functional.py

# mypy: allow-untyped-defs
import itertools
import operator
from collections.abc import Sequence
from typing import Any, Optional, TYPE_CHECKING, Union

import torch
import torch.nn.functional as F
from torch import _VF, Tensor
from torch._C import _add_docstr
from torch._jit_internal import _overload as overload, boolean_dispatch
from torch._lowrank import pca_lowrank, svd_lowrank
from torch.overrides import (
    handle_torch_function,
    has_torch_function,
    has_torch_function_unary,
    has_torch_function_variadic,
)


__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",
    "unravel_index",
]


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]])
    """
    # This wrapper exists to support variadic args.
    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.
    """
    # This wrapper exists to support variadic args.
    # TODO Move this to C++ once the jit has better support for torch.Size.
    if not torch.jit.is_tracing():
        max_len = 0
        for shape in shapes:
            if isinstance(shape, (int, torch.SymInt)):
                if max_len < 1:
                    max_len = 1
            elif isinstance(shape, (tuple, list)):
                s = len(shape)
                if max_len < s:
                    max_len = s
        result = [1] * max_len

        from torch.fx.experimental.symbolic_shapes import (
            guard_size_oblivious,
            is_nested_int,
        )

        for shape in shapes:
            if isinstance(shape, (int, torch.SymInt)):
                shape = (shape,)
            if isinstance(shape, (tuple, list)):
                for i in range(-1, -1 - len(shape), -1):
                    if shape[i] < 0:
                        raise RuntimeError(
                            f"Trying to create tensor with negative dimension ({shape[i]}): ({shape[i]})"
                        )

                    # NB: handle nested ints specially to avoid invalid guarding on Ne(j0, 1).
                    if is_nested_int(shape[i]):
                        # Broadcasting is allowed for (j0, 1) or (j0, j0);
                        # not (j0, j1), (j0, 5), etc.
                        if is_nested_int(result[i]) and guard_size_oblivious(
                            shape[i] == result[i]
                        ):
                            continue
                    else:
                        # NB: result is initialized to 1 so this is effectively an
                        # equals one test
                        if guard_size_oblivious(shape[i] == 1) or guard_size_oblivious(
                            shape[i] == result[i]
                        ):
                            continue

                    if result[i] != 1:
                        raise RuntimeError(
                            "Shape mismatch: objects cannot be broadcast to a single shape"
                        )
                    result[i] = shape[i]
            else:
                raise RuntimeError(
                    "Input shapes should be of type ints, a tuple of ints, or a list of ints, got ",
                    shape,
                )
        return torch.Size(result)
    else:
        # with implementation above, torch.jit.trace hardcodes the sizes which makes subsequent replays fail
        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: Tensor,
    split_size_or_sections: Union[int, list[int]],
    dim: int = 0,
) -> tuple[Tensor, ...]:
    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: Any) -> Tensor:
    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 subscripts 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::

        Please install opt-einsum (https://optimized-einsum.readthedocs.io/en/stable/) in order to enroll into a more
        performant einsum. You can install when installing torch like so: `pip install torch[opt-einsum]` or by itself
        with `pip install opt-einsum`.

        If opt-einsum is available, this function will automatically speed up computation and/or consume less memory
        by optimizing contraction order through our opt_einsum backend :mod:`torch.backends.opt_einsum` (The _ vs - is
        confusing, I know). This optimization occurs when there are at least three inputs, since the order does not matter
        otherwise. Note that finding `the` optimal path is an NP-hard problem, thus, opt-einsum relies on different
        heuristics to achieve near-optimal results. If opt-einsum is not available, the default order is to contract
        from left to right.

        To bypass this default behavior, add the following to disable opt_einsum and skip path calculation:
        ``torch.backends.opt_einsum.enabled = False``

        To specify which strategy you'd like for opt_einsum to compute the contraction path, add the following line:
        ``torch.backends.opt_einsum.strategy = 'auto'``. The default strategy is 'auto', and we also support 'greedy' and
        'optimal'. Disclaimer that the runtime of 'optimal' is factorial in the number of inputs! See more details in
        the opt_einsum documentation (https://optimized-einsum.readthedocs.io/en/stable/path_finding.html).

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

    Examples::

        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> # trace
        >>> torch.einsum('ii', torch.randn(4, 4))
        tensor(-1.2104)

        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> # diagonal
        >>> torch.einsum('ii->i', torch.randn(4, 4))
        tensor([-0.1034,  0.7952, -0.2433,  0.4545])

        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> # 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]])

        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> # 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]]])

        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> # 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]])
    """
    import torch.backends.opt_einsum as opt_einsum

    # This wrapper exists to support variadic args.
    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)

    if len(operands) <= 2 or not opt_einsum.enabled:
        # the path for contracting 0 or 1 time(s) is already optimized
        # or the user has disabled using opt_einsum
        return _VF.einsum(equation, operands)  # type: ignore[attr-defined]

    path = None
    if opt_einsum.is_available():
        _opt_einsum = opt_einsum.get_opt_einsum()
        tupled_path = _opt_einsum.contract_path(
            equation, *operands, optimize=opt_einsum.strategy
        )[0]
        # flatten path for dispatching to C++
        path = [*itertools.chain.from_iterable(tupled_path)]
    return _VF.einsum(equation, operands, path=path)  # type: ignore[attr-defined]


# This wrapper exists to support variadic args.
if TYPE_CHECKING:
    # The JIT doesn't understand Union, so only add type annotation for mypy
    def meshgrid(
        *tensors: Union[Tensor, list[Tensor]], indexing: Optional[str] = None
    ) -> tuple[Tensor, ...]:
        return _meshgrid(*tensors, indexing=indexing)

else:

    def meshgrid(*tensors, indexing: Optional[str] = None) -> tuple[Tensor, ...]:
        r"""Creates grids of coordinates specified by the 1D inputs in `attr`:tensors.

        This is helpful when you want to visualize data over some
        range of inputs. See below for a plotting example.

        Given :math:`N` 1D tensors :math:`T_0 \ldots T_{N-1}` as
        inputs with corresponding sizes :math:`S_0 \ldots S_{N-1}`,
        this creates :math:`N` N-dimensional tensors :math:`G_0 \ldots
        G_{N-1}`, each with shape :math:`(S_0, ..., S_{N-1})` where
        the output :math:`G_i` is constructed by expanding :math:`T_i`
        to the result shape.

        .. note::
            0D inputs are treated equivalently to 1D inputs of a
            single element.

        .. warning::
            `torch.meshgrid(*tensors)` currently has the same behavior
            as calling `numpy.meshgrid(*arrays, indexing='ij')`.

            In the future `torch.meshgrid` will transition to
            `indexing='xy'` as the default.

            https://github.com/pytorch/pytorch/issues/50276 tracks
            this issue with the goal of migrating to NumPy's behavior.

        .. seealso::

            :func:`torch.cartesian_prod` has the same effect but it
            collects the data in a tensor of vectors.

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

            indexing: (str, optional): the indexing mode, either "xy"
                or "ij", defaults to "ij". See warning for future changes.

                If "xy" is selected, the first dimension corresponds
                to the cardinality of the second input and the second
                dimension corresponds to the cardinality of the first
                input.

                If "ij" is selected, the dimensions are in the same
                order as the cardinality of the inputs.

        Returns:
            seq (sequence of Tensors): If the input has :math:`N`
            tensors of size :math:`S_0 \ldots S_{N-1}``, then the
            output will also have :math:`N` tensors, where each tensor
            is of shape :math:`(S_0, ..., S_{N-1})`.

        Example::

            >>> x = torch.tensor([1, 2, 3])
            >>> y = torch.tensor([4, 5, 6])

            Observe the element-wise pairings across the grid, (1, 4),
            (1, 5), ..., (3, 6). This is the same thing as the
            cartesian product.
            >>> grid_x, grid_y = torch.meshgrid(x, y, indexing='ij')
            >>> grid_x
            tensor([[1, 1, 1],
                    [2, 2, 2],
                    [3, 3, 3]])
            >>> grid_y
            tensor([[4, 5, 6],
                    [4, 5, 6],
                    [4, 5, 6]])

            This correspondence can be seen when these grids are
            stacked properly.
            >>> torch.equal(torch.cat(tuple(torch.dstack([grid_x, grid_y]))),
            ...             torch.cartesian_prod(x, y))
            True

            `torch.meshgrid` is commonly used to produce a grid for
            plotting.
            >>> # xdoctest: +REQUIRES(module:matplotlib)
            >>> # xdoctest: +REQUIRES(env:DOCTEST_SHOW)
            >>> import matplotlib.pyplot as plt
            >>> xs = torch.linspace(-5, 5, steps=100)
            >>> ys = torch.linspace(-5, 5, steps=100)
            >>> x, y = torch.meshgrid(xs, ys, indexing='xy')
            >>> z = torch.sin(torch.sqrt(x * x + y * y))
            >>> ax = plt.axes(projection='3d')
            >>> ax.plot_surface(x.numpy(), y.numpy(), z.numpy())
            >>> plt.show()

        .. image:: ../_static/img/meshgrid.png
            :width: 512

        """
        return _meshgrid(*tensors, indexing=indexing)


def _meshgrid(*tensors, indexing: Optional[str]):
    if has_torch_function(tensors):
        return handle_torch_function(meshgrid, tensors, *tensors, indexing=indexing)
    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]

    # Continue allowing call of old method that takes no indexing
    # kwarg for forward compatibility reasons.
    #
    # Remove this two weeks after landing.
    kwargs = {} if indexing is None else {"indexing": indexing}
    return _VF.meshgrid(tensors, **kwargs)  # 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,
    align_to_window: 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.

    .. warning::
        From version 2.1, a warning will be provided if a :attr:`window` is
        not specified. In a future release, this attribute will be required.
        Not providing a window currently defaults to using a rectangular window,
        which may result in undesirable artifacts. Consider using tapered windows,
        such as :func:`torch.hann_window`.

    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 (but *not* a drop-in
    replacement for) 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{n\_fft}}\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 of shape `(B?, L)` where `B?` is an optional
            batch dimension
        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.
            Shape must be 1d and `<= n_fft`
            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 (str, 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.

            .. versionchanged:: 2.0
               ``return_complex`` is now a required argument for real inputs,
               as the default is being transitioned to ``True``.

            .. deprecated:: 2.0
               ``return_complex=False`` is deprecated, instead use ``return_complex=True``
               Note that calling :func:`torch.view_as_real` on the output will
               recover the deprecated output format.

    Returns:
        Tensor: A tensor containing the STFT result with shape `(B?, N, T, C?)` where
           - `B?` is an optional batch dimension from the input.
           - `N` is the number of frequency samples, `(n_fft // 2) + 1` for
             `onesided=True`, or otherwise `n_fft`.
           - `T` is the number of frames, `1 + L // hop_length`
             for `center=True`, or `1 + (L - n_fft) // hop_length` otherwise.
           - `C?` is an optional length-2 dimension of real and imaginary
             components, present when `return_complex=False`.

    """
    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,
            align_to_window=align_to_window,
        )
    if center and align_to_window is not None:
        raise RuntimeError(
            "stft align_to_window should only be set when center = false"
        )
    # NOTE: Do not edit. This code will be removed once the forward-compatibility
    #       period is over for PR #73432
    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(  # type: ignore[attr-defined]
        input,
        n_fft,
        hop_length,
        win_length,
        window,
        normalized,
        onesided,
        return_complex,
        align_to_window,
    )


istft = _add_docstr(
    torch.istft,
    "istft(input, n_fft, hop_length=None, win_length=None, window=None, center=True, "
    "normalized=False, onesided=None, length=None, return_complex=False) -> Tensor:\n"
    r"""
Inverse short time Fourier Transform. This is expected to be the inverse of :func:`~torch.stft`.

.. warning::
    From version 2.1, a warning will be provided if a :attr:`window` is
    not specified. In a future release, this attribute will be required.
    Please provide the same window used in the stft call.

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 envelope
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 `length` is given in the arguments and is longer than expected,
``istft`` will pad zeros to the end of the returned signal.

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 in the format of :func:`~torch.stft`,
        output. That is a complex tensor of shape `(B?, N, T)` where

        - `B?` is an optional batch dimension
        - `N` is the number of frequency samples, `(n_fft // 2) + 1`
          for onesided input, or otherwise `n_fft`.
        - `T` is the number of frames, `1 + length // hop_length` for centered stft,
          or `1 + (length - n_fft) // hop_length` otherwise.

        .. versionchanged:: 2.0
            Real datatype inputs are no longer supported. Input must now have a
            complex datatype, as returned by ``stft(..., return_complex=True)``.
    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.
        Shape must be 1d and `<= n_fft`
        (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). Defaults to `(T - 1) * hop_length` for
        centered stft, or `n_fft + (T - 1) * hop_length` otherwise, where `T`
        is the number of input frames.
    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 shape `(B?, length)` where
        `B?` is an optional batch dimension from the input tensor.
""",
)


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"""unique(input, sorted=True, return_inverse=False, return_counts=False, dim=None) -> tuple[Tensor, Tensor, Tensor]

    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,
        `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, optional): the dimension to operate upon. If ``None``, the
            unique of the flattened input is returned. Otherwise, each of the
            tensors indexed by the given dimension is treated as one of the
            elements to apply the unique operation upon. See examples for more
            details. 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([1, 2, 3])

        >>> 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]])

        >>> a = torch.tensor([
        ...     [
        ...         [1, 1, 0, 0],
        ...         [1, 1, 0, 0],
        ...         [0, 0, 1, 1],
        ...     ],
        ...     [
        ...         [0, 0, 1, 1],
        ...         [0, 0, 1, 1],
        ...         [1, 1, 1, 1],
        ...     ],
        ...     [
        ...         [1, 1, 0, 0],
        ...         [1, 1, 0, 0],
        ...         [0, 0, 1, 1],
        ...     ],
        ... ])

        >>> # If we call `torch.unique(a, dim=0)`, each of the tensors `a[idx, :, :]`
        >>> # will be compared. We can see that `a[0, :, :]` and `a[2, :, :]` match
        >>> # each other, so one of them will be removed.
        >>> (a[0, :, :] == a[2, :, :]).all()
        tensor(True)
        >>> a_unique_dim0 = torch.unique(a, dim=0)
        >>> a_unique_dim0
        tensor([[[0, 0, 1, 1],
                 [0, 0, 1, 1],
                 [1, 1, 1, 1]],
                [[1, 1, 0, 0],
                 [1, 1, 0, 0],
                 [0, 0, 1, 1]]])

        >>> # Notice which sub-tensors from `a` match with the sub-tensors from
        >>> # `a_unique_dim0`:
        >>> (a_unique_dim0[0, :, :] == a[1, :, :]).all()
        tensor(True)
        >>> (a_unique_dim0[1, :, :] == a[0, :, :]).all()
        tensor(True)

        >>> # For `torch.unique(a, dim=1)`, each of the tensors `a[:, idx, :]` are
        >>> # compared. `a[:, 0, :]` and `a[:, 1, :]` match each other, so one of
        >>> # them will be removed.
        >>> (a[:, 0, :] == a[:, 1, :]).all()
        tensor(True)
        >>> torch.unique(a, dim=1)
        tensor([[[0, 0, 1, 1],
                 [1, 1, 0, 0]],
                [[1, 1, 1, 1],
                 [0, 0, 1, 1]],
                [[0, 0, 1, 1],
                 [1, 1, 0, 0]]])

        >>> # For `torch.unique(a, dim=2)`, the tensors `a[:, :, idx]` are compared.
        >>> # `a[:, :, 0]` and `a[:, :, 1]` match each other. Also, `a[:, :, 2]` and
        >>> # `a[:, :, 3]` match each other as well. So in this case, two of the
        >>> # sub-tensors will be removed.
        >>> (a[:, :, 0] == a[:, :, 1]).all()
        tensor(True)
        >>> (a[:, :, 2] == a[:, :, 3]).all()
        tensor(True)
        >>> torch.unique(a, dim=2)
        tensor([[[0, 1],
                 [0, 1],
                 [1, 0]],
                [[1, 0],
                 [1, 0],
                 [1, 1]],
                [[0, 1],
                 [0, 1],
                 [1, 0]]])
    """
    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(  # noqa: F811
        a,
        b,
        dims: tuple[list[int], list[int]],
        out: Optional[torch.Tensor] = None,
    ):
        pass

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

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


def tensordot(  # noqa: F811
    a,
    b,
    dims=2,
    out: Optional[torch.Tensor] = None,
):
    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.]])

        >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_CUDA)
        >>> 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, out=out)

    if not isinstance(dims, (tuple, list, torch.Tensor, int, torch.SymInt)):
        raise RuntimeError(
            "tensordot expects dims to be int or "
            + "tuple[list[int], list[int]] or "
            + "list[list[int]] containing two lists, but got "
            + f"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, torch.SymInt)):
        if dims < 0:
            raise RuntimeError(f"tensordot expects dims >= 0, but got dims={dims}")
        if dims > min(a.dim(), b.dim()):
            raise RuntimeError(
                f"tensordot expects dims < ndim_a or ndim_b, but got dims={dims}"
            )
        dims_a = list(range(-dims, 0))
        dims_b = list(range(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: Tensor) -> Tensor:
    """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::

        >>> import itertools
        >>> 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]])
    """
    # This wrapper exists to support variadic args.
    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]])
    """
    # This wrapper exists to support variadic args.
    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=2.0, 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.arange(2)
        >>> x
        tensor([0, 1])
        >>> torch.atleast_1d(x)
        tensor([0, 1])
        >>> 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.]))
    """
    # This wrapper exists to support variadic args.
    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.arange(4).view(2, 2)
        >>> x
        tensor([[0, 1],
                [2, 3]])
        >>> torch.atleast_2d(x)
        tensor([[0, 1],
                [2, 3]])
        >>> x = torch.tensor(0.5)
        >>> y = torch.tensor(1.)
        >>> torch.atleast_2d((x, y))
        (tensor([[0.5000]]), tensor([[1.]]))
    """
    # This wrapper exists to support variadic args.
    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.arange(4).view(2, 2)
        >>> y
        tensor([[0, 1],
                [2, 3]])
        >>> torch.atleast_3d(y)
        tensor([[[0],
                 [1]],
                <BLANKLINE>
                [[2],
                 [3]]])
        >>> x = torch.tensor(1).view(1, 1, 1)
        >>> x
        tensor([[[1]]])
        >>> torch.atleast_3d(x)
        tensor([[[1]]])
        >>> x = torch.tensor(0.5)
        >>> y = torch.tensor(1.0)
        >>> torch.atleast_3d((x, y))
        (tensor([[[0.5000]]]), tensor([[[1.]]]))
    """
    # This wrapper exists to support variadic args.
    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(  # noqa: F811
        input,
        p="fro",
        dim=None,
        keepdim=False,
        out=None,
        dtype=None,
    ):
        # type: (Tensor, Optional[number], Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
        pass

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

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


def norm(  # noqa: F811
    input,
    p: Optional[Union[float, str]] = "fro",
    dim=None,
    keepdim=False,
    out=None,
    dtype=None,
):
    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.
        Its documentation and behavior may be incorrect, and it is no longer
        actively maintained.

        Use :func:`torch.linalg.vector_norm` when computing vector norms and
        :func:`torch.linalg.matrix_norm` when computing matrix norms.
        For a function with a similar behavior as this one see :func:`torch.linalg.norm`.
        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.matrix_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
        )

    # NB. All the repeated code and weird python is to please TorchScript.
    #     For a more compact implementation see the relevant function in `_refs/__init__.py`

    # We don't do this for MPS or sparse tensors
    if input.layout == torch.strided and input.device.type in (
        "cpu",
        "cuda",
        "meta",
        torch.utils.backend_registration._privateuse1_backend_name,
    ):
        if dim is not None:
            if isinstance(dim, (int, torch.SymInt)):
                _dim = [dim]
            else:
                _dim = dim
        else:
            _dim = None  # type: ignore[assignment]

        if isinstance(p, str):
            if p == "fro" and (
                dim is None or isinstance(dim, (int, torch.SymInt)) or len(dim) <= 2
            ):
                if out is None:
                    return torch.linalg.vector_norm(
                        input, 2, _dim, keepdim, dtype=dtype
                    )
                else:
                    return torch.linalg.vector_norm(
                        input, 2, _dim, keepdim, dtype=dtype, out=out
                    )

            # Here we either call the nuclear norm, or we call matrix_norm with some arguments
            # that will throw an error
            if _dim is None:
                _dim = list(range(input.ndim))
            if out is None:
                return torch.linalg.matrix_norm(input, p, _dim, keepdim, dtype=dtype)
            else:
                return torch.linalg.matrix_norm(
                    input, p, _dim, keepdim, dtype=dtype, out=out
                )
        else:
            # NB. p should be Union[str, number], not Optional!
            _p = 2.0 if p is None else p
            if out is None:
                return torch.linalg.vector_norm(input, _p, _dim, keepdim, dtype=dtype)
            else:
                return torch.linalg.vector_norm(
                    input, _p, _dim, keepdim, dtype=dtype, out=out
                )

    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 = list(range(ndim))
            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, torch.SymInt)):
            _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)  # type: ignore[arg-type]
            else:
                return _VF.frobenius_norm(input, _dim, keepdim=keepdim, out=out)  # type: ignore[arg-type]
        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)  # type: ignore[arg-type]
                else:
                    return _VF.nuclear_norm(input, keepdim=keepdim, out=out)  # type: ignore[arg-type]
            else:
                if out is None:
                    return _VF.nuclear_norm(input, _dim, keepdim=keepdim)  # type: ignore[arg-type]
                else:
                    return _VF.nuclear_norm(input, _dim, keepdim=keepdim, out=out)  # type: ignore[arg-type]
        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 unravel_index(
    indices: Tensor,
    shape: Union[int, Sequence[int], torch.Size],
) -> tuple[Tensor, ...]:
    r"""Converts a tensor of flat indices into a tuple of coordinate tensors that
    index into an arbitrary tensor of the specified shape.

    Args:
        indices (Tensor): An integer tensor containing indices into the
            flattened version of an arbitrary tensor of shape :attr:`shape`.
            All elements must be in the range ``[0, prod(shape) - 1]``.

        shape (int, sequence of ints, or torch.Size): The shape of the arbitrary
            tensor. All elements must be non-negative.

    Returns:
        tuple of Tensors: Each ``i``-th tensor in the output corresponds with
        dimension ``i`` of :attr:`shape`. Each tensor has the same shape as
        ``indices`` and contains one index into dimension ``i`` for each of the
        flat indices given by ``indices``.

    Example::

        >>> import torch
        >>> torch.unravel_index(torch.tensor(4), (3, 2))
        (tensor(2),
         tensor(0))

        >>> torch.unravel_index(torch.tensor([4, 1]), (3, 2))
        (tensor([2, 0]),
         tensor([0, 1]))

        >>> torch.unravel_index(torch.tensor([0, 1, 2, 3, 4, 5]), (3, 2))
        (tensor([0, 0, 1, 1, 2, 2]),
         tensor([0, 1, 0, 1, 0, 1]))

        >>> torch.unravel_index(torch.tensor([1234, 5678]), (10, 10, 10, 10))
        (tensor([1, 5]),
         tensor([2, 6]),
         tensor([3, 7]),
         tensor([4, 8]))

        >>> torch.unravel_index(torch.tensor([[1234], [5678]]), (10, 10, 10, 10))
        (tensor([[1], [5]]),
         tensor([[2], [6]]),
         tensor([[3], [7]]),
         tensor([[4], [8]]))

        >>> torch.unravel_index(torch.tensor([[1234], [5678]]), (100, 100))
        (tensor([[12], [56]]),
         tensor([[34], [78]]))
    """
    if has_torch_function_unary(indices):
        return handle_torch_function(unravel_index, (indices,), indices, shape=shape)
    res_tensor = _unravel_index(indices, shape)
    return res_tensor.unbind(-1)


def _unravel_index(indices: Tensor, shape: Union[int, Sequence[int]]) -> Tensor:
    torch._check_type(
        not indices.is_complex()
        and not indices.is_floating_point()
        and not indices.dtype == torch.bool,
        lambda: f"expected 'indices' to be integer dtype, but got {indices.dtype}",
    )

    torch._check_type(
        isinstance(shape, (int, torch.SymInt, Sequence)),
        lambda: f"expected 'shape' to be int or sequence of ints, but got {type(shape)}",
    )

    if isinstance(shape, (int, torch.SymInt)):
        shape = torch.Size([shape])
    else:
        for dim in shape:
            torch._check_type(
                isinstance(dim, (int, torch.SymInt)),
                lambda: f"expected 'shape' sequence to only contain ints, but got {type(dim)}",
            )
        shape = torch.Size(shape)

    torch._check_value(
        all(dim >= 0 for dim in shape),
        lambda: f"'shape' cannot have negative values, but got {tuple(shape)}",
    )

    coefs = list(
        reversed(
            list(
                itertools.accumulate(
                    reversed(shape[1:] + torch.Size([1])), func=operator.mul
                )
            )
        )
    )
    return indices.unsqueeze(-1).floor_divide(
        torch.tensor(coefs, device=indices.device, dtype=torch.int64)
    ) % torch.tensor(shape, device=indices.device, dtype=torch.int64)


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::

        >>> # xdoctest: +SKIP
        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> a = torch.randn(3, 4)
        >>> b = torch.randn(4, 5)
        >>> c = torch.randn(5, 6)
        >>> d = torch.randn(6, 7)
        >>> # will raise a deprecation warning
        >>> 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
    """
    # This wrapper exists to support variadic args.
    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``.

    .. warning::

        :func:`torch.lu` is deprecated in favor of :func:`torch.linalg.lu_factor`
        and :func:`torch.linalg.lu_factor_ex`. :func:`torch.lu` will be removed in a
        future PyTorch release.
        ``LU, pivots, info = torch.lu(A, compute_pivots)`` should be replaced with

        .. code:: python

            LU, pivots = torch.linalg.lu_factor(A, compute_pivots)

        ``LU, pivots, info = torch.lu(A, compute_pivots, get_infos=True)`` should be replaced with

        .. code:: python

            LU, pivots, info = torch.linalg.lu_factor_ex(A, compute_pivots)

    .. note::
        * The returned permutation matrix for every matrix in the batch is
          represented by a 1-indexed vector of size ``min(A.shape[-2], A.shape[-1])``.
          ``pivots[i] == j`` represents that in the ``i``-th step of the algorithm,
          the ``i``-th row was permuted with the ``j-1``-th row.
        * 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.
        * 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.
        * 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).
        * ``L``, ``U``, and ``P`` can be derived using :func:`torch.lu_unpack`.

    .. warning::
        The gradients of this function will only be finite when :attr:`A` is full rank.
        This is because the LU decomposition is just differentiable at full rank matrices.
        Furthermore, if :attr:`A` is close to not being full rank,
        the gradient will be numerically unstable as it depends on the computation of :math:`L^{-1}` and :math:`U^{-1}`.

    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::

        >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
        >>> # xdoctest: +IGNORE_WANT("non-deterministic")
        >>> 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))


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.")