indices of the embeddings to extract.
                                When `input` is 1D Tensor of shape `N`, an `offsets` Tensor is given, that contains the
                                starting position of each new sequence in the mini-batch.
            offsets (B or None): LongTensor containing the starting positions of each sample in a mini-batch of variable
                                 length sequences. If `input` is 2D (BxN), then offsets does not need to be given,
                                 as the `input` is treated as a mini-batch of fixed length sequences of length `N` each.
            max_norm (float, optional): If given, will renormalize the embeddings to always have a norm lesser than this
            norm_type (float, optional): The p of the p-norm to compute for the max_norm option
            scale_grad_by_freq (boolean, optional): if given, this will scale gradients by the frequency of
                                                    the words in the dictionary.
            mode (string, optional): 'sum' | 'mean'. Specifies the way to reduce the bag. Default: 'mean'
            sparse (boolean, optional): if ``True``, gradient w.r.t. weight matrix will be a sparse tensor. See Notes
                                        for more details regarding sparse gradients.

        Shape:
            - Embedding_matrix: FloatTensor `(V, embedding_dim)`,
                                V = number of embeddings, embedding_dim = embedding size
            - Input: LongTensor `N`, N = number of embeddings to extract
                     (or) LongTensor `BxN`, B = number of sequences in mini-batch,
                                            N = number of embeddings per sequence
            - Offsets: LongTensor `B`, B = number of bags. The values are the
                       offsets in `input` for each bag, i.e. the cumsum of lengths.
                       Offsets is not given if Input is 2D `BxN` Tensor,
                       the input is considered to be of fixed-length sequences
            - Output: `(B, embedding_dim)`

        Examples::

            >>> # an Embedding module containing 10 tensors of size 3
            >>> embedding_matrix = torch.rand(10, 3)
            >>> # a batch of 2 samples of 4 indices each
            >>> input = torch.LongTensor([1,2,4,5,4,3,2,9])
            >>> offsets = torch.LongTensor([0,4])
            >>> embedding_bag(embedding_matrix, input, offsets)

            -1.1840 -0.2547 -0.5860
            -0.7126  0.0002 -0.3411
            [torch.FloatTensor of size (2,3)]

        """
    if indices.dim() == 2:
        if offsets is not None:
            raise ValueError("if input is 2D, then offsets has to be None"
                             ", as input is treated is a mini-batch of"
                             " fixed length sequences. However, found "
                             "offsets of type {}".format(type(offsets)))
        else:
            offsets = Variable(torch.arange(0, indices.numel(), indices.size(1),
                                            out=indices.data.new().long()))
            indices = indices.view(-1)
    elif indices.dim() == 1:
        if offsets is None:
            raise ValueError("offsets has to be a 1D Tensor but got None")
        if offsets.dim() != 1:
            raise ValueError("offsets has to be a 1D Tensor")
        if offsets[0] != 0:
            raise ValueError("offsets[0] has to be 0, i.e. the first sequence"
                             " in the mini-batch has to start from position 0."
                             "However, got {}".format(offsets[0]))
        if offsets[-1] > indices.size(0):
            raise ValueError("offsets[-1] has to be smaller than indices's length"
                             " ({}), but got offsets[-1] of {}"
                             .format(indices.size(0), offsets[-1]))
    else:
        raise ValueError("input has to be 1D or 2D Tensor,"
                         " but got Tensor of dimension {}".format(indices.dim()))

    if mode == 'sum':
        mode = 0
    elif mode == 'mean':
        mode = 1
    else:
        raise ValueError("mode has to be one of sum or mean")

    if max_norm is not None:
        with torch.no_grad():
            torch.embedding_renorm_(weight, input, max_norm, norm_type)

    ret, _, _ = torch.embedding_bag(
        embedding_matrix,
        indices,
        offsets,
        scale_grad_by_freq,
        mode,
        sparse)
    return ret


def batch_norm(input, running_mean, running_var, weight=None, bias=None,
               training=False, momentum=0.1, eps=1e-5):
    r"""Applies Batch Normalization for each channel across a batch of data.

    See :class:`~torch.nn.BatchNorm1d`, :class:`~torch.nn.BatchNorm2d`,
    :class:`~torch.nn.BatchNorm3d` for details.
    """
    if training:
        size = list(input.size())
        if reduce(mul, size[2:], size[0]) == 1:
            raise ValueError('Expected more than 1 value per channel when training, got input size {}'.format(size))
    return torch.batch_norm(
        input, weight, bias, running_mean, running_var,
        training, momentum, eps, torch.backends.cudnn.enabled
    )


def instance_norm(input, running_mean=None, running_var=None, weight=None,
                  bias=None, use_input_stats=True, momentum=0.1, eps=1e-5):
    r"""Applies Instance Normalization for each channel in each data sample in a
    batch.

    See :class:`~torch.nn.InstanceNorm1d`, :class:`~torch.nn.InstanceNorm2d`,
    :class:`~torch.nn.InstanceNorm3d` for details.
    """
    if not use_input_stats and (running_mean is None or running_var is None):
        raise ValueError('Expected running_mean and running_var to be not None when use_input_stats=False')

    b, c = input.size(0), input.size(1)
    if weight is not None:
        weight = weight.repeat(b)
    if bias is not None:
        bias = bias.repeat(b)

    import torch.onnx.symbolic

    @torch.onnx.symbolic_override_first_arg_based(torch.onnx.symbolic.instance_norm)
    def _instance_norm(input, running_mean=None, running_var=None, weight=None,
                       bias=None, use_input_stats=None, momentum=None, eps=None):
        # Repeat stored stats and affine transform params if necessary
        if running_mean is not None:
            running_mean_orig = running_mean
            running_mean = running_mean_orig.repeat(b)
        if running_var is not None:
            running_var_orig = running_var
            running_var = running_var_orig.repeat(b)

        # Apply instance norm
        input_reshaped = input.contiguous().view(1, b * c, *input.size()[2:])

        out = batch_norm(
            input_reshaped, running_mean, running_var, weight=weight, bias=bias,
            training=use_input_stats, momentum=momentum, eps=eps)

        # Reshape and copy back
        if running_mean is not None:
            running_mean_orig.copy_(running_mean.view(b, c).mean(0, keepdim=False))
        if running_var is not None:
            running_var_orig.copy_(running_var.view(b, c).mean(0, keepdim=False))

        return out.view(b, c, *input.size()[2:])
    return _instance_norm(input, running_mean=running_mean,
                          running_var=running_var, weight=weight, bias=bias,
                          use_input_stats=use_input_stats, momentum=momentum,
                          eps=eps)


def layer_norm(input, normalized_shape, running_mean=None, running_var=None,
               weight=None, bias=None, use_input_stats=True,
               momentum=0.1, eps=1e-5):
    r"""Applies Layer Normalization for last certain number of dimensions.

    See :class:`~torch.nn.LayerNorm` for details.
    """
    if not use_input_stats and (running_mean is None or running_var is None):
        raise ValueError('Expected running_mean and running_var to be not None when use_input_stats=False')

    if weight is not None and weight.size() != normalized_shape:
        raise ValueError('Expected weight to be of same shape as '
                         'normalized_shape, but got {} weight and '
                         'normalized_shape={}'.format(weight.size(), normalized_shape))

    if bias is not None and bias.size() != normalized_shape:
        raise ValueError('Expected bias to be of same shape as '
                         'normalized_shape, but got {} bias and '
                         'normalized_shape={}'.format(bias.size(), normalized_shape))

    normalized_ndim = len(normalized_shape)
    input_shape = input.size()

    if input_shape[-normalized_ndim:] != torch.Size(normalized_shape):
        raise ValueError('Expected input with shape [*, {}], but got {} input'
                         .format(', '.join(normalized_shape), list(input_shape)))

    n = reduce(mul, input_shape[:-normalized_ndim], 1)

    # Repeat stored stats if necessary
    if running_mean is not None:
        running_mean_orig = running_mean
        running_mean = running_mean_orig.repeat(n)
    if running_var is not None:
        running_var_orig = running_var
        running_var = running_var_orig.repeat(n)

    # Apply layer norm
    input_reshaped = input.contiguous().view(1, n, -1)

    out = batch_norm(
        input_reshaped, running_mean, running_var, None, None,
        use_input_stats, momentum, eps)

    # Copy back
    if running_mean is not None:
        running_mean_orig.fill_(running_mean.mean())
    if running_var is not None:
        running_var_orig.fill_(running_var.mean())

    out = out.view(*input_shape)

    if weight is not None and bias is not None:
        return torch.addcmul(bias, 1, out, weight)
    elif weight is not None:
        return torch.mul(out, weight)
    elif bias is not None:
        return torch.add(out, bias)
    else:
        return out


def group_norm(input, num_groups, weight=None, bias=None, eps=1e-5):
    r"""Applies Group Normalization for last certain number of dimensions.

    See :class:`~torch.nn.GroupNorm` for details.
    """
    return torch.group_norm(input, num_groups, weight, bias, eps)


def local_response_norm(input, size, alpha=1e-4, beta=0.75, k=1):
    r"""Applies local response normalization over an input signal composed of
    several input planes, where channels occupy the second dimension.
    Applies normalization across channels.

    See :class:`~torch.nn.LocalResponseNorm` for details.
    """
    dim = input.dim()
    if dim < 3:
        raise ValueError('Expected 3D or higher dimensionality \
                         input (got {} dimensions)'.format(dim))
    div = input.mul(input).unsqueeze(1)
    if dim == 3:
        div = pad(div, (0, 0, size // 2, (size - 1) // 2))
        div = avg_pool2d(div, (size, 1), stride=1).squeeze(1)
    else:
        sizes = input.size()
        div = div.view(sizes[0], 1, sizes[1], sizes[2], -1)
        div = pad(div, (0, 0, 0, 0, size // 2, (size - 1) // 2))
        div = avg_pool3d(div, (size, 1, 1), stride=1).squeeze(1)
        div = div.view(sizes)
    div = div.mul(alpha).add(k).pow(beta)
    return input / div


# loss


def nll_loss(input, target, weight=None, size_average=True, ignore_index=-100, reduce=True):
    r"""The negative log likelihood loss.

    See :class:`~torch.nn.NLLLoss` for details.

    Args:
        input: :math:`(N, C)` where `C = number of classes` or :math:`(N, C, H, W)`
            in case of 2D Loss, or :math:`(N, C, d_1, d_2, ..., d_K)` where :math:`K > 1`
            in the case of K-dimensional loss.
        target: :math:`(N)` where each value is :math:`0 \leq \text{targets}[i] \leq C-1`,
            or :math:`(N, d_1, d_2, ..., d_K)` where :math:`K \geq 1` for
            K-dimensional loss.
        weight (Tensor, optional): a manual rescaling weight given to each
            class. If given, has to be a Tensor of size `C`
        size_average (bool, optional): By default, the losses are averaged
            over observations for each minibatch. If :attr:`size_average`
            is ``False``, the losses are summed for each minibatch. Default: ``True``
        ignore_index (int, optional): Specifies a target value that is ignored
            and does not contribute to the input gradient. When :attr:`size_average` is
            ``True``, the loss is averaged over non-ignored targets. Default: -100

    Example::

        >>> # input is of size N x C = 3 x 5
        >>> input = torch.randn(3, 5)
        >>> # each element in target has to have 0 <= value < C
        >>> target = torch.LongTensor([1, 0, 4])
        >>> output = F.nll_loss(F.log_softmax(input), target)
        >>> output.backward()
    """
    dim = input.dim()
    if torch.is_tensor(weight):
        weight = weight
    if dim < 2:
        raise ValueError('Expected 2 or more dimensions (got {})'.format(dim))

    if input.size(0) != target.size(0):
        raise ValueError('Expected input batch_size ({}) to match target batch_size ({}).'
                         .format(input.size(0), target.size(0)))
    if dim == 2:
        return torch._C._nn.nll_loss(input, target, weight, size_average, ignore_index, reduce)
    elif dim == 4:
        return torch._C._nn.nll_loss2d(input, target, weight, size_average, ignore_index, reduce)
    elif dim == 3 or dim > 4:
        n = input.size(0)
        c = input.size(1)
        out_size = (n,) + input.size()[2:]
        if target.size()[1:] != input.size()[2:]:
            raise ValueError('Expected target size {}, got {}'.format(
                out_size, input.size()))
        input = input.contiguous().view(n, c, 1, -1)
        target = target.contiguous().view(n, 1, -1)
        if reduce:
            return torch._C._nn.nll_loss2d(input, target, weight, size_average, ignore_index, reduce)
        out = torch._C._nn.nll_loss2d(input, target, weight, size_average, ignore_index, reduce)
        return out.view(out_size)


def poisson_nll_loss(input, target, log_input=True, full=False, size_average=True, eps=1e-8, reduce=True):
    r"""Poisson negative log likelihood loss.

    See :class:`~torch.nn.PoissonNLLLoss` for details.

    Args:
        input: expectation of underlying Poisson distribution.
        target: random sample :math:`target \sim \text{Poisson}(input)`.
        log_input: if ``True`` the loss is computed as
            :math:`\exp(\text{input}) - \text{target} * \text{input}`, if ``False`` then loss is
            :math:`\text{input} - \text{target} * \log(\text{input}+\text{eps})`. Default: ``True``
        full: whether to compute full loss, i. e. to add the Stirling
            approximation term. Default: ``False``
            :math:`\text{target} * \log(\text{target}) - \text{target} + 0.5 * \log(2 * \pi * \text{target})`.
        size_average: By default, the losses are averaged over observations for
            each minibatch. However, if the field :attr:`size_average` is set to ``False``,
            the losses are instead summed for each minibatch. Default: ``True``
        eps (float, optional): Small value to avoid evaluation of :math:`\log(0)` when
            :attr:`log_input`=``False``. Default: 1e-8
        reduce (bool, optional): By default, the losses are averaged
            over observations for each minibatch, or summed, depending on
            :attr:`size_average`. When reduce is ``False``, returns a loss per batch
            instead and ignores :attr:`size_average`. Default: ``True``
    """
    if log_input:
        loss = torch.exp(input) - target * input
    else:
        loss = input - target * torch.log(input + eps)
    if full:
        mask = target > 1
        loss[mask] += (target * torch.log(target) - target + 0.5 * torch.log(2 * math.pi * target))[mask]
    if not reduce:
        return loss
    if size_average:
        return torch.mean(loss)
    return torch.sum(loss)


kl_div = _add_docstr(torch._C._nn.kl_div, r"""
kl_div(input, target, size_average=True) -> Tensor

The `Kullback-Leibler divergence`_ Loss.

See :class:`~torch.nn.KLDivLoss` for details.

Args:
    input: Variable of arbitrary shape
    target: Variable of the same shape as input
    size_average: if ``True`` the output is divided by the number of elements
        in input tensor. Default: ``True``
    reduce (bool, optional): By default, the losses are averaged
        over observations for each minibatch, or summed, depending on
        size_average. When reduce is ``False``, returns a loss per input/target
        element instead and ignores :attr:`size_average`. Default: ``True``

""")


def cross_entropy(input, target, weight=None, size_average=True, ignore_index=-100, reduce=True):
    r"""This criterion combines `log_softmax` and `nll_loss` in a single
    function.

    See :class:`~torch.nn.CrossEntropyLoss` for details.

    Args:
        input: Variable :math:`(N, C)` where `C = number of classes`
        target: Variable :math:`(N)` where each value is
            :math:`0 \leq \text{targets}[i] \leq C-1`
        weight (Tensor, optional): a manual rescaling weight given to each
                class. If given, has to be a Tensor of size `C`
        size_average (bool, optional): By default, the losses are averaged
                over observations for each minibatch. However, if the field
                :attr:`size_average` is set to ``False``, the losses are instead summed
                for each minibatch. Ignored if :attr:`reduce` is ``False``. Default: ``True``
        ignore_index (int, optional): Specifies a target value that is ignored
                and does not contribute to the input gradient. When :attr:`size_average` is
                ``True``, the loss is averaged over non-ignored targets. Default: -100
        reduce (bool, optional): By default, the losses are averaged or summed over
                observations for each minibatch depending on :attr:`size_average`. When :attr:`reduce`
                is ``False``, returns a loss per batch instead and ignores
                :attr:`size_average`. Default: ``True``

    Examples::

        >>> input = torch.randn(3, 5, requires_grad=True)
        >>> target = torch.LongTensor(3).random_(5)
        >>> loss = F.cross_entropy(input, target)
        >>> loss.backward()
    """
    return nll_loss(log_softmax(input, 1), target, weight, size_average, ignore_index, reduce)


def binary_cross_entropy(input, target, weight=None, size_average=True, reduce=True):
    r"""Function that measures the Binary Cross Entropy
    between the target and the output.

    See :class:`~torch.nn.BCELoss` for details.

    Args:
        input: Variable of arbitrary shape
        target: Variable of the same shape as input
        weight (Variable, optional): a manual rescaling weight
                if provided it's repeated to match input tensor shape
        size_average (bool, optional): By default, the losses are averaged
                over observations for each minibatch. However, if the field
                :attr:`size_average` is set to ``False``, the losses are instead summed
                for each minibatch. Default: ``True``
        reduce (bool, optional): By default, the losses are averaged or summed over
                observations for each minibatch depending on :attr:`size_average`. When :attr:`reduce`
                is ``False``, returns a loss per input/target element instead and ignores
                :attr:`size_average`. Default: ``True``

    Examples::

        >>> input = torch.randn(3, requires_grad=True)
        >>> target = torch.LongTensor(3).random_(2)
        >>> loss = F.binary_cross_entropy(F.sigmoid(input), target)
        >>> loss.backward()
    """
    if not (target.size() == input.size()):
        warnings.warn("Using a target size ({}) that is different to the input size ({}) is deprecated. "
                      "Please ensure they have the same size.".format(target.size(), input.size()))
    if input.nelement() != target.nelement():
        raise ValueError("Target and input must have the same number of elements. target nelement ({}) "
                         "!= input nelement ({})".format(target.nelement(), input.nelement()))

    if weight is not None:
        new_size = _infer_size(target.size(), weight.size())
        weight = weight.expand(new_size)
        if torch.is_tensor(weight):
            weight = weight

    return torch._C._nn.binary_cross_entropy(input, target, weight, size_average, reduce)


def binary_cross_entropy_with_logits(input, target, weight=None, size_average=True, reduce=True):
    r"""Function that measures Binary Cross Entropy between target and output
    logits.

    See :class:`~torch.nn.BCEWithLogitsLoss` for details.

    Args:
        input: Variable of arbitrary shape
        target: Variable of the same shape as input
        weight (Variable, optional): a manual rescaling weight
                if provided it's repeated to match input tensor shape
        size_average (bool, optional): By default, the losses are averaged
                over observations for each minibatch. However, if the field
                :attr:`size_average` is set to ``False``, the losses are instead summed
                for each minibatch. Default: ``True``
        reduce (bool, optional): By default, the losses are averaged or summed over
                observations for each minibatch depending on :attr:`size_average`. When :attr:`reduce`
                is ``False``, returns a loss per input/target element instead and ignores
                :attr:`size_average`. Default: ``True``

    Examples::

         >>> input = torch.randn(3, requires_grad=True)
         >>> target = torch.FloatTensor(3).random_(2)
         >>> loss = F.binary_cross_entropy_with_logits(input, target)
         >>> loss.backward()
    """
    if not (target.size() == input.size()):
        raise ValueError("Target size ({}) must be the same as input size ({})".format(target.size(), input.size()))

    max_val = (-input).clamp(min=0)
    loss = input - input * target + max_val + ((-max_val).exp() + (-input - max_val).exp()).log()

    if weight is not None:
        loss = loss * weight

    if not reduce:
        return loss
    elif size_average:
        return loss.mean()
    else:
        return loss.sum()


def _pointwise_loss(lambd, lambd_optimized, input, target, size_average=True, reduce=True):
    if target.requires_grad:
        d = lambd(input, target)
        if not reduce:
            return d
        return torch.mean(d) if size_average else torch.sum(d)
    else:
        return lambd_optimized(input, target, size_average, reduce)


smooth_l1_loss = _add_docstr(torch._C._nn.smooth_l1_loss, r"""
smooth_l1_loss(input, target, size_average=True, reduce=True) -> Tensor

Function that uses a squared term if the absolute
element-wise error falls below 1 and an L1 term otherwise.

See :class:`~torch.nn.SmoothL1Loss` for details.
""")


def l1_loss(input, target, size_average=True, reduce=True):
    r"""l1_loss(input, target, size_average=True, reduce=True) -> Tensor

    Function that takes the mean element-wise absolute value difference.

    See :class:`~torch.nn.L1Loss` for details.
    """
    return _pointwise_loss(lambda a, b: torch.abs(a - b), torch._C._nn.l1_loss,
                           input, target, size_average, reduce)


def mse_loss(input, target, size_average=True, reduce=True):
    r"""mse_loss(input, target, size_average=True, reduce=True) -> Tensor

    Measures the element-wise mean squared error.

    See :class:`~torch.nn.MSELoss` for details.
    """
    return _pointwise_loss(lambda a, b: (a - b) ** 2, torch._C._nn.mse_loss,
                           input, target, size_average, reduce)


def margin_ranking_loss(input1, input2, target, margin=0, size_average=True, reduce=True):
    r"""margin_ranking_loss(input1, input2, target, margin=0, size_average=True, reduce=True) -> Tensor

    See :class:`~torch.nn.MarginRankingLoss` for details.
    """
    if input1.dim() == 0 or input2.dim() == 0 or target.dim() == 0:
        raise RuntimeError(("margin_ranking_loss does not support scalars, got sizes: "
                            "input1: {}, input2: {}, target: {} ".format(input1.size(), input2.size(), target.size())))
    return torch._C._VariableFunctions.margin_ranking_loss(input1, input2, target, margin, size_average, reduce)


def hinge_embedding_loss(input, target, margin=1.0, size_average=True, reduce=True):
    r"""hinge_embedding_loss(input, target, margin=1.0, size_average=True, reduce=True) -> Tensor

    See :class:`~torch.nn.HingeEmbeddingLoss` for details.
    """
    return torch.hinge_embedding_loss(input, target, margin, size_average, reduce)


multilabel_margin_loss = _add_docstr(torch._C._nn.multilabel_margin_loss, r"""
multilabel_margin_loss(input, target, size_average=True, reduce=True) -> Tensor

See :class:`~torch.nn.MultiLabelMarginLoss` for details.
""")

soft_margin_loss = _add_docstr(torch._C._nn.soft_margin_loss, r"""
soft_margin_loss(input, target, size_average=True, reduce=True) -> Tensor

See :class:`~torch.nn.SoftMarginLoss` for details.
""")


def multilabel_soft_margin_loss(input, target, weight=None, size_average=True, reduce=True):
    r"""multilabel_soft_margin_loss(input, target, weight=None, size_average=True) -> Tensor

    See :class:`~torch.nn.MultiLabelSoftMarginLoss` for details.
    """
    input = torch.sigmoid(input)
    return binary_cross_entropy(input, target, weight, size_average, reduce)


def cosine_embedding_loss(input1, input2, target, margin=0, size_average=True, reduce=True):
    r"""cosine_embedding_loss(input1, input2, target, margin=0, size_average=True, reduce=True) -> Tensor

    See :class:`~torch.nn.CosineEmbeddingLoss` for details.
    """
    return torch._C._VariableFunctions.cosine_embedding_loss(input1, input2, target, margin, size_average, reduce)


def multi_margin_loss(input, target, p=1, margin=1, weight=None, size_average=True, reduce=True):
    r"""multi_margin_loss(input, target, p=1, margin=1, weight=None, size_average=True, reduce=True) -> Tensor

    See :class:`~torch.nn.MultiMarginLoss` for details.
    """
    if p != 1 and p != 2:
        raise ValueError('only p == 1 and p == 2 supported')
    if weight is not None and weight.dim() != 1:
        raise ValueError('weight must be one-dimensional')

    return torch._C._nn.multi_margin_loss(input, target, p, margin, weight, size_average, reduce)


def pixel_shuffle(input, upscale_factor):
    r"""Rearranges elements in a tensor of shape :math:`[*, C*r^2, H, W]` to a
    tensor of shape :math:`[C, H*r, W*r]`.

    See :class:`~torch.nn.PixelShuffle` for details.

    Args:
        input (Variable): Input
        upscale_factor (int): factor to increase spatial resolution by

    Examples::

        >>> ps = nn.PixelShuffle(3)
        >>> input = torch.Tensor(1, 9, 4, 4)
        >>> output = ps(input)
        >>> print(output.size())
        torch.Size([1, 1, 12, 12])
    """
    batch_size, channels, in_height, in_width = input.size()
    channels //= upscale_factor ** 2

    out_height = in_height * upscale_factor
    out_width = in_width * upscale_factor

    input_view = input.contiguous().view(
        batch_size, channels, upscale_factor, upscale_factor,
        in_height, in_width)

    shuffle_out = input_view.permute(0, 1, 4, 2, 5, 3).contiguous()
    return shuffle_out.view(batch_size, channels, out_height, out_width)


def upsample(input, size=None, scale_factor=None, mode='nearest', align_corners=None):
    r"""Upsamples the input to either the given :attr:`size` or the given
    :attr:`scale_factor`

    The algorithm used for upsampling is determined by :attr:`mode`.

    Currently temporal, spatial and volumetric upsampling are supported, i.e.
    expected inputs are 3-D, 4-D or 5-D in shape.

    The input dimensions are interpreted in the form:
    `mini-batch x channels x [optional depth] x [optional height] x width`.

    The modes available for upsampling are: `nearest`, `linear` (3D-only),
    `bilinear` (4D-only), `trilinear` (5D-only)

    Args:
        input (Tensor): the input tensor
        size (int or Tuple[int] or Tuple[int, int] or Tuple[int, int, int]):
            output spatial size.
        scale_factor (int): multiplier for spatial size. Has to be an integer.
        mode (string): algorithm used for upsampling:
            'nearest' | 'linear' | 'bilinear' | 'trilinear'. Default: 'nearest'
        align_corners (bool, optional): if True, the corner pixels of the input
            and output tensors are aligned, and thus preserving the values at
            those pixels. This only has effect when :attr:`mode` is `linear`,
            `bilinear`, or `trilinear`. Default: False

    .. warning::
        With ``align_corners = True``, the linearly interpolating modes
        (`linear`, `bilinear`, and `trilinear`) don't proportionally align the
        output and input pixels, and thus the output values can depend on the
        input size. This was the default behavior for these modes up to version
        0.3.1. Since then, the default behavior is ``align_corners = False``.
        See :class:`~torch.nn.Upsample` for concrete examples on how this
        affects the outputs.

    """
    from numbers import Integral
    from .modules.utils import _ntuple

    def _check_size_scale_factor():
        if size is None and scale_factor is None:
            raise ValueError('either size or scale_factor should be defined')
        if size is not None and scale_factor is not None:
            raise ValueError('only one of size or scale_factor should be defined')
        if scale_factor is not None and not isinstance(scale_factor, (Integral, tuple)):
            raise ValueError('scale_factor must be of integer type or a tuple of integer types')

    def _scale_factor(dim):
        _check_size_scale_factor()
        if scale_factor is not None and not isinstance(scale_factor, Integral):
            raise ValueError('scale_factor must be a single Integer value for nearest neighbor sampling')
        if scale_factor is not None:
            return scale_factor
        sizes = _ntuple(dim)(size)
        computed_scale_factor = sizes[0] // input.size(2)
        for d in range(dim):
            if sizes[d] % input.size(d + 2) != 0:
                raise RuntimeError("output size specified in UpsamplingNearest "
                                   "({}) has to be divisible by the input size, but got: "
                                   "{}".format('x'.join(map(str, sizes)),
                                               'x'.join(map(str, input.size()))))
            if sizes[d] // input.size(d + 2) != computed_scale_factor:
                raise RuntimeError("input aspect ratio doesn't match the output ratio")

        return computed_scale_factor

    def _output_size(dim):
        _check_size_scale_factor()
        if size is not None:
            return size
        scale_factors = _ntuple(dim)(scale_factor)
        return [input.size(i + 2) * scale_factors[i] for i in range(dim)]

    if mode == 'nearest':
        if align_corners is not None:
            raise ValueError("align_corners option can only be set with the "
                             "interpolating modes: linear | bilinear | trilinear")
    else:
        if align_corners is None:
            warnings.warn("Default upsampling behavior when mode={} is changed "
                          "to align_corners=False since 0.4.0. Please specify "
                          "align_corners=True if the old behavior is desired. "
                          "See the documentation of nn.Upsample for details.".format(mode))
            align_corners = False

    if input.dim() == 3 and mode == 'nearest':
        return torch._C._nn.upsample_nearest1d(input, _scale_factor(1))
    elif input.dim() == 4 and mode == 'nearest':
        return torch._C._nn.upsample_nearest2d(input, _scale_factor(2))
    elif input.dim() == 5 and mode == 'nearest':
        return torch._C._nn.upsample_nearest3d(input, _scale_factor(3))
    elif input.dim() == 3 and mode == 'linear':
        return torch._C._nn.upsample_linear1d(input, _output_size(1), align_corners)
    elif input.dim() == 3 and mode == 'bilinear':
        raise NotImplementedError("Got 3D input, but bilinear mode needs 4D input")
    elif input.dim() == 3 and mode == 'trilinear':
        raise NotImplementedError("Got 3D input, but trilinear mode needs 5D input")
    elif input.dim() == 4 and mode == 'linear':
        raise NotImplementedError("Got 4D input, but linear mode needs 3D input")
    elif input.dim() == 4 and mode == 'bilinear':
        return torch._C._nn.upsample_bilinear2d(input, _output_size(2), align_corners)
    elif input.dim() == 4 and mode == 'trilinear':
        raise NotImplementedError("Got 4D input, but trilinear mode needs 5D input")
    elif input.dim() == 5 and mode == 'linear':
        raise NotImplementedError("Got 5D input, but linear mode needs 3D input")
    elif input.dim() == 5 and mode == 'bilinear':
        raise NotImplementedError("Got 5D input, but bilinear mode needs 4D input")
    elif input.dim() == 5 and mode == 'trilinear':
        return torch._C._nn.upsample_trilinear3d(input, _output_size(3), align_corners)
    else:
        raise NotImplementedError("Input Error: Only 3D, 4D and 5D input Tensors supported"
                                  " (got {}D) for the modes: nearest | linear | bilinear | trilinear"
                                  " (got {})".format(input.dim(), mode))


def upsample_nearest(input, size=None, scale_factor=None):
    r"""Upsamples the input, using nearest neighbours' pixel values.

    .. warning::
        This function is deprecated in favor of :meth:`torch.nn.functional.upsample`.

    Currently spatial and volumetric upsampling are supported (i.e. expected
    inputs are 4 or 5 dimensional).

    Args:
        input (Variable): input
        size (int or Tuple[int, int] or Tuple[int, int, int]): output spatia
            size.
        scale_factor (int): multiplier for spatial size. Has to be an integer.
    """
    # DeprecationWarning is ignored by default
    warnings.warn("nn.functional.upsample_nearest is deprecated. Use nn.functional.upsample instead.")
    return upsample(input, size, scale_factor, mode='nearest')


def upsample_bilinear(input, size=None, scale_factor=None):
    r"""Upsamples the input, using bilinear upsampling.

    .. warning::
        This function is deprecated in favor of :meth:`torch.nn.functional.upsample`.
        This is equivalent with
        ``nn.functional.upsample(..., mode='bilinear', align_corners=True)``.

    Expected inputs are spatial (4 dimensional). Use `upsample_trilinear` fo
    volumetric (5 dimensional) inputs.

    Args:
        input (Variable): input
        size (int or Tuple[int, int]): output spatial size.
        scale_factor (int or Tuple[int, int]): multiplier for spatial size
    """
    # DeprecationWarning is ignored by default
    warnings.warn("nn.functional.upsample_bilinear is deprecated. Use nn.functional.upsample instead.")
    return upsample(input, size, scale_factor, mode='bilinear', align_corners=True)


def grid_sample(input, grid, mode='bilinear', padding_mode='zeros'):
    r"""Given an :attr:`input` and a flow-field :attr:`grid`, computes the
    `output` using input pixel locations from the grid.

    Uses bilinear interpolation to sample the input pixels.
    Currently, only spatial (4 dimensional) and volumetric (5 dimensional)
    inputs are supported.

    For each output location, :attr:`grid` has `x`, `y`
    input pixel locations which are used to compute output.
    In the case of 5D inputes, :attr:`grid` has `x`, `y`, `z` pixel locations.

    .. Note::
        To avoid confusion in notation, let's note that `x` corresponds to the `width` dimension `IW`,
        `y` corresponds to the height dimension `IH` and `z` corresponds to the `depth` dimension `ID`.

    :attr:`grid` has values in the range of `[-1, 1]`. This is because the
    pixel locations are normalized by the input height and width.

    For example, values: x: -1, y: -1 is the left-top pixel of the input, and
    values: x: 1, y: 1 is the right-bottom pixel of the input.

    If :attr:`grid` has values outside the range of `[-1, 1]`, those locations
    are handled as defined by `padding_mode`. Options are `zeros` or `border`,
    defining those locations to use 0 or image border values as contribution
    to the bilinear interpolation.

    .. Note:: This function is used in building Spatial Transformer Networks

    Args:
        input (Variable): input batch (N x C x IH x IW) or (N x C x ID x IH x IW)
        grid (Variable): flow-field of size (N x OH x OW x 2) or (N x OD x OH x OW x 3)
        padding_mode (str): padding mode for outside grid values
            'zeros' | 'border'. Default: 'zeros'

    Returns:
        output (Variable): output Tensor

    """
    return vision.grid_sampler(input, grid, padding_mode)


def affine_grid(theta, size):
    r"""Generates a 2d flow field, given a batch of affine matrices :attr:`theta`
    Generally used in conjunction with :func:`grid_sample` to
    implement Spatial Transformer Networks.

    Args:
        theta (Variable): input batch of affine matrices (:math:`N \times 2 \times 3`)
        size (torch.Size): the target output image size (:math:`N \times C \times H \times W`)
                           Example: torch.Size((32, 3, 24, 24))

    Returns:
        output (Variable): output Tensor of size (:math:`N \times H \times W \times 2`)
    """
    return vision.affine_grid_generator(theta, size)


def pad(input, pad, mode='constant', value=0):
    r"""Pads tensor.

    `Nd` constant padding:  The number of dimensions to pad is
        :math:`\left\lfloor\frac{len(padding)}{2}\right\rfloor` and the dimensions that gets padded begins with the
        last dimension and moves forward. See below for examples.

    `1D`, `2D` and `3D` "reflect" / "replicate" padding:
        for 1D:
                3D input tensor with padding of the form `(padLeft, padRight)`
        for 2D:
                4D input tensor with padding of the form `(padLeft, padRight, padTop, padBottom)`.
        for 3D:
                5D input tensor with padding of the form
                `(padLeft, padRight, padTop, padBottom, padFront, padBack)`. No "reflect" implementation.

    Args:
        input (Variable): `Nd` tensor
        pad (tuple): m-elem tuple, where :math:`\frac{m}{2} \leq` input dimensions and :math:`m` is even.
        mode: 'constant', 'reflect' or 'replicate'. Default: 'constant'
        value: fill value for 'constant' padding. Default: 0

    Examples::

        >>> t4d = torch.Tensor(3, 3, 4, 2)
        >>> p1d = (1, 1) # pad last dim by 1 on each side
        >>> out = F.pad(t4d, p1d, "constant", 0)
        >>> print(out.data.size())
        torch.Size([3, 3, 4, 4])
        >>> p2d = (1, 1, 2, 2) # pad last dim by (1, 1) and 2nd to last by (2, 2)
        >>> out = F.pad(t4d, p2d, "constant", 0)
        >>> print(out.data.size())
        torch.Size([3, 3, 8, 4])
        >>> t4d = torch.Tensor(3, 3, 4, 2)
        >>> p3d = (0, 1, 2, 1, 3, 3) # pad by (0, 1), (2, 1), and (3, 3)
        >>> out = F.pad(t4d, p3d, "constant", 0)
        >>> print(out.data.size())
        torch.Size([3, 9, 7, 3])
    """
    assert len(pad) % 2 == 0, 'Padding length must be divisible by 2'
    assert len(pad) // 2 <= input.dim(), 'Padding length too large'
    if mode == 'constant':
        return ConstantPadNd.apply(input, pad, value)
    elif input.dim() == 3:
        assert len(pad) == 2, '3D tensors expect 2 values for padding'
        if mode == 'reflect':
            return torch._C._nn.reflection_pad1d(input, pad)
        elif mode == 'replicate':
            return torch._C._nn.replication_pad1d(input, pad)
    elif input.dim() == 4:
        assert len(pad) == 4, '4D tensors expect 4 values for padding'
        if mode == 'reflect':
            return torch._C._nn.reflection_pad2d(input, pad)
        elif mode == 'replicate':
            return torch._C._nn.replication_pad2d(input, pad)
    elif input.dim() == 5:
        assert len(pad) == 6, '5D tensors expect 6 values for padding'
        if mode == 'reflect':
            raise NotImplementedError
        elif mode == 'replicate':
            return torch._C._nn.replication_pad3d(input, pad)
    else:
        raise NotImplementedError("Only 3D, 4D, 5D padding with non-constant padding are supported for now")


# distance

def pairwise_distance(x1, x2, p=2, eps=1e-6, keepdim=False):
    r"""
    See :class:`torch.nn.PairwiseDistance` for details
    """
    return torch._C._VariableFunctions.pairwise_distance(x1, x2, p, eps, keepdim)


def cosine_similarity(x1, x2, dim=1, eps=1e-8):
    r"""Returns cosine similarity between x1 and x2, computed along dim.

    .. math ::
        \text{similarity} = \dfrac{x_1 \cdot x_2}{\max(\Vert x_1 \Vert _2 \cdot \Vert x_2 \Vert _2, \epsilon)}

    Args:
        x1 (Variable): First input.
        x2 (Variable): Second input (of size matching x1).
        dim (int, optional): Dimension of vectors. Default: 1
        eps (float, optional): Small value to avoid division by zero.
            Default: 1e-8

    Shape:
        - Input: :math:`(\ast_1, D, \ast_2)` where D is at position `dim`.
        - Output: :math:`(\ast_1, \ast_2)` where 1 is at position `dim`.

    Example::

        >>> input1 = torch.randn(100, 128)
        >>> input2 = torch.randn(100, 128)
        >>> output = F.cosine_similarity(input1, input2)
        >>> print(output)
    """
    w12 = torch.sum(x1 * x2, dim)
    w1 = torch.norm(x1, 2, dim)
    w2 = torch.norm(x2, 2, dim)
    return w12 / (w1 * w2).clamp(min=eps)


def triplet_margin_loss(anchor, positive, negative, margin=1.0, p=2, eps=1e-6, swap=False, size_average=True,
                        reduce=True):
    r"""
    See :class:`~torch.nn.TripletMarginLoss` for details
    """
    return torch._C._VariableFunctions.triplet_margin_loss(anchor, positive, negative, margin, p, eps, swap,
                                                           size_average, reduce)


def normalize(input, p=2, dim=1, eps=1e-12):
    r"""Performs :math:`L_p` normalization of inputs over specified dimension.

    Does:

    .. math::
        v = \frac{v}{\max(\lVert v \rVert_p, \epsilon)}

    for each subtensor v over dimension dim of input. Each subtensor is
    flattened into a vector, i.e. :math:`\lVert v \rVert_p` is not a matrix
    norm.

    With default arguments normalizes over the second dimension with Euclidean
    norm.

    Args:
        input: input tensor of any shape
        p (float): the exponent value in the norm formulation. Default: 2
        dim (int): the dimension to reduce. Default: 1
        eps (float): small value to avoid division by zero. Default: 1e-12
    """
    return input / input.norm(p, dim, True).clamp(min=eps).expand_as(input)


def assert_int_or_pair(arg, arg_name, message):
    assert isinstance(arg, int) or len(arg) == 2, message.format(arg_name)


def unfold(input, kernel_size, dilation=1, padding=0, stride=1):
    r"""
    See :class:`torch.nn.Unfold` for details
    """

    if input is not None and input.dim() == 4:
        msg = '{} must be int or 2-tuple for 4D input'
        assert_int_or_pair(kernel_size, 'kernel_size', msg)
        assert_int_or_pair(dilation, 'dilation', msg)
        assert_int_or_pair(padding, 'padding', msg)
        assert_int_or_pair(stride, 'stride', msg)

        return Im2Col.apply(input, _pair(kernel_size),
                            _pair(dilation), _pair(padding), _pair(stride))
    else:
        raise NotImplementedError("Input Error: Only 4D input Tensors supported (got {}D)".format(input.dim()))


def fold(input, output_size, kernel_size, dilation=1, padding=0, stride=1):
    r"""
    See :class:`torch.nn.Fold` for details
    """
    if input is not None and input.dim() == 3:
        msg = '{} must be int or 2-tuple for 3D input'
        assert_int_or_pair(output_size, 'output_size', msg)
        assert_int_or_pair(kernel_size, 'kernel_size', msg)
        assert_int_or_pair(dilation, 'dilation', msg)
        assert_int_or_pair(padding, 'padding', msg)
        assert_int_or_pair(stride, 'stride', msg)

        return Col2Im.apply(input, _pair(output_size), _pair(kernel_size),
                            _pair(dilation), _pair(padding), _pair(stride))
    else:
        raise NotImplementedError("Input Error: Only 3D input Tensors supported (got {}D)".format(input.dim()))
