functional.py

import torch
import torch.nn.functional as F
from ._lowrank import svd_lowrank, pca_lowrank
from ._overrides import has_torch_function, handle_torch_function
from ._jit_internal import boolean_dispatch, List
from ._jit_internal import _overload as overload
from torch._six import PY2

Tensor = torch.Tensor
from torch import _VF

__all__ = [
    'align_tensors',
    'broadcast_tensors',
    'cartesian_prod',
    'cdist',
    'chain_matmul',
    'einsum',
    'lu',
    'lu_unpack',
    'norm',
    'meshgrid',
    'pca_lowrank',
    'split',
    'stft',
    'svd_lowrank',
    'tensordot',
    'unique',
    'unique_consecutive',
]


def broadcast_tensors(*tensors):
    r"""broadcast_tensors(*tensors) -> List of Tensors

    Broadcasts the given tensors according to :ref:`broadcasting-semantics`.

    Args:
        *tensors: any number of tensors of the same type

    .. warning::

        More than one element of a broadcasted tensor may refer to a single
        memory location. As a result, in-place operations (especially ones that
        are vectorized) may result in incorrect behavior. If you need to write
        to the tensors, please clone them first.

    Example::

        >>> x = torch.arange(3).view(1, 3)
        >>> y = torch.arange(2).view(2, 1)
        >>> a, b = torch.broadcast_tensors(x, y)
        >>> a.size()
        torch.Size([2, 3])
        >>> a
        tensor([[0, 1, 2],
                [0, 1, 2]])
    """
    if not torch.jit.is_scripting():
        if any(type(t) is not Tensor for t in tensors) and has_torch_function(tensors):
            return handle_torch_function(broadcast_tensors, tensors, *tensors)
    return _VF.broadcast_tensors(tensors)


def split(tensor, split_size_or_sections, dim=0):
    r"""Splits the tensor into chunks. Each chunk is a view of the original tensor.

    If :attr:`split_size_or_sections` is an integer type, then :attr:`tensor` will
    be split into equally sized chunks (if possible). Last chunk will be smaller if
    the tensor size along the given dimension :attr:`dim` is not divisible by
    :attr:`split_size`.

    If :attr:`split_size_or_sections` is a list, then :attr:`tensor` will be split
    into ``len(split_size_or_sections)`` chunks with sizes in :attr:`dim` according
    to :attr:`split_size_or_sections`.

    Arguments:
        tensor (Tensor): tensor to split.
        split_size_or_sections (int) or (list(int)): size of a single chunk or
            list of sizes for each chunk
        dim (int): dimension along which to split the tensor.
    """
    if not torch.jit.is_scripting():
        if type(tensor) is not Tensor and has_torch_function((tensor,)):
            return handle_torch_function(split, (tensor,), tensor, split_size_or_sections,
                                         dim=dim)
    # Overwriting reason:
    # This dispatches to two ATen functions depending on the type of
    # split_size_or_sections. The branching code is in tensor.py, which we
    # call here.
    return tensor.split(split_size_or_sections, dim)

# equivalent to itertools.product(indices)
def _indices_product(indices):
    # type: (List[int]) -> (List[List[int]])
    empty_list = torch.jit.annotate(List[int], [])
    result = [empty_list]
    for idx in indices:
        result_temp = torch.jit.annotate(List[List[int]], [])
        for res in result:
            for i in range(idx):
                result_temp.append(res + [i])
        result = result_temp
    return result

def _index_tensor_with_indices_list(tensor, indices):
    # type: (Tensor, List[int]) -> Tensor
    out = tensor
    for index in indices:
        out = out[index]
    return out

def lu_unpack(LU_data, LU_pivots, unpack_data=True, unpack_pivots=True):
    # type: (Tensor, Tensor, bool, bool) ->  (Tuple[Optional[Tensor], Optional[Tensor], Optional[Tensor]])
    r"""Unpacks the data and pivots from a LU factorization of a tensor.

    Returns a tuple of tensors as ``(the pivots, the L tensor, the U tensor)``.

    Arguments:
        LU_data (Tensor): the packed LU factorization data
        LU_pivots (Tensor): the packed LU factorization pivots
        unpack_data (bool): flag indicating if the data should be unpacked
        unpack_pivots (bool): flag indicating if the pivots should be unpacked

    Examples::

        >>> A = torch.randn(2, 3, 3)
        >>> A_LU, pivots = A.lu()
        >>> P, A_L, A_U = torch.lu_unpack(A_LU, pivots)
        >>>
        >>> # can recover A from factorization
        >>> A_ = torch.bmm(P, torch.bmm(A_L, A_U))

        >>> # LU factorization of a rectangular matrix:
        >>> A = torch.randn(2, 3, 2)
        >>> A_LU, pivots = A.lu()
        >>> P, A_L, A_U = torch.lu_unpack(A_LU, pivots)
        >>> P
        tensor([[[1., 0., 0.],
                 [0., 1., 0.],
                 [0., 0., 1.]],

                [[0., 0., 1.],
                 [0., 1., 0.],
                 [1., 0., 0.]]])
        >>> A_L
        tensor([[[ 1.0000,  0.0000],
                 [ 0.4763,  1.0000],
                 [ 0.3683,  0.1135]],

                [[ 1.0000,  0.0000],
                 [ 0.2957,  1.0000],
                 [-0.9668, -0.3335]]])
        >>> A_U
        tensor([[[ 2.1962,  1.0881],
                 [ 0.0000, -0.8681]],

                [[-1.0947,  0.3736],
                 [ 0.0000,  0.5718]]])
        >>> A_ = torch.bmm(P, torch.bmm(A_L, A_U))
        >>> torch.norm(A_ - A)
        tensor(2.9802e-08)
    """
    if not torch.jit.is_scripting():
        tens_ops = (LU_data, LU_pivots)
        if any([type(t) is not Tensor for t in tens_ops]) and has_torch_function(tens_ops):
            return handle_torch_function(
                lu_unpack, tens_ops, LU_data, LU_pivots, unpack_data=unpack_data,
                unpack_pivots=unpack_pivots)
    shape = LU_data.shape
    # In generalized LU factorization, the following shape relations hold:
    #   A.shape[-2:] == (m, n)
    #   P.shape[-2:] == (m, m)
    #   L.shape[-2:] == (m, k)
    #   U.shape[-2:] == (k, n)
    # where k = min(m, n)
    m, n = shape[-2:]
    k = min(m, n)
    if unpack_data:
        U = LU_data.triu()
        if m != k:
            U = U.narrow(-2, 0, k)
        L = LU_data.tril()
        if k != n:
            L = L.narrow(-1, 0, k)
        L.diagonal(dim1=-2, dim2=-1).fill_(1)
    else:
        L = U = None

    if unpack_pivots:
        LU_pivots_zero_idx = LU_pivots - 1
        if LU_data.dim() > 2:
            P = torch.eye(m, device=LU_data.device, dtype=LU_data.dtype) \
                     .expand(shape[:-1] + (m,)) \
                     .clone(memory_format=torch.contiguous_format)

            # TODO: rewrite when TorchScript supports product and map as
            # product(*map(lambda x: list(range(x)), shape[:-2])) when issue 33781 is fixed
            indices = _indices_product(shape[:-2])
            for idx in indices:
                final_order = [i for i in range(m)]  # noqa: C416 TODO: rewrite as list(range(m))
                for k, j in enumerate(_index_tensor_with_indices_list(LU_pivots_zero_idx, idx)):
                    final_order[k], final_order[j] = final_order[j], final_order[k]
                # TODO: remove _index_tensor_with_indices_list when TorchScript supports indexing Tensor with list
                p_idx = _index_tensor_with_indices_list(P, idx)
                p_idx.copy_(p_idx.index_select(1, torch.as_tensor(final_order, device=LU_pivots.device)))
        else:
            P = torch.eye(m, device=LU_data.device, dtype=LU_data.dtype)
            final_order = [i for i in range(m)]  # noqa: C416 TODO: rewrite as list(range(m))
            for k, j, in enumerate(LU_pivots_zero_idx):
                final_order[k], final_order[j] = final_order[j], final_order[k]
            P = P.index_select(1, torch.as_tensor(final_order, device=LU_pivots.device))
    else:
        P = None

    return P, L, U


def einsum(equation, *operands):
    r"""einsum(equation, *operands) -> Tensor

This function provides a way of computing multilinear expressions (i.e. sums of products) using the
Einstein summation convention.

Args:
    equation (string): The equation is given in terms of lower case letters (indices) to be associated
           with each dimension of the operands and result. The left hand side lists the operands
           dimensions, separated by commas. There should be one index letter per tensor dimension.
           The right hand side follows after `->` and gives the indices for the output.
           If the `->` and right hand side are omitted, it implicitly defined as the alphabetically
           sorted list of all indices appearing exactly once in the left hand side.
           The indices not apprearing in the output are summed over after multiplying the operands
           entries.
           If an index appears several times for the same operand, a diagonal is taken.
           Ellipses `...` represent a fixed number of dimensions. If the right hand side is inferred,
           the ellipsis dimensions are at the beginning of the output.
    operands (Tensor): The operands to compute the Einstein sum of.

Examples::

    >>> x = torch.randn(5)
    >>> y = torch.randn(4)
    >>> torch.einsum('i,j->ij', x, y)  # outer product
    tensor([[-0.0570, -0.0286, -0.0231,  0.0197],
            [ 1.2616,  0.6335,  0.5113, -0.4351],
            [ 1.4452,  0.7257,  0.5857, -0.4984],
            [-0.4647, -0.2333, -0.1883,  0.1603],
            [-1.1130, -0.5588, -0.4510,  0.3838]])


    >>> A = torch.randn(3,5,4)
    >>> l = torch.randn(2,5)
    >>> r = torch.randn(2,4)
    >>> torch.einsum('bn,anm,bm->ba', l, A, r) # compare torch.nn.functional.bilinear
    tensor([[-0.3430, -5.2405,  0.4494],
            [ 0.3311,  5.5201, -3.0356]])


    >>> As = torch.randn(3,2,5)
    >>> Bs = torch.randn(3,5,4)
    >>> torch.einsum('bij,bjk->bik', As, Bs) # batch matrix multiplication
    tensor([[[-1.0564, -1.5904,  3.2023,  3.1271],
             [-1.6706, -0.8097, -0.8025, -2.1183]],

            [[ 4.2239,  0.3107, -0.5756, -0.2354],
             [-1.4558, -0.3460,  1.5087, -0.8530]],

            [[ 2.8153,  1.8787, -4.3839, -1.2112],
             [ 0.3728, -2.1131,  0.0921,  0.8305]]])

    >>> A = torch.randn(3, 3)
    >>> torch.einsum('ii->i', A) # diagonal
    tensor([-0.7825,  0.8291, -0.1936])

    >>> A = torch.randn(4, 3, 3)
    >>> torch.einsum('...ii->...i', A) # batch diagonal
    tensor([[-1.0864,  0.7292,  0.0569],
            [-0.9725, -1.0270,  0.6493],
            [ 0.5832, -1.1716, -1.5084],
            [ 0.4041, -1.1690,  0.8570]])

    >>> A = torch.randn(2, 3, 4, 5)
    >>> torch.einsum('...ij->...ji', A).shape # batch permute
    torch.Size([2, 3, 5, 4])
"""
    if not torch.jit.is_scripting():
        if any(type(t) is not Tensor for t in operands) and has_torch_function(operands):
            return handle_torch_function(einsum, operands, *operands)
    if len(operands) == 1 and isinstance(operands[0], (list, tuple)):
        # the old interface of passing the operands as one list argument
        operands = operands[0]
    return _VF.einsum(equation, operands)


def meshgrid(*tensors):
    r"""Take :math:`N` tensors, each of which can be either scalar or 1-dimensional
vector, and create :math:`N` N-dimensional grids, where the :math:`i` :sup:`th` grid is defined by
expanding the :math:`i` :sup:`th` input over dimensions defined by other inputs.


    Args:
        tensors (list of Tensor): list of scalars or 1 dimensional tensors. Scalars will be
        treated as tensors of size :math:`(1,)` automatically

    Returns:
        seq (sequence of Tensors): If the input has :math:`k` tensors of size
        :math:`(N_1,), (N_2,), \ldots , (N_k,)`, then the output would also have :math:`k` tensors,
        where all tensors are of size :math:`(N_1, N_2, \ldots , N_k)`.

    Example::

        >>> x = torch.tensor([1, 2, 3])
        >>> y = torch.tensor([4, 5, 6])
        >>> grid_x, grid_y = torch.meshgrid(x, y)
        >>> grid_x
        tensor([[1, 1, 1],
                [2, 2, 2],
                [3, 3, 3]])
        >>> grid_y
        tensor([[4, 5, 6],
                [4, 5, 6],
                [4, 5, 6]])
    """
    if not torch.jit.is_scripting():
        if any(type(t) is not Tensor for t in tensors) and has_torch_function(tensors):
            return handle_torch_function(meshgrid, tensors, *tensors)
    if len(tensors) == 1 and isinstance(tensors[0], (list, tuple)):
        # the old interface of passing the operands as one list argument
        tensors = tensors[0]
    return _VF.meshgrid(tensors)


def stft(input, n_fft, hop_length=None, win_length=None, window=None,
         center=True, pad_mode='reflect', normalized=False, onesided=True):
    # type: (Tensor, int, Optional[int], Optional[int], Optional[Tensor], bool, str, bool, bool) -> Tensor
    r"""Short-time Fourier transform (STFT).

    Ignoring the optional batch dimension, this method computes the following
    expression:

    .. math::
        X[m, \omega] = \sum_{k = 0}^{\text{win\_length-1}}%
                            \text{window}[k]\ \text{input}[m \times \text{hop\_length} + k]\ %
                            \exp\left(- j \frac{2 \pi \cdot \omega k}{\text{win\_length}}\right),

    where :math:`m` is the index of the sliding window, and :math:`\omega` is
    the frequency that :math:`0 \leq \omega < \text{n\_fft}`. When
    :attr:`onesided` is the default value ``True``,

    * :attr:`input` must be either a 1-D time sequence or a 2-D batch of time
      sequences.

    * If :attr:`hop_length` is ``None`` (default), it is treated as equal to
      ``floor(n_fft / 4)``.

    * If :attr:`win_length` is ``None`` (default), it is treated as equal to
      :attr:`n_fft`.

    * :attr:`window` can be a 1-D tensor of size :attr:`win_length`, e.g., from
      :meth:`torch.hann_window`. If :attr:`window` is ``None`` (default), it is
      treated as if having :math:`1` everywhere in the window. If
      :math:`\text{win\_length} < \text{n\_fft}`, :attr:`window` will be padded on
      both sides to length :attr:`n_fft` before being applied.

    * If :attr:`center` is ``True`` (default), :attr:`input` will be padded on
      both sides so that the :math:`t`-th frame is centered at time
      :math:`t \times \text{hop\_length}`. Otherwise, the :math:`t`-th frame
      begins at time  :math:`t \times \text{hop\_length}`.

    * :attr:`pad_mode` determines the padding method used on :attr:`input` when
      :attr:`center` is ``True``. See :meth:`torch.nn.functional.pad` for
      all available options. Default is ``"reflect"``.

    * If :attr:`onesided` is ``True`` (default), only values for :math:`\omega`
      in :math:`\left[0, 1, 2, \dots, \left\lfloor \frac{\text{n\_fft}}{2} \right\rfloor + 1\right]`
      are returned because the real-to-complex Fourier transform satisfies the
      conjugate symmetry, i.e., :math:`X[m, \omega] = X[m, \text{n\_fft} - \omega]^*`.

    * If :attr:`normalized` is ``True`` (default is ``False``), the function
      returns the normalized STFT results, i.e., multiplied by :math:`(\text{frame\_length})^{-0.5}`.

    Returns the real and the imaginary parts together as one tensor of size
    :math:`(* \times N \times T \times 2)`, where :math:`*` is the optional
    batch size of :attr:`input`, :math:`N` is the number of frequencies where
    STFT is applied, :math:`T` is the total number of frames used, and each pair
    in the last dimension represents a complex number as the real part and the
    imaginary part.

    .. warning::
      This function changed signature at version 0.4.1. Calling with the
      previous signature may cause error or return incorrect result.

    Arguments:
        input (Tensor): the input tensor
        n_fft (int): size of Fourier transform
        hop_length (int, optional): the distance between neighboring sliding window
            frames. Default: ``None`` (treated as equal to ``floor(n_fft / 4)``)
        win_length (int, optional): the size of window frame and STFT filter.
            Default: ``None``  (treated as equal to :attr:`n_fft`)
        window (Tensor, optional): the optional window function.
            Default: ``None`` (treated as window of all :math:`1` s)
        center (bool, optional): whether to pad :attr:`input` on both sides so
            that the :math:`t`-th frame is centered at time :math:`t \times \text{hop\_length}`.
            Default: ``True``
        pad_mode (string, optional): controls the padding method used when
            :attr:`center` is ``True``. Default: ``"reflect"``
        normalized (bool, optional): controls whether to return the normalized STFT results
             Default: ``False``
        onesided (bool, optional): controls whether to return half of results to
            avoid redundancy Default: ``True``

    Returns:
        Tensor: A tensor containing the STFT result with shape described above

    """
    if not torch.jit.is_scripting():
        if type(input) is not Tensor and has_torch_function((input,)):
            return handle_torch_function(
                stft, (input,), input, n_fft, hop_length=hop_length, win_length=win_length,
                window=window, center=center, pad_mode=pad_mode, normalized=normalized,
                onesided=onesided)
    # TODO: after having proper ways to map Python strings to ATen Enum, move
    #       this and F.pad to ATen.
    if center:
        signal_dim = input.dim()
        extended_shape = [1] * (3 - signal_dim) + list(input.size())
        pad = int(n_fft // 2)
        input = F.pad(input.view(extended_shape), (pad, pad), pad_mode)
        input = input.view(input.shape[-signal_dim:])
    return _VF.stft(input, n_fft, hop_length, win_length, window, normalized, onesided)


del torch.unique_dim


def unique(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
    r"""Returns the unique elements of the input tensor.

    .. note:: This function is different from :func:`torch.unique_consecutive` in the sense that
        this function also eliminates non-consecutive duplicate values.

    .. note:: Currently in the CUDA implementation and the CPU implementation when dim is specified,
        `torch.unique` always sort the tensor at the beginning regardless of the `sort` argument.
        Sorting could be slow, so if your input tensor is already sorted, it is recommended to use
        :func:`torch.unique_consecutive` which avoids the sorting.

    Arguments:
        input (Tensor): the input tensor
        sorted (bool): Whether to sort the unique elements in ascending order
            before returning as output.
        return_inverse (bool): Whether to also return the indices for where
            elements in the original input ended up in the returned unique list.
        return_counts (bool): Whether to also return the counts for each unique
            element.
        dim (int): the dimension to apply unique. If ``None``, the unique of the
            flattened input is returned. default: ``None``

    Returns:
        (Tensor, Tensor (optional), Tensor (optional)): A tensor or a tuple of tensors containing

            - **output** (*Tensor*): the output list of unique scalar elements.
            - **inverse_indices** (*Tensor*): (optional) if
              :attr:`return_inverse` is True, there will be an additional
              returned tensor (same shape as input) representing the indices
              for where elements in the original input map to in the output;
              otherwise, this function will only return a single tensor.
            - **counts** (*Tensor*): (optional) if
              :attr:`return_counts` is True, there will be an additional
              returned tensor (same shape as output or output.size(dim),
              if dim was specified) representing the number of occurrences
              for each unique value or tensor.

    Example::

        >>> output = torch.unique(torch.tensor([1, 3, 2, 3], dtype=torch.long))
        >>> output
        tensor([ 2,  3,  1])

        >>> output, inverse_indices = torch.unique(
                torch.tensor([1, 3, 2, 3], dtype=torch.long), sorted=True, return_inverse=True)
        >>> output
        tensor([ 1,  2,  3])
        >>> inverse_indices
        tensor([ 0,  2,  1,  2])

        >>> output, inverse_indices = torch.unique(
                torch.tensor([[1, 3], [2, 3]], dtype=torch.long), sorted=True, return_inverse=True)
        >>> output
        tensor([ 1,  2,  3])
        >>> inverse_indices
        tensor([[ 0,  2],
                [ 1,  2]])

    """
    if not torch.jit.is_scripting():
        if type(input) is not Tensor and has_torch_function((input,)):
            return handle_torch_function(
                unique, (input,), input, sorted=sorted, return_inverse=return_inverse,
                return_counts=return_counts, dim=dim)
    if dim is not None:
        output, inverse_indices, counts = _VF.unique_dim(
            input,
            dim,
            sorted=sorted,
            return_inverse=return_inverse,
            return_counts=return_counts,
        )
    else:
        output, inverse_indices, counts = torch._unique2(
            input,
            sorted=sorted,
            return_inverse=return_inverse,
            return_counts=return_counts,
        )
    if return_inverse and return_counts:
        return output, inverse_indices, counts
    elif return_inverse:
        return output, inverse_indices
    elif return_counts:
        return output, counts
    else:
        return output


def unique_consecutive(input, return_inverse=False, return_counts=False, dim=None):
    r"""Eliminates all but the first element from every consecutive group of equivalent elements.

    .. note:: This function is different from :func:`torch.unique` in the sense that this function
        only eliminates consecutive duplicate values. This semantics is similar to `std::unique`
        in C++.

    Arguments:
        input (Tensor): the input tensor
        return_inverse (bool): Whether to also return the indices for where
            elements in the original input ended up in the returned unique list.
        return_counts (bool): Whether to also return the counts for each unique
            element.
        dim (int): the dimension to apply unique. If ``None``, the unique of the
            flattened input is returned. default: ``None``

    Returns:
        (Tensor, Tensor (optional), Tensor (optional)): A tensor or a tuple of tensors containing

            - **output** (*Tensor*): the output list of unique scalar elements.
            - **inverse_indices** (*Tensor*): (optional) if
              :attr:`return_inverse` is True, there will be an additional
              returned tensor (same shape as input) representing the indices
              for where elements in the original input map to in the output;
              otherwise, this function will only return a single tensor.
            - **counts** (*Tensor*): (optional) if
              :attr:`return_counts` is True, there will be an additional
              returned tensor (same shape as output or output.size(dim),
              if dim was specified) representing the number of occurrences
              for each unique value or tensor.

    Example::

        >>> x = torch.tensor([1, 1, 2, 2, 3, 1, 1, 2])
        >>> output = torch.unique_consecutive(x)
        >>> output
        tensor([1, 2, 3, 1, 2])

        >>> output, inverse_indices = torch.unique_consecutive(x, return_inverse=True)
        >>> output
        tensor([1, 2, 3, 1, 2])
        >>> inverse_indices
        tensor([0, 0, 1, 1, 2, 3, 3, 4])

        >>> output, counts = torch.unique_consecutive(x, return_counts=True)
        >>> output
        tensor([1, 2, 3, 1, 2])
        >>> counts
        tensor([2, 2, 1, 2, 1])
    """
    if not torch.jit.is_scripting():
        if type(input) is not Tensor and has_torch_function((input,)):
            return handle_torch_function(
                unique_consecutive, (input,), input, return_inverse=return_inverse,
                return_counts=return_counts, dim=dim)
    output, inverse_indices, counts = _VF.unique_consecutive(
        input, return_inverse=return_inverse, return_counts=return_counts, dim=dim)
    if return_inverse and return_counts:
        return output, inverse_indices, counts
    if return_inverse:
        return output, inverse_indices
    if return_counts:
        return output, counts
    return output


def tensordot(a, b, dims=2):
    r"""Returns a contraction of a and b over multiple dimensions.

    :attr:`tensordot` implements a generalized matrix product.

    Args:
      a (Tensor): Left tensor to contract
      b (Tensor): Right tensor to contract
      dims (int or tuple of two lists of integers): number of dimensions to
         contract or explicit lists of dimensions for :attr:`a` and
         :attr:`b` respectively

    When called with a non-negative integer argument :attr:`dims` = :math:`d`, and
    the number of dimensions of :attr:`a` and :attr:`b` is :math:`m` and :math:`n`,
    respectively, :func:`~torch.tensordot` computes

    .. math::
        r_{i_0,...,i_{m-d}, i_d,...,i_n}
          = \sum_{k_0,...,k_{d-1}} a_{i_0,...,i_{m-d},k_0,...,k_{d-1}} \times b_{k_0,...,k_{d-1}, i_d,...,i_n}.

    When called with :attr:`dims` of the list form, the given dimensions will be contracted
    in place of the last :math:`d` of :attr:`a` and the first :math:`d` of :math:`b`. The sizes
    in these dimensions must match, but :func:`~torch.tensordot` will deal with broadcasted
    dimensions.

    Examples::

        >>> a = torch.arange(60.).reshape(3, 4, 5)
        >>> b = torch.arange(24.).reshape(4, 3, 2)
        >>> torch.tensordot(a, b, dims=([1, 0], [0, 1]))
        tensor([[4400., 4730.],
                [4532., 4874.],
                [4664., 5018.],
                [4796., 5162.],
                [4928., 5306.]])

        >>> a = torch.randn(3, 4, 5, device='cuda')
        >>> b = torch.randn(4, 5, 6, device='cuda')
        >>> c = torch.tensordot(a, b, dims=2).cpu()
        tensor([[ 8.3504, -2.5436,  6.2922,  2.7556, -1.0732,  3.2741],
                [ 3.3161,  0.0704,  5.0187, -0.4079, -4.3126,  4.8744],
                [ 0.8223,  3.9445,  3.2168, -0.2400,  3.4117,  1.7780]])

    """
    if not torch.jit.is_scripting():
        if (type(a) is not Tensor or type(b) is not Tensor) and has_torch_function((a, b)):
            return handle_torch_function(tensordot, (a, b), a, b, dims=dims)
    if isinstance(dims, (list, tuple)) or \
       (isinstance(dims, torch.Tensor) and dims.numel() > 1):
        dims_a, dims_b = dims
    else:
        if isinstance(dims, torch.Tensor):
            dims = dims.item()
        if dims < 0:
            raise RuntimeError("tensordot expects dims >= 0, but got dims={}".format(dims))
        dims_a = list(range(-dims, 0))
        dims_b = list(range(dims))
    return _VF.tensordot(a, b, dims_a, dims_b)


def cartesian_prod(*tensors):
    """Do cartesian product of the given sequence of tensors. The behavior is similar to
    python's `itertools.product`.

    Arguments:
        *tensors: any number of 1 dimensional tensors.

    Returns:
        Tensor: A tensor equivalent to converting all the input tensors into lists,
            do `itertools.product` on these lists, and finally convert the resulting list
            into tensor.

    Example::

        >>> a = [1, 2, 3]
        >>> b = [4, 5]
        >>> list(itertools.product(a, b))
        [(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)]
        >>> tensor_a = torch.tensor(a)
        >>> tensor_b = torch.tensor(b)
        >>> torch.cartesian_prod(tensor_a, tensor_b)
        tensor([[1, 4],
                [1, 5],
                [2, 4],
                [2, 5],
                [3, 4],
                [3, 5]])
    """
    if not torch.jit.is_scripting():
        if any(type(t) is not Tensor for t in tensors) and has_torch_function(tensors):
            return handle_torch_function(cartesian_prod, tensors, *tensors)
    return _VF.cartesian_prod(tensors)


def cdist(x1, x2, p=2., compute_mode='use_mm_for_euclid_dist_if_necessary'):
    # type: (Tensor, Tensor, float, str) -> (Tensor)
    r"""Computes batched the p-norm distance between each pair of the two collections of row vectors.

    Args:
        x1 (Tensor): input tensor of shape :math:`B \times P \times M`.
        x2 (Tensor): input tensor of shape :math:`B \times R \times M`.
        p: p value for the p-norm distance to calculate between each vector pair
            :math:`\in [0, \infty]`.
        compute_mode:
            'use_mm_for_euclid_dist_if_necessary' - will use matrix multiplication approach to calculate
            euclidean distance (p = 2) if P > 25 or R > 25
            'use_mm_for_euclid_dist' - will always use matrix multiplication approach to calculate
            euclidean distance (p = 2)
            'donot_use_mm_for_euclid_dist' - will never use matrix multiplication approach to calculate
            euclidean distance (p = 2)
            Default: use_mm_for_euclid_dist_if_necessary.

    If x1 has shape :math:`B \times P \times M` and x2 has shape :math:`B \times R \times M` then the
    output will have shape :math:`B \times P \times R`.

    This function is equivalent to `scipy.spatial.distance.cdist(input,'minkowski', p=p)`
    if :math:`p \in (0, \infty)`. When :math:`p = 0` it is equivalent to
    `scipy.spatial.distance.cdist(input, 'hamming') * M`. When :math:`p = \infty`, the closest
    scipy function is `scipy.spatial.distance.cdist(xn, lambda x, y: np.abs(x - y).max())`.

    Example:

        >>> a = torch.tensor([[0.9041,  0.0196], [-0.3108, -2.4423], [-0.4821,  1.059]])
        >>> a
        tensor([[ 0.9041,  0.0196],
                [-0.3108, -2.4423],
                [-0.4821,  1.0590]])
        >>> b = torch.tensor([[-2.1763, -0.4713], [-0.6986,  1.3702]])
        >>> b
        tensor([[-2.1763, -0.4713],
                [-0.6986,  1.3702]])
        >>> torch.cdist(a, b, p=2)
        tensor([[3.1193, 2.0959],
                [2.7138, 3.8322],
                [2.2830, 0.3791]])
    """
    if not torch.jit.is_scripting():
        if (type(x1) is not Tensor or type(x2) is not Tensor) and has_torch_function((x1, x2)):
            return handle_torch_function(
                cdist, (x1, x2), x1, x2, p=p, compute_mode=compute_mode)
    if compute_mode == 'use_mm_for_euclid_dist_if_necessary':
        return _VF.cdist(x1, x2, p, None)
    elif compute_mode == 'use_mm_for_euclid_dist':
        return _VF.cdist(x1, x2, p, 1)
    elif compute_mode == 'donot_use_mm_for_euclid_dist':
        return _VF.cdist(x1, x2, p, 2)
    else:
        raise ValueError("{} is not a valid value for compute_mode".format(compute_mode))

# TODO: type dim as BroadcastingList when https://github.com/pytorch/pytorch/issues/33782 is fixed
@overload  # noqa: 749
def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: 749
    # type: (Tensor, str, Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
    pass

@overload  # noqa: 749
def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: 749
    # type: (Tensor, Optional[number], Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
    pass

@overload  # noqa: 749
def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: 749
    # type: (Tensor, Optional[number], Optional[int], bool, Optional[Tensor], Optional[int]) -> Tensor
    pass

@overload  # noqa: 749
def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: 749
    # type: (Tensor, str, Optional[int], bool, Optional[Tensor], Optional[int]) -> Tensor
    pass

def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: 749
    r"""Returns the matrix norm or vector norm of a given tensor.

    Args:
        input (Tensor): the input tensor
        p (int, float, inf, -inf, 'fro', 'nuc', optional): the order of norm. Default: ``'fro'``
            The following norms can be calculated:

            =====  ============================  ==========================
            ord    matrix norm                   vector norm
            =====  ============================  ==========================
            None   Frobenius norm                2-norm
            'fro'  Frobenius norm                --
            'nuc'  nuclear norm                  --
            Other  as vec norm when dim is None  sum(abs(x)**ord)**(1./ord)
            =====  ============================  ==========================

        dim (int, 2-tuple of ints, 2-list of ints, optional): If it is an int,
            vector norm will be calculated, if it is 2-tuple of ints, matrix norm
            will be calculated. If the value is None, matrix norm will be calculated
            when the input tensor only has two dimensions, vector norm will be
            calculated when the input tensor only has one dimension. If the input
            tensor has more than two dimensions, the vector norm will be applied to
            last dimension.
        keepdim (bool, optional): whether the output tensors have :attr:`dim`
            retained or not. Ignored if :attr:`dim` = ``None`` and
            :attr:`out` = ``None``. Default: ``False``
        out (Tensor, optional): the output tensor. Ignored if
            :attr:`dim` = ``None`` and :attr:`out` = ``None``.
        dtype (:class:`torch.dtype`, optional): the desired data type of
            returned tensor. If specified, the input tensor is casted to
            :attr:'dtype' while performing the operation. Default: None.


    Example::

        >>> import torch
        >>> a = torch.arange(9, dtype= torch.float) - 4
        >>> b = a.reshape((3, 3))
        >>> torch.norm(a)
        tensor(7.7460)
        >>> torch.norm(b)
        tensor(7.7460)
        >>> torch.norm(a, float('inf'))
        tensor(4.)
        >>> torch.norm(b, float('inf'))
        tensor(4.)
        >>> c = torch.tensor([[ 1, 2, 3],[-1, 1, 4]] , dtype= torch.float)
        >>> torch.norm(c, dim=0)
        tensor([1.4142, 2.2361, 5.0000])
        >>> torch.norm(c, dim=1)
        tensor([3.7417, 4.2426])
        >>> torch.norm(c, p=1, dim=1)
        tensor([6., 6.])
        >>> d = torch.arange(8, dtype= torch.float).reshape(2,2,2)
        >>> torch.norm(d, dim=(1,2))
        tensor([ 3.7417, 11.2250])
        >>> torch.norm(d[0, :, :]), torch.norm(d[1, :, :])
        (tensor(3.7417), tensor(11.2250))
    """
    if not torch.jit.is_scripting():
        if type(input) is not Tensor and has_torch_function((input,)):
            return handle_torch_function(
                norm, (input,), input, p=p, dim=dim, keepdim=keepdim, out=out, dtype=dtype)
        # py2 considers isinstance(unicodestr, str) == False
        if PY2 and isinstance(p, unicode):
            p = str(p)

    ndim = input.dim()


    # catch default case
    if dim is None and out is None and dtype is None and p is not None:
        if isinstance(p, str):
            if p == "fro":
                return _VF.frobenius_norm(input)
        if not isinstance(p, str):
            return _VF.norm(input, p)

    # TODO: when https://github.com/pytorch/pytorch/issues/33782 is fixed
    # remove the overloads where dim is an int and replace with BraodcastingList1
    # and remove next four lines, replace _dim with dim
    if dim is not None:
        if isinstance(dim, int):
            _dim = [dim]
        else:
            _dim = dim
    else:
        _dim = None

    if isinstance(p, str):
        if p == "fro":
            if dtype is not None:
                raise ValueError("dtype argument is not supported in frobenius norm")

            if _dim is None:
                _dim = [i for i in range(ndim)]  # noqa: C416 TODO: rewrite as list(range(m))
            if out is None:
                return _VF.frobenius_norm(input, _dim, keepdim=keepdim)
            else:
                return _VF.frobenius_norm(input, _dim, keepdim=keepdim, out=out)
        elif p == "nuc":
            if dtype is not None:
                raise ValueError("dtype argument is not supported in nuclear norm")
            if _dim is None:
                if out is None:
                    return _VF.nuclear_norm(input, keepdim=keepdim)
                else:
                    return _VF.nuclear_norm(input, keepdim=keepdim, out=out)
            else:
                if out is None:
                    return _VF.nuclear_norm(input, _dim, keepdim=keepdim)
                else:
                    return _VF.nuclear_norm(input, _dim, keepdim=keepdim, out=out)
        raise RuntimeError("only valid string values are 'fro' and 'nuc', found {}".format(p))
    else:
        if _dim is None:
            _dim = [i for i in range(ndim)]  # noqa: C416 TODO: rewrite as list(range(m))

        if out is None:
            if dtype is None:
                return _VF.norm(input, p, _dim, keepdim=keepdim)
            else:
                return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype)
        else:
            if dtype is None:
                return _VF.norm(input, p, _dim, keepdim=keepdim, out=out)
            else:
                return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype, out=out)

def chain_matmul(*matrices):
    r"""Returns the matrix product of the :math:`N` 2-D tensors. This product is efficiently computed
    using the matrix chain order algorithm which selects the order in which incurs the lowest cost in terms
    of arithmetic operations (`[CLRS]`_). Note that since this is a function to compute the product, :math:`N`
    needs to be greater than or equal to 2; if equal to 2 then a trivial matrix-matrix product is returned.
    If :math:`N` is 1, then this is a no-op - the original matrix is returned as is.


    Args:
        matrices (Tensors...): a sequence of 2 or more 2-D tensors whose product is to be determined.


    Returns:
        Tensor: if the :math:`i^{th}` tensor was of dimensions :math:`p_{i} \times p_{i + 1}`, then the product
        would be of dimensions :math:`p_{1} \times p_{N + 1}`.

    Example::

        >>> a = torch.randn(3, 4)
        >>> b = torch.randn(4, 5)
        >>> c = torch.randn(5, 6)
        >>> d = torch.randn(6, 7)
        >>> torch.chain_matmul(a, b, c, d)
        tensor([[ -2.3375,  -3.9790,  -4.1119,  -6.6577,   9.5609, -11.5095,  -3.2614],
                [ 21.4038,   3.3378,  -8.4982,  -5.2457, -10.2561,  -2.4684,   2.7163],
                [ -0.9647,  -5.8917,  -2.3213,  -5.2284,  12.8615, -12.2816,  -2.5095]])

    .. _`[CLRS]`: https://mitpress.mit.edu/books/introduction-algorithms-third-edition
    """
    if not torch.jit.is_scripting():
        if any(type(t) is not Tensor for t in matrices) and has_torch_function(matrices):
            return handle_torch_function(chain_matmul, matrices, *matrices)
    return _VF.chain_matmul(matrices)


def _lu_impl(A, pivot=True, get_infos=False, out=None):
    # type: (Tensor, bool, bool, Any) -> Tuple[Tensor, Tensor, Tensor] 
    r"""Computes the LU factorization of a matrix or batches of matrices
    :attr:`A`. Returns a tuple containing the LU factorization and
    pivots of :attr:`A`.  Pivoting is done if :attr:`pivot` is set to
    ``True``.

    .. note::
        The pivots returned by the function are 1-indexed. If :attr:`pivot` is ``False``,
        then the returned pivots is a tensor filled with zeros of the appropriate size.

    .. note::
        LU factorization with :attr:`pivot` = ``False`` is not available for CPU, and attempting
        to do so will throw an error. However, LU factorization with :attr:`pivot` = ``False`` is
        available for CUDA.

    .. note::
        This function does not check if the factorization was successful or not if
        :attr:`get_infos` is ``True`` since the status of the factorization is present in the
        third element of the return tuple.

    .. note::
        In the case of batches of square matrices with size less or
        equal to 32 on a CUDA device, the LU factorization is repeated
        for singular matrices due to the bug in the MAGMA library (see
        magma issue 13).

    Arguments:
        A (Tensor): the tensor to factor of size :math:`(*, m, n)`
        pivot (bool, optional): controls whether pivoting is done. Default: ``True``
        get_infos (bool, optional): if set to ``True``, returns an info IntTensor.
                                    Default: ``False``
        out (tuple, optional): optional output tuple. If :attr:`get_infos` is ``True``,
                               then the elements in the tuple are Tensor, IntTensor,
                               and IntTensor. If :attr:`get_infos` is ``False``, then the
                               elements in the tuple are Tensor, IntTensor. Default: ``None``

    Returns:
        (Tensor, IntTensor, IntTensor (optional)): A tuple of tensors containing

            - **factorization** (*Tensor*): the factorization of size :math:`(*, m, n)`

            - **pivots** (*IntTensor*): the pivots of size :math:`(*, m)`

            - **infos** (*IntTensor*, *optional*): if :attr:`get_infos` is ``True``, this is a tensor of
              size :math:`(*)` where non-zero values indicate whether factorization for the matrix or
              each minibatch has succeeded or failed

    Example::

        >>> A = torch.randn(2, 3, 3)
        >>> A_LU, pivots = torch.lu(A)
        >>> A_LU
        tensor([[[ 1.3506,  2.5558, -0.0816],
                 [ 0.1684,  1.1551,  0.1940],
                 [ 0.1193,  0.6189, -0.5497]],

                [[ 0.4526,  1.2526, -0.3285],
                 [-0.7988,  0.7175, -0.9701],
                 [ 0.2634, -0.9255, -0.3459]]])
        >>> pivots
        tensor([[ 3,  3,  3],
                [ 3,  3,  3]], dtype=torch.int32)
        >>> A_LU, pivots, info = torch.lu(A, get_infos=True)
        >>> if info.nonzero().size(0) == 0:
        ...   print('LU factorization succeeded for all samples!')
        LU factorization succeeded for all samples!
    """
    # If get_infos is True, then we don't need to check for errors and vice versa
    return torch._lu_with_info(A, pivot=pivot, check_errors=(not get_infos))

def _check_list_size(out_len, get_infos, out):
    # type: (int, bool, List[Tensor]) -> None   
    get_infos_int = 1 if get_infos else 0
    if out_len - get_infos_int != 2:
        raise TypeError("expected tuple of {} elements but got {}"
                        .format(2 + int(get_infos), len(out_len)))
    if not isinstance(out, (tuple, list)):
        raise TypeError("argument 'out' must be tuple of Tensors, not {}"
                        .format(type(out).__name__))

def _lu_with_infos(A, pivot=True, get_infos=False, out=None):
    # type: (Tensor, bool, bool, Optional[Tuple[Tensor, Tensor, Tensor]]) -> Tuple[Tensor, Tensor, Tensor]
    if not torch.jit.is_scripting():
        if type(A) is not Tensor and has_torch_function((A,)):
            return handle_torch_function(
                lu, (A,), A, pivot=pivot, get_infos=get_infos, out=out)
    result = _lu_impl(A, pivot, get_infos, out)
    if out is not None:
        _check_list_size(len(out), get_infos, out)
        for i in range(len(out)):
            out[i].resize_as_(result[i]).copy_(result[i])
        return out
    else:
        return result  # A_LU, pivots, infos

def _lu_no_infos(A, pivot=True, get_infos=False, out=None):
    # type: (Tensor, bool, bool, Optional[Tuple[Tensor, Tensor]]) -> Tuple[Tensor, Tensor] 
    # need to check for torch_function here so that we exit if 
    if not torch.jit.is_scripting():
        if type(A) is not Tensor and has_torch_function((A,)):
            return handle_torch_function(
                lu, (A,), A, pivot=pivot, get_infos=get_infos, out=out)
    result = _lu_impl(A, pivot, get_infos, out)
    if out is not None:
        _check_list_size(len(out), get_infos, out)
        for i in range(len(out)):
            out[i].resize_as_(result[i]).copy_(result[i])
        return out
    else:
        return result[0], result[1]  # A_LU, pivots

# The return type of lu depends on `get_infos`, so in order to resolve the output type
# of lu in TorchScript we need to statically know the value of `get_infos` 
lu = boolean_dispatch(
    arg_name='get_infos',
    arg_index=2,
    default=False,
    if_true=_lu_with_infos,
    if_false=_lu_no_infos,
    module_name=__name__,
    func_name='lu')
lu.__doc__ = _lu_impl.__doc__

def align_tensors(*tensors):
    raise RuntimeError('`align_tensors` not yet implemented.')