functional.py

from typing import (
    Tuple, Optional, Union, Any, Sequence, TYPE_CHECKING
)

import torch
import torch.nn.functional as F
from ._lowrank import svd_lowrank, pca_lowrank
from .overrides import (
    has_torch_function, has_torch_function_unary, has_torch_function_variadic,
    handle_torch_function)
from ._jit_internal import boolean_dispatch, List
from ._jit_internal import _overload as overload
from torch._autograd_functions import _LU

Tensor = torch.Tensor
from torch import _VF

__all__ = [
    'atleast_1d',
    'atleast_2d',
    'atleast_3d',
    'align_tensors',
    'broadcast_shapes',
    'broadcast_tensors',
    'cartesian_prod',
    'block_diag',
    'cdist',
    'chain_matmul',
    'einsum',
    'istft',
    'lu',
    '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 has_torch_function(tensors):
        return handle_torch_function(broadcast_tensors, tensors, *tensors)
    return _VF.broadcast_tensors(tensors)  # type: ignore[attr-defined]


def broadcast_shapes(*shapes):
    r"""broadcast_shapes(*shapes) -> Size

    Similar to :func:`broadcast_tensors` but for shapes.

    This is equivalent to
    ``torch.broadcast_tensors(*map(torch.empty, shapes))[0].shape``
    but avoids the need create to intermediate tensors. This is useful for
    broadcasting tensors of common batch shape but different rightmost shape,
    e.g. to broadcast mean vectors with covariance matrices.

    Example::

        >>> torch.broadcast_shapes((2,), (3, 1), (1, 1, 1))
        torch.Size([1, 3, 2])

    Args:
        \*shapes (torch.Size): Shapes of tensors.

    Returns:
        shape (torch.Size): A shape compatible with all input shapes.

    Raises:
        RuntimeError: If shapes are incompatible.
    """
    # TODO Movie this to C++ once the jit has better support for torch.Size.
    with torch.no_grad():
        scalar = torch.zeros((), device="cpu")
        tensors = [scalar.expand(shape) for shape in shapes]
        tensors = broadcast_tensors(*tensors)
        return tensors[0].shape


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`.

    Args:
        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.

    Example::

        >>> a = torch.arange(10).reshape(5,2)
        >>> a
        tensor([[0, 1],
                [2, 3],
                [4, 5],
                [6, 7],
                [8, 9]])
        >>> torch.split(a, 2)
        (tensor([[0, 1],
                 [2, 3]]),
         tensor([[4, 5],
                 [6, 7]]),
         tensor([[8, 9]]))
        >>> torch.split(a, [1,4])
        (tensor([[0, 1]]),
         tensor([[2, 3],
                 [4, 5],
                 [6, 7],
                 [8, 9]]))
    """
    if has_torch_function_unary(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)


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

    Sums the product of the elements of the input :attr:`operands` along dimensions specified using a notation
    based on the Einstein summation convention.

    Einsum allows computing many common multi-dimensional linear algebraic array operations by representing them
    in a short-hand format based on the Einstein summation convention, given by :attr:`equation`. The details of
    this format are described below, but the general idea is to label every dimension of the input :attr:`operands`
    with some subscript and define which subscripts are part of the output. The output is then computed by summing
    the product of the elements of the :attr:`operands` along the dimensions whose subscripts are not part of the
    output. For example, matrix multiplication can be computed using einsum as `torch.einsum("ij,jk->ik", A, B)`.
    Here, j is the summation subscript and i and k the output subscripts (see section below for more details on why).

    Equation:

        The :attr:`equation` string specifies the subscripts (letters in `[a-zA-Z]`) for each dimension of
        the input :attr:`operands` in the same order as the dimensions, separating subcripts for each operand by a
        comma (','), e.g. `'ij,jk'` specify subscripts for two 2D operands. The dimensions labeled with the same subscript
        must be broadcastable, that is, their size must either match or be `1`. The exception is if a subscript is
        repeated for the same input operand, in which case the dimensions labeled with this subscript for this operand
        must match in size and the operand will be replaced by its diagonal along these dimensions. The subscripts that
        appear exactly once in the :attr:`equation` will be part of the output, sorted in increasing alphabetical order.
        The output is computed by multiplying the input :attr:`operands` element-wise, with their dimensions aligned based
        on the subscripts, and then summing out the dimensions whose subscripts are not part of the output.

        Optionally, the output subscripts can be explicitly defined by adding an arrow ('->') at the end of the equation
        followed by the subscripts for the output. For instance, the following equation computes the transpose of a
        matrix multiplication: 'ij,jk->ki'. The output subscripts must appear at least once for some input operand and
        at most once for the output.

        Ellipsis ('...') can be used in place of subscripts to broadcast the dimensions covered by the ellipsis.
        Each input operand may contain at most one ellipsis which will cover the dimensions not covered by subscripts,
        e.g. for an input operand with 5 dimensions, the ellipsis in the equation `'ab...c'` cover the third and fourth
        dimensions. The ellipsis does not need to cover the same number of dimensions across the :attr:`operands` but the
        'shape' of the ellipsis (the size of the dimensions covered by them) must broadcast together. If the output is not
        explicitly defined with the arrow ('->') notation, the ellipsis will come first in the output (left-most dimensions),
        before the subscript labels that appear exactly once for the input operands. e.g. the following equation implements
        batch matrix multiplication `'...ij,...jk'`.

        A few final notes: the equation may contain whitespaces between the different elements (subscripts, ellipsis,
        arrow and comma) but something like `'. . .'` is not valid. An empty string `''` is valid for scalar operands.

    .. note::

        ``torch.einsum`` handles ellipsis ('...') differently from NumPy in that it allows dimensions
        covered by the ellipsis to be summed over, that is, ellipsis are not required to be part of the output.

    .. note::

        This function does not optimize the given expression, so a different formula for the same computation may
        run faster or consume less memory. Projects like opt_einsum (https://optimized-einsum.readthedocs.io/en/stable/)
        can optimize the formula for you.

    .. note::

        As of PyTorch 1.10 :func:`torch.einsum` also supports the sublist format (see examples below). In this format,
        subscripts for each operand are specified by sublists, list of integers in the range [0, 52). These sublists
        follow their operands, and an extra sublist can appear at the end of the input to specify the output's
        subscripts., e.g. `torch.einsum(op1, sublist1, op2, sublist2, ..., [subslist_out])`. Python's `Ellipsis` object
        may be provided in a sublist to enable broadcasting as described in the Equation section above.

    Args:
        equation (string): The subscripts for the Einstein summation.
        operands (List[Tensor]): The tensors to compute the Einstein summation of.

    Examples::

        # trace
        >>> torch.einsum('ii', torch.randn(4, 4))
        tensor(-1.2104)

        # diagonal
        >>> torch.einsum('ii->i', torch.randn(4, 4))
        tensor([-0.1034,  0.7952, -0.2433,  0.4545])

        # outer product
        >>> x = torch.randn(5)
        >>> y = torch.randn(4)
        >>> torch.einsum('i,j->ij', x, y)
        tensor([[ 0.1156, -0.2897, -0.3918,  0.4963],
                [-0.3744,  0.9381,  1.2685, -1.6070],
                [ 0.7208, -1.8058, -2.4419,  3.0936],
                [ 0.1713, -0.4291, -0.5802,  0.7350],
                [ 0.5704, -1.4290, -1.9323,  2.4480]])

        # batch matrix multiplication
        >>> As = torch.randn(3,2,5)
        >>> Bs = torch.randn(3,5,4)
        >>> torch.einsum('bij,bjk->bik', As, Bs)
        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]]])

        # with sublist format and ellipsis
        >>> torch.einsum(As, [..., 0, 1], Bs, [..., 1, 2], [..., 0, 2])
        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]]])

        # batch permute
        >>> A = torch.randn(2, 3, 4, 5)
        >>> torch.einsum('...ij->...ji', A).shape
        torch.Size([2, 3, 5, 4])

        # equivalent to torch.nn.functional.bilinear
        >>> 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)
        tensor([[-0.3430, -5.2405,  0.4494],
                [ 0.3311,  5.5201, -3.0356]])
    """
    if len(args) < 2:
        raise ValueError('einsum(): must specify the equation string and at least one operand, '
                         'or at least one operand and its subscripts list')

    equation = None
    operands = None

    if isinstance(args[0], torch.Tensor):
        # Convert the subscript list format which is an interleaving of operand and its subscripts
        # list with an optional output subscripts list at the end (see documentation for more details on this)
        # to the equation string format by creating the equation string from the subscripts list and grouping the
        # input operands into a tensorlist (List[Tensor]).
        def parse_subscript(n: int) -> str:
            if n == Ellipsis:
                return '...'
            if n >= 0 and n < 26:
                return chr(ord('A') + n)
            if n >= 26 and n < 52:
                return chr(ord('a') + n - 26)
            raise ValueError('einsum(): subscript in subscript list is not within the valid range [0, 52)')

        # Parse subscripts for input operands
        equation = ','.join(''.join(parse_subscript(s) for s in l) for l in args[1::2])

        # Parse optional output subscripts (provided when the number of arguments is odd)
        if len(args) % 2 == 1:
            equation += '->' + ''.join(parse_subscript(s) for s in args[-1])
            operands = args[:-1:2]
        else:
            operands = args[::2]
    else:
        equation = args[0]
        operands = args[1:]

    if has_torch_function(operands):
        return handle_torch_function(einsum, operands, equation, *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]
        # recurse incase operands contains value that has torch function
        # in the original implementation this line is omitted
        return einsum(equation, *_operands)

    return _VF.einsum(equation, operands)  # type: ignore[attr-defined]


if TYPE_CHECKING:
    # The JIT doesn't understand Union, so only add type annotation for mypy
    def meshgrid(*tensors: Union[Tensor, List[Tensor]]) -> Tuple[Tensor, ...]:
        return _meshgrid(*tensors)
else:
    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]])
        """
        return _meshgrid(*tensors)


def _meshgrid(*tensors):
    if 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]  # type: ignore[assignment]
    return _VF.meshgrid(tensors)  # type: ignore[attr-defined]


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

    .. warning::
        From version 1.8.0, :attr:`return_complex` must always be given
        explicitly for real inputs and `return_complex=False` has been
        deprecated. Strongly prefer `return_complex=True` as in a future
        pytorch release, this function will only return complex tensors.

        Note that :func:`torch.view_as_real` can be used to recover a real
        tensor with an extra last dimension for real and imaginary components.

    The STFT computes the Fourier transform of short overlapping windows of the
    input. This giving frequency components of the signal as they change over
    time. The interface of this function is modeled after the librosa_ stft function.

    .. _librosa: https://librosa.org/doc/latest/generated/librosa.stft.html

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

    .. math::
        X[\omega, m] = \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 :math:`0 \leq \omega < \text{n\_fft}` for ``onesided=False``,
    or :math:`0 \leq \omega < \lfloor \text{n\_fft} / 2 \rfloor + 1` for ``onesided=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 for real input), 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]^*`.
      Note if the input or window tensors are complex, then :attr:`onesided`
      output is not possible.

    * 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}`.

    * If :attr:`return_complex` is ``True`` (default if input is complex), the
      return is a ``input.dim() + 1`` dimensional complex tensor. If ``False``,
      the output is a ``input.dim() + 2`` dimensional real tensor where the last
      dimension represents the real and imaginary components.

    Returns either a complex tensor of size :math:`(* \times N \times T)` if
    :attr:`return_complex` is true, or a real 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
    and :math:`T` is the total number of frames used.

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

    Args:
        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 for real inputs.
            Default: ``True`` for real :attr:`input` and :attr:`window`, ``False`` otherwise.
        return_complex (bool, optional): whether to return a complex tensor, or
            a real tensor with an extra last dimension for the real and
            imaginary components.

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

    """
    if has_torch_function_unary(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, return_complex=return_complex)
    # 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,  # type: ignore[attr-defined]
                    normalized, onesided, return_complex)

def istft(input: Tensor, n_fft: int, hop_length: Optional[int] = None,
          win_length: Optional[int] = None, window: Optional[Tensor] = None,
          center: bool = True, normalized: bool = False,
          onesided: Optional[bool] = None, length: Optional[int] = None,
          return_complex: bool = False) -> Tensor:
    r"""Inverse short time Fourier Transform. This is expected to be the inverse of :func:`~torch.stft`.
    It has the same parameters (+ additional optional parameter of :attr:`length`) and it should return the
    least squares estimation of the original signal. The algorithm will check using the NOLA condition (
    nonzero overlap).

    Important consideration in the parameters :attr:`window` and :attr:`center` so that the envelop
    created by the summation of all the windows is never zero at certain point in time. Specifically,
    :math:`\sum_{t=-\infty}^{\infty} |w|^2[n-t\times hop\_length] \cancel{=} 0`.

    Since :func:`~torch.stft` discards elements at the end of the signal if they do not fit in a frame,
    ``istft`` may return a shorter signal than the original signal (can occur if :attr:`center` is False
    since the signal isn't padded).

    If :attr:`center` is ``True``, then there will be padding e.g. ``'constant'``, ``'reflect'``, etc.
    Left padding can be trimmed off exactly because they can be calculated but right padding cannot be
    calculated without additional information.

    Example: Suppose the last window is:
    ``[17, 18, 0, 0, 0]`` vs ``[18, 0, 0, 0, 0]``

    The :attr:`n_fft`, :attr:`hop_length`, :attr:`win_length` are all the same which prevents the calculation
    of right padding. These additional values could be zeros or a reflection of the signal so providing
    :attr:`length` could be useful. If :attr:`length` is ``None`` then padding will be aggressively removed
    (some loss of signal).

    [1] D. W. Griffin and J. S. Lim, "Signal estimation from modified short-time Fourier transform,"
    IEEE Trans. ASSP, vol.32, no.2, pp.236-243, Apr. 1984.

    Args:
        input (Tensor): The input tensor. Expected to be output of :func:`~torch.stft`,
            can either be complex (``channel``, ``fft_size``, ``n_frame``), or real
            (``channel``, ``fft_size``, ``n_frame``, 2) where the ``channel``
            dimension is optional.

            .. deprecated:: 1.8.0
               Real input is deprecated, use complex inputs as returned by
               ``stft(..., return_complex=True)`` instead.
        n_fft (int): Size of Fourier transform
        hop_length (Optional[int]): The distance between neighboring sliding window frames.
            (Default: ``n_fft // 4``)
        win_length (Optional[int]): The size of window frame and STFT filter. (Default: ``n_fft``)
        window (Optional[torch.Tensor]): The optional window function.
            (Default: ``torch.ones(win_length)``)
        center (bool): Whether :attr:`input` was padded on both sides so that the :math:`t`-th frame is
            centered at time :math:`t \times \text{hop\_length}`.
            (Default: ``True``)
        normalized (bool): Whether the STFT was normalized. (Default: ``False``)
        onesided (Optional[bool]): Whether the STFT was onesided.
            (Default: ``True`` if ``n_fft != fft_size`` in the input size)
        length (Optional[int]): The amount to trim the signal by (i.e. the
            original signal length). (Default: whole signal)
        return_complex (Optional[bool]):
            Whether the output should be complex, or if the input should be
            assumed to derive from a real signal and window.
            Note that this is incompatible with ``onesided=True``.
            (Default: ``False``)

    Returns:
        Tensor: Least squares estimation of the original signal of size (..., signal_length)
    """
    if has_torch_function_unary(input):
        return handle_torch_function(
            istft, (input,), input, n_fft, hop_length=hop_length, win_length=win_length,
            window=window, center=center, normalized=normalized, onesided=onesided,
            length=length, return_complex=return_complex)

    return _VF.istft(input, n_fft, hop_length, win_length, window, center,  # type: ignore[attr-defined]
                     normalized, onesided, length, return_complex)


del torch.unique_dim


if TYPE_CHECKING:
    # These _impl functions return a variable number of tensors as output with
    # __torch_function__; tuple unpacking is done already rather than being
    # done by the caller of the _impl function
    _unique_impl_out = Any
else:
    _unique_impl_out = Tuple[Tensor, Tensor, Tensor]


def _unique_impl(input: Tensor, sorted: bool = True,
                 return_inverse: bool = False, return_counts: bool = False,
                 dim: Optional[int] = None) -> _unique_impl_out:
    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.

    Args:
        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 has_torch_function_unary(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,
        )
    return output, inverse_indices, counts


def _unique_consecutive_impl(input: Tensor, return_inverse: bool = False,
                             return_counts: bool = False,
                             dim: Optional[int] = None) -> _unique_impl_out:
    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++.

    Args:
        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 has_torch_function_unary(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(  # type: ignore[attr-defined]
        input, return_inverse=return_inverse, return_counts=return_counts, dim=dim)
    return output, inverse_indices, counts


def _return_counts(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

    if has_torch_function_unary(input):
        return _unique_impl(input, sorted, return_inverse, return_counts, dim)

    output, _, counts = _unique_impl(input, sorted, return_inverse, return_counts, dim)
    return output, counts


def _return_output(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, bool, Optional[int]) -> Tensor

    if has_torch_function_unary(input):
        return _unique_impl(input, sorted, return_inverse, return_counts, dim)

    output, _, _ = _unique_impl(input, sorted, return_inverse, return_counts, dim)
    return output


def _return_inverse(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

    if has_torch_function_unary(input):
        return _unique_impl(input, sorted, return_inverse, return_counts, dim)

    output, inverse_indices, _ = _unique_impl(input, sorted, return_inverse, return_counts, dim)
    return output, inverse_indices


_return_inverse_false = boolean_dispatch(
    arg_name='return_counts',
    arg_index=3,
    default=False,
    if_true=_return_counts,
    if_false=_return_output,
    module_name=__name__,
    func_name='unique')

_return_inverse_true = boolean_dispatch(
    arg_name='return_counts',
    arg_index=3,
    default=False,
    if_true=_unique_impl,
    if_false=_return_inverse,
    module_name=__name__,
    func_name='unique')

# The return type of unique depends on `return_inverse`, and `return_counts` so in order to
# resolve the output type in TorchScript we need to statically know the value of both parameters

unique = boolean_dispatch(
    arg_name='return_inverse',
    arg_index=2,
    default=False,
    if_true=_return_inverse_true,
    if_false=_return_inverse_false,
    module_name=__name__,
    func_name='unique')
unique.__doc__ = _unique_impl.__doc__


def _consecutive_return_counts(input, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

    if has_torch_function_unary(input):
        return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

    output, _, counts = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
    return output, counts


def _consecutive_return_output(input, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, Optional[int]) -> Tensor

    if has_torch_function_unary(input):
        return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

    output, _, _ = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
    return output


def _consecutive_return_inverse(input, return_inverse=False, return_counts=False, dim=None):
    # type: (Tensor, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

    if has_torch_function_unary(input):
        return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

    output, inverse_indices, _ = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
    return output, inverse_indices


_consecutive_return_inverse_false = boolean_dispatch(
    arg_name='return_counts',
    arg_index=1,
    default=False,
    if_true=_consecutive_return_counts,
    if_false=_consecutive_return_output,
    module_name=__name__,
    func_name='unique_consecutive')

_consecutive_return_inverse_true = boolean_dispatch(
    arg_name='return_counts',
    arg_index=1,
    default=False,
    if_true=_unique_consecutive_impl,
    if_false=_consecutive_return_inverse,
    module_name=__name__,
    func_name='unique_consecutive')

# The return type of unique depends on `return_inverse`, and `return_counts` so in order to
# resolve the output type in TorchScript we need to statically know the value of both parameters

unique_consecutive = boolean_dispatch(
    arg_name='return_inverse',
    arg_index=2,
    default=False,
    if_true=_consecutive_return_inverse_true,
    if_false=_consecutive_return_inverse_false,
    module_name=__name__,
    func_name='unique_consecutive')
unique_consecutive.__doc__ = _unique_consecutive_impl.__doc__

if TYPE_CHECKING:
    pass
    # There's no good way to use this type annotation without breaking JIT
    # overloads. So leave untyped for mypy for now.
else:
    @overload
    def tensordot(a, b, dims: int = 2, out: Optional[torch.Tensor] = None):
        pass

    @overload
    def tensordot(a, b, dims: Tuple[List[int], List[int]], out: Optional[torch.Tensor] = None):  # noqa: F811
        pass

    @overload
    def tensordot(a, b, dims: List[List[int]], out: Optional[torch.Tensor] = None):  # noqa: F811
        pass

    @overload
    def tensordot(a, b, dims: torch.Tensor, out: Optional[torch.Tensor] = None):  # noqa: F811
        pass

def tensordot(a, b, dims=2, out: Optional[torch.Tensor] = None):  # noqa: F811
    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[List[int], List[int]] or List[List[int]] containing two lists or Tensor): 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]])

        >>> a = torch.randn(3, 5, 4, 6)
        >>> b = torch.randn(6, 4, 5, 3)
        >>> torch.tensordot(a, b, dims=([2, 1, 3], [1, 2, 0]))
        tensor([[  7.7193,  -2.4867, -10.3204],
                [  1.5513, -14.4737,  -6.5113],
                [ -0.2850,   4.2573,  -3.5997]])
    """
    if has_torch_function_variadic(a, b):
        return handle_torch_function(tensordot, (a, b), a, b, dims=dims)

    dims_a: List[int] = []
    dims_b: List[int] = []

    if isinstance(dims, (tuple, list)):
        dims_a, dims_b = dims

    if isinstance(dims, torch.Tensor):
        num_elements = dims.numel()
        if num_elements > 1:
            assert dims.size()[0] == 2
            dims_a = torch.jit.annotate(List[int], dims[0].tolist())
            dims_b = torch.jit.annotate(List[int], dims[1].tolist())
        else:
            dims_val = int(dims.item())
            if dims_val < 0:
                raise RuntimeError(f"tensordot expects dims >= 0, but got dims={dims}")
            dims_a = list(range(-dims_val, 0))
            dims_b = list(range(dims_val))

    if isinstance(dims, int):
        if dims < 0:
            raise RuntimeError(f"tensordot expects dims >= 0, but got dims={dims}")
        dims_a = list(range(-dims, 0))
        dims_b = list(range(dims))

    if len(dims_a) == 0 or len(dims_b) == 0:
        raise RuntimeError(f"unsupported input to tensordot, got dims={dims}")

    if out is None:
        return _VF.tensordot(a, b, dims_a, dims_b)  # type: ignore[attr-defined]
    else:
        return _VF.tensordot(a, b, dims_a, dims_b, out=out)  # type: ignore[attr-defined]

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

    Args:
        *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 has_torch_function(tensors):
        return handle_torch_function(cartesian_prod, tensors, *tensors)
    return _VF.cartesian_prod(tensors)  # type: ignore[attr-defined]

def block_diag(*tensors):
    """Create a block diagonal matrix from provided tensors.

    Args:
        *tensors: One or more tensors with 0, 1, or 2 dimensions.

    Returns:
        Tensor: A 2 dimensional tensor with all the input tensors arranged in
        order such that their upper left and lower right corners are
        diagonally adjacent. All other elements are set to 0.

    Example::

        >>> import torch
        >>> A = torch.tensor([[0, 1], [1, 0]])
        >>> B = torch.tensor([[3, 4, 5], [6, 7, 8]])
        >>> C = torch.tensor(7)
        >>> D = torch.tensor([1, 2, 3])
        >>> E = torch.tensor([[4], [5], [6]])
        >>> torch.block_diag(A, B, C, D, E)
        tensor([[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
                [1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                [0, 0, 3, 4, 5, 0, 0, 0, 0, 0],
                [0, 0, 6, 7, 8, 0, 0, 0, 0, 0],
                [0, 0, 0, 0, 0, 7, 0, 0, 0, 0],
                [0, 0, 0, 0, 0, 0, 1, 2, 3, 0],
                [0, 0, 0, 0, 0, 0, 0, 0, 0, 4],
                [0, 0, 0, 0, 0, 0, 0, 0, 0, 5],
                [0, 0, 0, 0, 0, 0, 0, 0, 0, 6]])
    """
    if has_torch_function(tensors):
        return handle_torch_function(block_diag, tensors, *tensors)
    return torch._C._VariableFunctions.block_diag(tensors)  # type: ignore[attr-defined]


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 has_torch_function_variadic(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)  # type: ignore[attr-defined]
    elif compute_mode == 'use_mm_for_euclid_dist':
        return _VF.cdist(x1, x2, p, 1)  # type: ignore[attr-defined]
    elif compute_mode == 'donot_use_mm_for_euclid_dist':
        return _VF.cdist(x1, x2, p, 2)  # type: ignore[attr-defined]
    else:
        raise ValueError(f"{compute_mode} is not a valid value for compute_mode")

def atleast_1d(*tensors):
    r"""
    Returns a 1-dimensional view of each input tensor with zero dimensions.
    Input tensors with one or more dimensions are returned as-is.

    Args:
        input (Tensor or list of Tensors)

    Returns:
        output (Tensor or tuple of Tensors)

    Example::

        >>> x = torch.randn(2)
        >>> x
        tensor([1.4584, 0.7583])
        >>> torch.atleast_1d(x)
        tensor([1.4584, 0.7583])
        >>> x = torch.tensor(1.)
        >>> x
        tensor(1.)
        >>> torch.atleast_1d(x)
        tensor([1.])
        >>> x = torch.tensor(0.5)
        >>> y = torch.tensor(1.)
        >>> torch.atleast_1d((x,y))
        (tensor([0.5000]), tensor([1.]))
    """
    if has_torch_function(tensors):
        return handle_torch_function(atleast_1d, tensors, *tensors)
    if len(tensors) == 1:
        tensors = tensors[0]
    return _VF.atleast_1d(tensors)  # type: ignore[attr-defined]

def atleast_2d(*tensors):
    r"""
    Returns a 2-dimensional view of each input tensor with zero dimensions.
    Input tensors with two or more dimensions are returned as-is.

    Args:
        input (Tensor or list of Tensors)

    Returns:
        output (Tensor or tuple of Tensors)

    Example::

        >>> x = torch.tensor(1.)
        >>> x
        tensor(1.)
        >>> torch.atleast_2d(x)
        tensor([[1.]])
        >>> x = torch.randn(2,2)
        >>> x
        tensor([[2.2086, 2.5165],
                [0.1757, 0.5194]])
        >>> torch.atleast_2d(x)
        tensor([[2.2086, 2.5165],
                [0.1757, 0.5194]])
        >>> x = torch.tensor(0.5)
        >>> y = torch.tensor(1.)
        >>> torch.atleast_2d((x,y))
        (tensor([[0.5000]]), tensor([[1.]]))
    """
    if has_torch_function(tensors):
        return handle_torch_function(atleast_2d, tensors, *tensors)
    if len(tensors) == 1:
        tensors = tensors[0]
    return _VF.atleast_2d(tensors)  # type: ignore[attr-defined]

def atleast_3d(*tensors):
    r"""
    Returns a 3-dimensional view of each input tensor with zero dimensions.
    Input tensors with three or more dimensions are returned as-is.

    Args:
        input (Tensor or list of Tensors)

    Returns:
        output (Tensor or tuple of Tensors)

    Example:

        >>> x = torch.tensor(0.5)
        >>> x
        tensor(0.5000)
        >>> torch.atleast_3d(x)
        tensor([[[0.5000]]])
        >>> y = torch.randn(2,2)
        >>> y
        tensor([[-0.8079,  0.7460],
                [-1.1647,  1.4734]])
        >>> torch.atleast_3d(y)
        tensor([[[-0.8079],
                [ 0.7460]],
                <BLANKLINE>
                [[-1.1647],
                [ 1.4734]]])
        >>> x = torch.randn(1,1,1)
        >>> x
        tensor([[[-1.5689]]])
        >>> torch.atleast_3d(x)
        tensor([[[-1.5689]]])
        >>> x = torch.tensor(0.5)
        >>> y = torch.tensor(1.)
        >>> torch.atleast_3d((x,y))
        (tensor([[[0.5000]]]), tensor([[[1.]]]))
    """
    if has_torch_function(tensors):
        return handle_torch_function(atleast_3d, tensors, *tensors)
    if len(tensors) == 1:
        tensors = tensors[0]
    return _VF.atleast_3d(tensors)  # type: ignore[attr-defined]


if TYPE_CHECKING:
    pass
    # There's no good way to use this type annotation; cannot rename norm() to
    # _norm_impl() in a way that doesn't break JIT overloads. So leave untyped
    # for mypy for now.
    #    def norm(input: Tensor,
    #             p: Optional[Union[str, Number]] = "fro",
    #             dim: Optional[Union[int, List[int]]] = None,
    #             keepdim: bool = False,
    #             out: Optional[Tensor] = None,
    #             dtype: _dtype = None) -> Tensor:
    #        return _norm_impl(input, p, dim, keepdim, out, dtype)
else:
    # TODO: type dim as BroadcastingList when
    # https://github.com/pytorch/pytorch/issues/33782 is fixed
    @overload
    def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):
        # type: (Tensor, str, Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
        pass

    @overload
    def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
        # type: (Tensor, Optional[number], Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
        pass

    @overload
    def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
        # type: (Tensor, Optional[number], Optional[int], bool, Optional[Tensor], Optional[int]) -> Tensor
        pass

    @overload
    def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
        # 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: F811
    r"""Returns the matrix norm or vector norm of a given tensor.

    .. warning::

        torch.norm is deprecated and may be removed in a future PyTorch release.

        Use :func:`torch.linalg.norm`, instead, or :func:`torch.linalg.vector_norm`
        when computing vector norms and :func:`torch.linalg.matrix_norm` when
        computing matrix norms. Note, however, the signature for these functions
        is slightly different than the signature for torch.norm.

    Args:
        input (Tensor): The input tensor. Its data type must be either a floating
            point or complex type. For complex inputs, the norm is calculated using the
            absolute value of each element. If the input is complex and neither
            :attr:`dtype` nor :attr:`out` is specified, the result's data type will
            be the corresponding floating point type (e.g. float if :attr:`input` is
            complexfloat).

        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
            ======  ==============  ==========================
            'fro'   Frobenius norm  --
            'nuc'   nuclear norm    --
            Number  --              sum(abs(x)**ord)**(1./ord)
            ======  ==============  ==========================

            The vector norm can be calculated across any number of dimensions.
            The corresponding dimensions of :attr:`input` are flattened into
            one dimension, and the norm is calculated on the flattened
            dimension.

            Frobenius norm produces the same result as ``p=2`` in all cases
            except when :attr:`dim` is a list of three or more dims, in which
            case Frobenius norm throws an error.

            Nuclear norm can only be calculated across exactly two dimensions.

        dim (int, tuple of ints, list of ints, optional):
            Specifies which dimension or dimensions of :attr:`input` to
            calculate the norm across. If :attr:`dim` is ``None``, the norm will
            be calculated across all dimensions of :attr:`input`. If the norm
            type indicated by :attr:`p` does not support the specified number of
            dimensions, an error will occur.
        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.

    .. note::
        Even though ``p='fro'`` supports any number of dimensions, the true
        mathematical definition of Frobenius norm only applies to tensors with
        exactly two dimensions. :func:`torch.linalg.norm` with ``ord='fro'`` aligns
        with the mathematical definition, since it can only be applied across
        exactly two dimensions.

    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 has_torch_function_unary(input):
        return handle_torch_function(
            norm, (input,), input, p=p, dim=dim, keepdim=keepdim, out=out, dtype=dtype)

    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, dim=(), keepdim=keepdim)
        if not isinstance(p, str):
            _dim = [i for i in range(ndim)]  # noqa: C416 TODO: rewrite as list(range(m))
            return _VF.norm(input, p, dim=_dim, keepdim=keepdim)  # type: ignore[attr-defined]

    # 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  # type: ignore[assignment]

    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 = list(range(ndim))
            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(f"only valid string values are 'fro' and 'nuc', found {p}")
    else:
        if _dim is None:
            _dim = list(range(ndim))

        if out is None:
            if dtype is None:
                return _VF.norm(input, p, _dim, keepdim=keepdim)  # type: ignore[attr-defined]
            else:
                return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype)  # type: ignore[attr-defined]
        else:
            if dtype is None:
                return _VF.norm(input, p, _dim, keepdim=keepdim, out=out)  # type: ignore[attr-defined]
            else:
                return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype, out=out)  # type: ignore[attr-defined]

def chain_matmul(*matrices, out=None):
    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.

    .. warning::

        :func:`torch.chain_matmul` is deprecated and will be removed in a future PyTorch release.
        Use :func:`torch.linalg.multi_dot` instead, which accepts a list of two or more tensors
        rather than multiple arguments.

    Args:
        matrices (Tensors...): a sequence of 2 or more 2-D tensors whose product is to be determined.
        out (Tensor, optional): the output tensor. Ignored if :attr:`out` = ``None``.

    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 has_torch_function(matrices):
        return handle_torch_function(chain_matmul, matrices, *matrices)

    if out is None:
        return _VF.chain_matmul(matrices)  # type: ignore[attr-defined]
    else:
        return _VF.chain_matmul(matrices, out=out)  # type: ignore[attr-defined]


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).

    .. note::
       ``L``, ``U``, and ``P`` can be derived using :func:`torch.lu_unpack`.

    .. warning::
        The LU factorization does have backward support,
        but only for square inputs of full rank.

    Args:
        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:`(*, \text{min}(m, n))`.
              ``pivots`` stores all the intermediate transpositions of rows.
              The final permutation ``perm`` could be reconstructed by
              applying ``swap(perm[i], perm[pivots[i] - 1])`` for ``i = 0, ..., pivots.size(-1) - 1``,
              where ``perm`` is initially the identity permutation of :math:`m` elements
              (essentially this is what :func:`torch.lu_unpack` is doing).

            - **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 not torch._jit_internal.is_scripting():
        if A.requires_grad:
            if not (A.size(-2) == A.size(-1) and (A.dtype.is_floating_point or A.is_complex)):
                raise ValueError(
                    'lu.backward works only with batches of squared full-rank matrices'
                    ' of floating or complex types.'
                )

            return _LU.apply(A, pivot, get_infos)
    else:
        if A.requires_grad:
            raise RuntimeError(
                'Script and require gradients is not supported at the moment.'
                'If you just want to do the forward, use .detach()'
                'on the input before calling the function.'
            )

    # 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))

if TYPE_CHECKING:
    _ListOrSeq = Sequence[Tensor]
else:
    _ListOrSeq = List[Tensor]

def _check_list_size(out_len: int, get_infos: bool, out: _ListOrSeq) -> None:
    get_infos_int = 1 if get_infos else 0
    if out_len - get_infos_int != 2:
        raise TypeError(f"expected tuple of {2 + int(get_infos)} elements but got {out_len}")
    if not isinstance(out, (tuple, list)):
        raise TypeError(f"argument 'out' must be tuple of Tensors, not {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 has_torch_function_unary(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 has_torch_function_unary(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.')