xhs-note-crawling/python/py/Lib/site-packages/einops/einops.py

import functools
import itertools
import string
import typing
from collections import OrderedDict
from typing import Any, Callable, Dict, List, Optional, Set, Tuple, TypeVar, Union, cast, overload

if typing.TYPE_CHECKING:
    # for docstrings in pycharm
    import numpy as np  # noqa E401

from . import EinopsError
from ._backends import get_backend
from .parsing import AnonymousAxis, ParsedExpression, _ellipsis

Tensor = TypeVar("Tensor")
ReductionCallable = Callable[[Tensor, Tuple[int, ...]], Tensor]
Reduction = Union[str, ReductionCallable]
Size = typing.Any

_reductions = ("min", "max", "sum", "mean", "prod", "any", "all")

# magic integers are required to stay within
# traceable subset of language
_unknown_axis_length = -999999
_expected_axis_length = -99999


def _product(sequence: List[int]) -> int:
    """minimalistic product that works both with numbers and symbols. Supports empty lists"""
    result = 1
    for element in sequence:
        result *= element
    return result


def _reduce_axes(tensor, reduction_type: Reduction, reduced_axes: List[int], backend):
    if callable(reduction_type):
        # custom callable
        return reduction_type(tensor, tuple(reduced_axes))
    else:
        # one of built-in operations
        assert reduction_type in _reductions
        if reduction_type == "mean":
            if not backend.is_float_type(tensor):
                raise NotImplementedError("reduce_mean is not available for non-floating tensors")
        return backend.reduce(tensor, reduction_type, tuple(reduced_axes))


def _optimize_transformation(init_shapes, reduced_axes, axes_reordering, final_shapes):
    # 'collapses' neighboring axes if those participate in the result pattern in the same order
    # TODO add support for added_axes
    assert len(axes_reordering) + len(reduced_axes) == len(init_shapes)
    # joining consecutive axes that will be reduced
    # possibly we can skip this if all backends can optimize this (not sure)
    reduced_axes = tuple(sorted(reduced_axes))
    for i in range(len(reduced_axes) - 1)[::-1]:
        if reduced_axes[i] + 1 == reduced_axes[i + 1]:
            removed_axis = reduced_axes[i + 1]
            removed_length = init_shapes[removed_axis]
            init_shapes = init_shapes[:removed_axis] + init_shapes[removed_axis + 1 :]
            init_shapes[removed_axis - 1] *= removed_length
            reduced_axes = reduced_axes[: i + 1] + tuple(axis - 1 for axis in reduced_axes[i + 2 :])

    # removing axes that are moved together during reshape
    def build_mapping():
        init_to_final = {}
        for axis in range(len(init_shapes)):
            if axis in reduced_axes:
                init_to_final[axis] = None
            else:
                after_reduction = sum(x is not None for x in init_to_final.values())
                init_to_final[axis] = list(axes_reordering).index(after_reduction)
        return init_to_final

    init_axis_to_final_axis = build_mapping()

    for init_axis in range(len(init_shapes) - 1)[::-1]:
        if init_axis_to_final_axis[init_axis] is None:
            continue
        if init_axis_to_final_axis[init_axis + 1] is None:
            continue
        if init_axis_to_final_axis[init_axis] + 1 == init_axis_to_final_axis[init_axis + 1]:
            removed_axis = init_axis + 1
            removed_length = init_shapes[removed_axis]
            removed_axis_after_reduction = sum(x not in reduced_axes for x in range(removed_axis))

            reduced_axes = tuple(axis if axis < removed_axis else axis - 1 for axis in reduced_axes)
            init_shapes = init_shapes[:removed_axis] + init_shapes[removed_axis + 1 :]
            init_shapes[removed_axis - 1] *= removed_length
            old_reordering = axes_reordering
            axes_reordering = []
            for axis in old_reordering:
                if axis == removed_axis_after_reduction:
                    pass
                elif axis < removed_axis_after_reduction:
                    axes_reordering.append(axis)
                else:
                    axes_reordering.append(axis - 1)
            init_axis_to_final_axis = build_mapping()

    return init_shapes, reduced_axes, axes_reordering, final_shapes


CookedRecipe = Tuple[Optional[List[int]], Optional[List[int]], List[int], Dict[int, int], Optional[List[int]], int]

# Actual type is tuple[tuple[str, int], ...]
# However torch.jit.script does not "understand" the correct type,
# and torch_specific will use list version.
HashableAxesLengths = Tuple[Tuple[str, int], ...]
FakeHashableAxesLengths = List[Tuple[str, int]]


class TransformRecipe:
    """
    Recipe describes actual computation pathway.
    Recipe can be applied to a tensor or variable.
    """

    # structure is non-mutable. In future, this can be non-mutable dataclass (python 3.7+)
    # update: pytorch 2.0 torch.jit.script seems to have problems with dataclasses unless they were explicitly provided

    def __init__(
        self,
        # list of sizes (or just sizes) for elementary axes as they appear in left expression.
        # this is what (after computing unknown parts) will be a shape after first transposition.
        # This does not include any ellipsis dimensions.
        elementary_axes_lengths: List[int],
        # if additional axes are provided, they should be set in prev array
        # This shows mapping from name to position
        axis_name2elementary_axis: Dict[str, int],
        # each dimension in input can help to reconstruct length of one elementary axis
        # or verify one of dimensions. Each element points to element of elementary_axes_lengths.
        input_composition_known_unknown: List[Tuple[List[int], List[int]]],
        # permutation applied to elementary axes, if ellipsis is absent
        axes_permutation: List[int],
        # permutation puts reduced axes in the end, we only need to know the first position.
        first_reduced_axis: int,
        # at which positions which of elementary axes should appear. Axis position -> axis index.
        added_axes: Dict[int, int],
        # ids of axes as they appear in result, again pointers to elementary_axes_lengths,
        # only used to infer result dimensions
        output_composite_axes: List[List[int]],
    ):
        self.elementary_axes_lengths: List[int] = elementary_axes_lengths
        self.axis_name2elementary_axis: Dict[str, int] = axis_name2elementary_axis
        self.input_composition_known_unknown: List[Tuple[List[int], List[int]]] = input_composition_known_unknown
        self.axes_permutation: List[int] = axes_permutation

        self.first_reduced_axis: int = first_reduced_axis
        self.added_axes: Dict[int, int] = added_axes
        self.output_composite_axes: List[List[int]] = output_composite_axes


def _reconstruct_from_shape_uncached(
    self: TransformRecipe, shape: List[int], axes_dims: FakeHashableAxesLengths
) -> CookedRecipe:
    """
    Reconstruct all actual parameters using shape.
    Shape is a tuple that may contain integers, shape symbols (tf, theano) and UnknownSize (tf, previously mxnet)
    known axes can be integers or symbols, but not Nones.
    """
    # magic number
    need_init_reshape = False

    # last axis is allocated for collapsed ellipsis
    axes_lengths: List[int] = list(self.elementary_axes_lengths)
    for axis, dim in axes_dims:
        axes_lengths[self.axis_name2elementary_axis[axis]] = dim

    for input_axis, (known_axes, unknown_axes) in enumerate(self.input_composition_known_unknown):
        length = shape[input_axis]
        if len(known_axes) == 0 and len(unknown_axes) == 1:
            # shortcut for the most common case
            axes_lengths[unknown_axes[0]] = length
            continue

        known_product = 1
        for axis in known_axes:
            known_product *= axes_lengths[axis]

        if len(unknown_axes) == 0:
            if isinstance(length, int) and isinstance(known_product, int) and length != known_product:
                raise EinopsError(f"Shape mismatch, {length} != {known_product}")
        else:
            # assert len(unknown_axes) == 1, 'this is enforced when recipe is created, so commented out'
            if isinstance(length, int) and isinstance(known_product, int) and length % known_product != 0:
                raise EinopsError(f"Shape mismatch, can't divide axis of length {length} in chunks of {known_product}")

            unknown_axis = unknown_axes[0]
            inferred_length: int = length // known_product
            axes_lengths[unknown_axis] = inferred_length

        if len(known_axes) + len(unknown_axes) != 1:
            need_init_reshape = True

    # at this point all axes_lengths are computed (either have values or variables, but not Nones)

    # elementary axes are ordered as they appear in input, then all added axes
    init_shapes: Optional[List[int]] = axes_lengths[: len(self.axes_permutation)] if need_init_reshape else None

    need_final_reshape = False
    final_shapes: List[int] = []
    for grouping in self.output_composite_axes:
        lengths = [axes_lengths[elementary_axis] for elementary_axis in grouping]
        final_shapes.append(_product(lengths))
        if len(lengths) != 1:
            need_final_reshape = True

    added_axes: Dict[int, int] = {
        pos: axes_lengths[pos_in_elementary] for pos, pos_in_elementary in self.added_axes.items()
    }

    # this list can be empty
    reduced_axes = list(range(self.first_reduced_axis, len(self.axes_permutation)))

    n_axes_after_adding_axes = len(added_axes) + len(self.axes_permutation)

    axes_reordering: Optional[List[int]] = self.axes_permutation
    if self.axes_permutation == list(range(len(self.axes_permutation))):
        axes_reordering = None

    _final_shapes = final_shapes if need_final_reshape else None
    return init_shapes, axes_reordering, reduced_axes, added_axes, _final_shapes, n_axes_after_adding_axes


_reconstruct_from_shape = functools.lru_cache(1024)(_reconstruct_from_shape_uncached)


def _apply_recipe(
    backend, recipe: TransformRecipe, tensor: Tensor, reduction_type: Reduction, axes_lengths: HashableAxesLengths
) -> Tensor:
    # this method implements actual work for all backends for 3 operations
    try:
        init_shapes, axes_reordering, reduced_axes, added_axes, final_shapes, n_axes_w_added = _reconstruct_from_shape(
            recipe, backend.shape(tensor), axes_lengths
        )
    except TypeError:
        # shape or one of passed axes lengths is not hashable (i.e. they are symbols)
        _result = _reconstruct_from_shape_uncached(recipe, backend.shape(tensor), axes_lengths)
        (init_shapes, axes_reordering, reduced_axes, added_axes, final_shapes, n_axes_w_added) = _result
    if init_shapes is not None:
        tensor = backend.reshape(tensor, init_shapes)
    if axes_reordering is not None:
        tensor = backend.transpose(tensor, axes_reordering)
    if len(reduced_axes) > 0:
        tensor = _reduce_axes(tensor, reduction_type=reduction_type, reduced_axes=reduced_axes, backend=backend)
    if len(added_axes) > 0:
        tensor = backend.add_axes(tensor, n_axes=n_axes_w_added, pos2len=added_axes)
    if final_shapes is not None:
        tensor = backend.reshape(tensor, final_shapes)
    return tensor


def _apply_recipe_array_api(
    xp, recipe: TransformRecipe, tensor: Tensor, reduction_type: Reduction, axes_lengths: HashableAxesLengths
) -> Tensor:
    # completely-inline implementation
    init_shapes, axes_reordering, reduced_axes, added_axes, final_shapes, n_axes_w_added = _reconstruct_from_shape(
        recipe, tensor.shape, axes_lengths
    )
    if init_shapes is not None:
        tensor = xp.reshape(tensor, init_shapes)
    if axes_reordering is not None:
        tensor = xp.permute_dims(tensor, axes_reordering)
    if len(reduced_axes) > 0:
        if callable(reduction_type):
            # custom callable
            tensor = reduction_type(tensor, tuple(reduced_axes))
        else:
            # one of built-in operations
            assert reduction_type in _reductions
            tensor = getattr(xp, reduction_type)(tensor, axis=tuple(reduced_axes))
    if len(added_axes) > 0:
        # we use broadcasting
        for axis_position, _axis_length in added_axes.items():
            tensor = xp.expand_dims(tensor, axis=axis_position)

        final_shape = list(tensor.shape)
        for axis_position, axis_length in added_axes.items():
            final_shape[axis_position] = axis_length

        tensor = xp.broadcast_to(tensor, final_shape)
    if final_shapes is not None:
        tensor = xp.reshape(tensor, final_shapes)
    return tensor


@functools.lru_cache(256)
def _prepare_transformation_recipe(
    pattern: str,
    operation: Reduction,
    axes_names: Tuple[str, ...],
    ndim: int,
) -> TransformRecipe:
    """Perform initial parsing of pattern and provided supplementary info
    axes_lengths is a tuple of tuples (axis_name, axis_length)
    """
    left_str, rght_str = pattern.split("->")
    left = ParsedExpression(left_str)
    rght = ParsedExpression(rght_str)

    # checking that axes are in agreement - new axes appear only in repeat, while disappear only in reduction
    if not left.has_ellipsis and rght.has_ellipsis:
        raise EinopsError(f"Ellipsis found in right side, but not left side of a pattern {pattern}")
    if left.has_ellipsis and left.has_ellipsis_parenthesized:
        raise EinopsError(f"Ellipsis inside parenthesis in the left side is not allowed: {pattern}")
    if operation == "rearrange":
        if left.has_non_unitary_anonymous_axes or rght.has_non_unitary_anonymous_axes:
            raise EinopsError("Non-unitary anonymous axes are not supported in rearrange (exception is length 1)")
        difference = set.symmetric_difference(left.identifiers, rght.identifiers)
        if len(difference) > 0:
            raise EinopsError(f"Identifiers only on one side of expression (should be on both): {difference}")
    elif operation == "repeat":
        difference = set.difference(left.identifiers, rght.identifiers)
        if len(difference) > 0:
            raise EinopsError(f"Unexpected identifiers on the left side of repeat: {difference}")
        axes_without_size = set.difference(
            {ax for ax in rght.identifiers if not isinstance(ax, AnonymousAxis)},
            {*left.identifiers, *axes_names},
        )
        if len(axes_without_size) > 0:
            raise EinopsError(f"Specify sizes for new axes in repeat: {axes_without_size}")
    elif operation in _reductions or callable(operation):
        difference = set.difference(rght.identifiers, left.identifiers)
        if len(difference) > 0:
            raise EinopsError(f"Unexpected identifiers on the right side of reduce {operation}: {difference}")
    else:
        raise EinopsError(f"Unknown reduction {operation}. Expect one of {_reductions}.")

    if left.has_ellipsis:
        n_other_dims = len(left.composition) - 1
        if ndim < n_other_dims:
            raise EinopsError(f"Wrong shape: expected >={n_other_dims} dims. Received {ndim}-dim tensor.")
        ellipsis_ndim = ndim - n_other_dims
        ell_axes = [_ellipsis + str(i) for i in range(ellipsis_ndim)]
        left_composition = []
        for composite_axis in left.composition:
            if composite_axis == _ellipsis:
                for axis in ell_axes:
                    left_composition.append([axis])
            else:
                left_composition.append(composite_axis)

        rght_composition = []
        for composite_axis in rght.composition:
            if composite_axis == _ellipsis:
                for axis in ell_axes:
                    rght_composition.append([axis])
            else:
                group = []
                for axis in composite_axis:
                    if axis == _ellipsis:
                        group.extend(ell_axes)
                    else:
                        group.append(axis)
                rght_composition.append(group)

        left.identifiers.update(ell_axes)
        left.identifiers.remove(_ellipsis)
        if rght.has_ellipsis:
            rght.identifiers.update(ell_axes)
            rght.identifiers.remove(_ellipsis)
    else:
        if ndim != len(left.composition):
            raise EinopsError(f"Wrong shape: expected {len(left.composition)} dims. Received {ndim}-dim tensor.")
        left_composition = left.composition
        rght_composition = rght.composition

    # parsing all dimensions to find out lengths
    axis_name2known_length: Dict[Union[str, AnonymousAxis], int] = OrderedDict()
    for composite_axis in left_composition:
        for axis_name in composite_axis:
            if isinstance(axis_name, AnonymousAxis):
                axis_name2known_length[axis_name] = axis_name.value
            else:
                axis_name2known_length[axis_name] = _unknown_axis_length

    # axis_ids_after_first_reshape = range(len(axis_name2known_length)) at this point

    repeat_axes_names = []
    for axis_name in rght.identifiers:
        if axis_name not in axis_name2known_length:
            if isinstance(axis_name, AnonymousAxis):
                axis_name2known_length[axis_name] = axis_name.value
            else:
                axis_name2known_length[axis_name] = _unknown_axis_length
            repeat_axes_names.append(axis_name)

    axis_name2position = {name: position for position, name in enumerate(axis_name2known_length)}

    # axes provided as kwargs
    for elementary_axis in axes_names:
        if not ParsedExpression.check_axis_name(elementary_axis):
            raise EinopsError("Invalid name for an axis", elementary_axis)
        if elementary_axis not in axis_name2known_length:
            raise EinopsError(f"Axis {elementary_axis} is not used in transform")
        axis_name2known_length[elementary_axis] = _expected_axis_length

    input_axes_known_unknown = []
    # some shapes are inferred later - all information is prepared for faster inference
    for composite_axis in left_composition:
        known: Set[str] = {axis for axis in composite_axis if axis_name2known_length[axis] != _unknown_axis_length}
        unknown: Set[str] = {axis for axis in composite_axis if axis_name2known_length[axis] == _unknown_axis_length}
        if len(unknown) > 1:
            raise EinopsError(f"Could not infer sizes for {unknown}")
        assert len(unknown) + len(known) == len(composite_axis)
        input_axes_known_unknown.append(
            ([axis_name2position[axis] for axis in known], [axis_name2position[axis] for axis in unknown])
        )

    axis_position_after_reduction: Dict[str, int] = {}
    for axis_name in itertools.chain(*left_composition):
        if axis_name in rght.identifiers:
            axis_position_after_reduction[axis_name] = len(axis_position_after_reduction)

    result_axes_grouping: List[List[int]] = [
        [axis_name2position[axis] for axis in composite_axis] for i, composite_axis in enumerate(rght_composition)
    ]

    ordered_axis_left = list(itertools.chain(*left_composition))
    ordered_axis_rght = list(itertools.chain(*rght_composition))
    reduced_axes = [axis for axis in ordered_axis_left if axis not in rght.identifiers]
    order_after_transposition = [axis for axis in ordered_axis_rght if axis in left.identifiers] + reduced_axes
    axes_permutation = [ordered_axis_left.index(axis) for axis in order_after_transposition]
    added_axes = {
        i: axis_name2position[axis_name]
        for i, axis_name in enumerate(ordered_axis_rght)
        if axis_name not in left.identifiers
    }

    first_reduced_axis = len(order_after_transposition) - len(reduced_axes)

    return TransformRecipe(
        elementary_axes_lengths=list(axis_name2known_length.values()),
        axis_name2elementary_axis={axis: axis_name2position[axis] for axis in axes_names},
        input_composition_known_unknown=input_axes_known_unknown,
        axes_permutation=axes_permutation,
        first_reduced_axis=first_reduced_axis,
        added_axes=added_axes,
        output_composite_axes=result_axes_grouping,
    )


def _prepare_recipes_for_all_dims(
    pattern: str, operation: Reduction, axes_names: Tuple[str, ...]
) -> Dict[int, TransformRecipe]:
    """
    Internal function, used in layers.
    Layer makes all recipe creation when it is initialized, thus to keep recipes simple we pre-compute for all dims
    """
    left_str, rght_str = pattern.split("->")
    left = ParsedExpression(left_str)
    dims = [len(left.composition)]
    if left.has_ellipsis:
        dims = [len(left.composition) - 1 + ellipsis_dims for ellipsis_dims in range(8)]
    return {ndim: _prepare_transformation_recipe(pattern, operation, axes_names, ndim=ndim) for ndim in dims}


@overload
def reduce(tensor: List[Tensor], pattern: str, reduction: Reduction, **axes_lengths: Size) -> Tensor: ...


@overload
def reduce(tensor: Tensor, pattern: str, reduction: Reduction, **axes_lengths: Size) -> Tensor: ...


def reduce(tensor: Union[Tensor, List[Tensor]], pattern: str, reduction: Reduction, **axes_lengths: Size) -> Tensor:
    """
    einops.reduce combines rearrangement and reduction using reader-friendly notation.

    Some examples:

    ```python
    >>> x = np.random.randn(100, 32, 64)

    # perform max-reduction on the first axis
    # Axis t does not appear on RHS - thus we reduced over t
    >>> y = reduce(x, 't b c -> b c', 'max')

    # same as previous, but using verbose names for axes
    >>> y = reduce(x, 'time batch channel -> batch channel', 'max')

    # let's pretend now that x is a batch of images
    # with 4 dims: batch=10, height=20, width=30, channel=40
    >>> x = np.random.randn(10, 20, 30, 40)

    # 2d max-pooling with kernel size = 2 * 2 for image processing
    >>> y1 = reduce(x, 'b c (h1 h2) (w1 w2) -> b c h1 w1', 'max', h2=2, w2=2)

    # same as previous, using anonymous axes,
    # note: only reduced axes can be anonymous
    >>> y1 = reduce(x, 'b c (h1 2) (w1 2) -> b c h1 w1', 'max')

    # adaptive 2d max-pooling to 3 * 4 grid,
    # each element is max of 10x10 tile in the original tensor.
    >>> reduce(x, 'b c (h1 h2) (w1 w2) -> b c h1 w1', 'max', h1=3, w1=4).shape
    (10, 20, 3, 4)

    # Global average pooling
    >>> reduce(x, 'b c h w -> b c', 'mean').shape
    (10, 20)

    # subtracting mean over batch for each channel;
    # similar to x - np.mean(x, axis=(0, 2, 3), keepdims=True)
    >>> y = x - reduce(x, 'b c h w -> 1 c 1 1', 'mean')

    # Subtracting per-image mean for each channel
    >>> y = x - reduce(x, 'b c h w -> b c 1 1', 'mean')

    # same as previous, but using empty compositions
    >>> y = x - reduce(x, 'b c h w -> b c () ()', 'mean')

    ```

    Parameters:
        tensor: tensor: tensor of any supported library (e.g. numpy.ndarray, tensorflow, pytorch).
            list of tensors is also accepted, those should be of the same type and shape
        pattern: string, reduction pattern
        reduction: one of available reductions ('min', 'max', 'sum', 'mean', 'prod', 'any', 'all').
            Alternatively, a callable f(tensor, reduced_axes) -> tensor can be provided.
            This allows using various reductions like: np.max, np.nanmean, tf.reduce_logsumexp, torch.var, etc.
        axes_lengths: any additional specifications for dimensions

    Returns:
        tensor of the same type as input
    """
    try:
        if isinstance(tensor, list):
            if len(tensor) == 0:
                raise TypeError("Rearrange/Reduce/Repeat can't be applied to an empty list")
            backend = get_backend(tensor[0])
            tensor = backend.stack_on_zeroth_dimension(tensor)
        else:
            backend = get_backend(tensor)

        hashable_axes_lengths = tuple(axes_lengths.items())
        shape = backend.shape(tensor)
        recipe = _prepare_transformation_recipe(pattern, reduction, axes_names=tuple(axes_lengths), ndim=len(shape))
        return _apply_recipe(
            backend, recipe, cast(Tensor, tensor), reduction_type=reduction, axes_lengths=hashable_axes_lengths
        )
    except EinopsError as e:
        message = f' Error while processing {reduction}-reduction pattern "{pattern}".'
        if not isinstance(tensor, list):
            message += f"\n Input tensor shape: {shape}. "
        else:
            message += "\n Input is list. "
        message += f"Additional info: {axes_lengths}."
        raise EinopsError(message + f"\n {e}") from None


@overload
def rearrange(tensor: List[Tensor], pattern: str, **axes_lengths: Size) -> Tensor: ...


@overload
def rearrange(tensor: Tensor, pattern: str, **axes_lengths: Size) -> Tensor: ...


def rearrange(tensor: Union[Tensor, List[Tensor]], pattern: str, **axes_lengths: Size) -> Tensor:
    """
    einops.rearrange is a reader-friendly smart element reordering for multidimensional tensors.
    This operation includes functionality of transpose (axes permutation), reshape (view), squeeze, unsqueeze,
    stack, concatenate and other operations.

    Examples:

    ```python
    # suppose we have a set of 32 images in "h w c" format (height-width-channel)
    >>> images = [np.random.randn(30, 40, 3) for _ in range(32)]

    # stack along first (batch) axis, output is a single array
    >>> rearrange(images, 'b h w c -> b h w c').shape
    (32, 30, 40, 3)

    # stacked and reordered axes to "b c h w" format
    >>> rearrange(images, 'b h w c -> b c h w').shape
    (32, 3, 30, 40)

    # concatenate images along height (vertical axis), 960 = 32 * 30
    >>> rearrange(images, 'b h w c -> (b h) w c').shape
    (960, 40, 3)

    # concatenated images along horizontal axis, 1280 = 32 * 40
    >>> rearrange(images, 'b h w c -> h (b w) c').shape
    (30, 1280, 3)

    # flattened each image into a vector, 3600 = 30 * 40 * 3
    >>> rearrange(images, 'b h w c -> b (c h w)').shape
    (32, 3600)

    # split each image into 4 smaller (top-left, top-right, bottom-left, bottom-right), 128 = 32 * 2 * 2
    >>> rearrange(images, 'b (h1 h) (w1 w) c -> (b h1 w1) h w c', h1=2, w1=2).shape
    (128, 15, 20, 3)

    # space-to-depth operation
    >>> rearrange(images, 'b (h h1) (w w1) c -> b h w (c h1 w1)', h1=2, w1=2).shape
    (32, 15, 20, 12)

    ```

    When composing axes, C-order enumeration used (consecutive elements have different last axis).
    Find more examples in einops tutorial.

    Parameters:
        tensor: tensor of any supported library (e.g. numpy.ndarray, tensorflow, pytorch).
                list of tensors is also accepted, those should be of the same type and shape
        pattern: string, rearrangement pattern
        axes_lengths: any additional specifications for dimensions

    Returns:
        tensor of the same type as input. If possible, a view to the original tensor is returned.

    """
    return reduce(tensor, pattern, reduction="rearrange", **axes_lengths)


@overload
def repeat(tensor: List[Tensor], pattern: str, **axes_lengths: Size) -> Tensor: ...


@overload
def repeat(tensor: Tensor, pattern: str, **axes_lengths: Size) -> Tensor: ...


def repeat(tensor: Union[Tensor, List[Tensor]], pattern: str, **axes_lengths: Size) -> Tensor:
    """
    einops.repeat allows reordering elements and repeating them in arbitrary combinations.
    This operation includes functionality of repeat, tile, and broadcast functions.

    Examples for repeat operation:

    ```python
    # a grayscale image (of shape height x width)
    >>> image = np.random.randn(30, 40)

    # change it to RGB format by repeating in each channel
    >>> repeat(image, 'h w -> h w c', c=3).shape
    (30, 40, 3)

    # repeat image 2 times along height (vertical axis)
    >>> repeat(image, 'h w -> (repeat h) w', repeat=2).shape
    (60, 40)

    # repeat image 2 time along height and 3 times along width
    >>> repeat(image, 'h w -> (h2 h) (w3 w)', h2=2, w3=3).shape
    (60, 120)

    # convert each pixel to a small square 2x2, i.e. upsample an image by 2x
    >>> repeat(image, 'h w -> (h h2) (w w2)', h2=2, w2=2).shape
    (60, 80)

    # 'pixelate' an image first by downsampling by 2x, then upsampling
    >>> downsampled = reduce(image, '(h h2) (w w2) -> h w', 'mean', h2=2, w2=2)
    >>> repeat(downsampled, 'h w -> (h h2) (w w2)', h2=2, w2=2).shape
    (30, 40)

    ```

    When composing axes, C-order enumeration used (consecutive elements have different last axis).
    Find more examples in einops tutorial.

    Parameters:
        tensor: tensor of any supported library (e.g. numpy.ndarray, tensorflow, pytorch).
            list of tensors is also accepted, those should be of the same type and shape
        pattern: string, rearrangement pattern
        axes_lengths: any additional specifications for dimensions

    Returns:
        Tensor of the same type as input. If possible, a view to the original tensor is returned.

    """
    return reduce(tensor, pattern, reduction="repeat", **axes_lengths)


def parse_shape(x: Tensor, pattern: str) -> dict:
    """
    Parse a tensor shape to dictionary mapping axes names to their lengths.

    ```python
    # Use underscore to skip the dimension in parsing.
    >>> x = np.zeros([2, 3, 5, 7])
    >>> parse_shape(x, 'batch _ h w')
    {'batch': 2, 'h': 5, 'w': 7}

    # `parse_shape` output can be used to specify axes_lengths for other operations:
    >>> y = np.zeros([700])
    >>> rearrange(y, '(b c h w) -> b c h w', **parse_shape(x, 'b _ h w')).shape
    (2, 10, 5, 7)

    ```

    For symbolic frameworks may return symbols, not integers.

    Parameters:
        x: tensor of any supported framework
        pattern: str, space separated names for axes, underscore means skip axis

    Returns:
        dict, maps axes names to their lengths
    """
    exp = ParsedExpression(pattern, allow_underscore=True)
    shape = get_backend(x).shape(x)
    if exp.has_composed_axes():
        raise RuntimeError(f"Can't parse shape with composite axes: {pattern} {shape}")
    if len(shape) != len(exp.composition):
        if exp.has_ellipsis:
            if len(shape) < len(exp.composition) - 1:
                raise RuntimeError(f"Can't parse shape with this number of dimensions: {pattern} {shape}")
        else:
            raise RuntimeError(f"Can't parse shape with different number of dimensions: {pattern} {shape}")
    if exp.has_ellipsis:
        ellipsis_idx = exp.composition.index(_ellipsis)
        composition = (
            exp.composition[:ellipsis_idx]
            + ["_"] * (len(shape) - len(exp.composition) + 1)
            + exp.composition[ellipsis_idx + 1 :]
        )
    else:
        composition = exp.composition
    result = {}
    for axes, axis_length in zip(composition, shape):  # type: ignore
        # axes either [], or [AnonymousAxis] or ['axis_name']
        if len(axes) == 0:
            if axis_length != 1:
                raise RuntimeError(f"Length of axis is not 1: {pattern} {shape}")
        else:
            [axis] = axes
            if isinstance(axis, str):
                if axis != "_":
                    result[axis] = axis_length
            else:
                if axis.value != axis_length:
                    raise RuntimeError(f"Length of anonymous axis does not match: {pattern} {shape}")
    return result


# _enumerate_directions is not exposed in the public API
def _enumerate_directions(x):
    """
    For an n-dimensional tensor, returns tensors to enumerate each axis.
    ```python
    x = np.zeros([2, 3, 4]) # or any other tensor
    i, j, k = _enumerate_directions(x)
    result = i + 2*j + 3*k
    ```

    `result[i, j, k] = i + 2j + 3k`, and also has the same shape as result
    Works very similarly to numpy.ogrid (open indexing grid)
    """
    backend = get_backend(x)
    shape = backend.shape(x)
    result = []
    for axis_id, axis_length in enumerate(shape):
        shape = [1] * len(shape)
        shape[axis_id] = axis_length
        result.append(backend.reshape(backend.arange(0, axis_length), shape))
    return result


# to avoid importing numpy
np_ndarray = Any


def asnumpy(tensor: Tensor) -> np_ndarray:
    """
    Convert a tensor of an imperative framework (i.e. numpy/cupy/torch/jax/etc.) to `numpy.ndarray`

    Parameters:
        tensor: tensor of any known imperative framework

    Returns:
        `numpy.ndarray`, converted to numpy
    """
    return get_backend(tensor).to_numpy(tensor)


def _validate_einsum_axis_name(axis_name):
    if len(axis_name) == 0:
        raise NotImplementedError("Singleton () axes are not yet supported in einsum.")
    if len(axis_name) > 1:
        raise NotImplementedError("Shape rearrangement is not yet supported in einsum.")

    axis_name = axis_name[0]

    if isinstance(axis_name, AnonymousAxis):
        raise NotImplementedError("Anonymous axes are not yet supported in einsum.")
    if len(axis_name) == 0:
        raise RuntimeError("Encountered empty axis name in einsum.")
    if not isinstance(axis_name, str):
        raise RuntimeError("Axis name in einsum must be a string.")


@functools.lru_cache(256)
def _compactify_pattern_for_einsum(pattern: str) -> str:
    if "->" not in pattern:
        # numpy allows this, so make sure users
        # don't accidentally do something like this.
        raise ValueError("Einsum pattern must contain '->'.")
    lefts_str, right_str = pattern.split("->")

    lefts = [ParsedExpression(left, allow_underscore=True, allow_duplicates=True) for left in lefts_str.split(",")]

    right = ParsedExpression(right_str, allow_underscore=True)

    # Start from 'a' and go up to 'Z'
    output_axis_names = string.ascii_letters
    i = 0
    axis_name_mapping = {}

    left_patterns = []
    for left in lefts:
        left_pattern = ""
        for raw_axis_name in left.composition:
            if raw_axis_name == _ellipsis:
                left_pattern += "..."
                continue

            _validate_einsum_axis_name(raw_axis_name)
            axis_name = raw_axis_name[0]
            if axis_name not in axis_name_mapping:
                if i >= len(output_axis_names):
                    raise RuntimeError("Too many axes in einsum.")
                axis_name_mapping[axis_name] = output_axis_names[i]
                i += 1

            left_pattern += axis_name_mapping[axis_name]
        left_patterns.append(left_pattern)

    compact_pattern = ",".join(left_patterns) + "->"

    for raw_axis_name in right.composition:
        if raw_axis_name == _ellipsis:
            compact_pattern += "..."
            continue

        _validate_einsum_axis_name(raw_axis_name)
        axis_name = raw_axis_name[0]

        if axis_name not in axis_name_mapping:
            raise EinopsError(f"Unknown axis {axis_name} on right side of einsum {pattern}.")

        compact_pattern += axis_name_mapping[axis_name]

    return compact_pattern


@typing.overload
def einsum(tensor: Tensor, pattern: str, /) -> Tensor: ...


@typing.overload
def einsum(tensor1: Tensor, tensor2: Tensor, pattern: str, /) -> Tensor: ...


@typing.overload
def einsum(tensor1: Tensor, tensor2: Tensor, tensor3: Tensor, pattern: str, /) -> Tensor: ...


@typing.overload
def einsum(tensor1: Tensor, tensor2: Tensor, tensor3: Tensor, tensor4: Tensor, pattern: str, /) -> Tensor: ...


def einsum(*tensors_and_pattern: Union[Tensor, str]) -> Tensor:
    r"""
    einops.einsum calls einsum operations with einops-style named
    axes indexing, computing tensor products with an arbitrary
    number of tensors. Unlike typical einsum syntax, here you must
    pass tensors first, and then the pattern.

    Also, note that rearrange operations such as `"(batch chan) out"`,
    or singleton axes `()`, are not currently supported.

    Examples:

    For a given pattern such as:
    ```python
    >>> x, y, z = np.random.randn(3, 20, 20, 20)
    >>> output = einsum(x, y, z, "a b c, c b d, a g k -> a b k")

    ```
    the following formula is computed:
    ```tex
    output[a, b, k] = \sum_{c, d, g} x[a, b, c] * y[c, b, d] * z[a, g, k]
    ```
    where the summation over `c`, `d`, and `g` is performed
    because those axes names do not appear on the right-hand side.

    Let's see some additional examples:
    ```python
    # Filter a set of images:
    >>> batched_images = np.random.randn(128, 16, 16)
    >>> filters = np.random.randn(16, 16, 30)
    >>> result = einsum(batched_images, filters,
    ...                 "batch h w, h w channel -> batch channel")
    >>> result.shape
    (128, 30)

    # Matrix multiplication, with an unknown input shape:
    >>> batch_shape = (50, 30)
    >>> data = np.random.randn(*batch_shape, 20)
    >>> weights = np.random.randn(10, 20)
    >>> result = einsum(weights, data,
    ...                 "out_dim in_dim, ... in_dim -> ... out_dim")
    >>> result.shape
    (50, 30, 10)

    # Matrix trace on a single tensor:
    >>> matrix = np.random.randn(10, 10)
    >>> result = einsum(matrix, "i i ->")
    >>> result.shape
    ()

    ```

    Parameters:
        tensors_and_pattern:
            tensors: tensors of any supported library (numpy, tensorflow, pytorch, jax).
            pattern: string, einsum pattern, with commas
                separating specifications for each tensor.
                pattern should be provided after all tensors.

    Returns:
        Tensor of the same type as input, after processing with einsum.

    """
    if len(tensors_and_pattern) <= 1:
        raise ValueError(
            "`einops.einsum` takes at minimum two arguments: the tensors (at least one), followed by the pattern."
        )
    pattern = tensors_and_pattern[-1]
    if not isinstance(pattern, str):
        raise ValueError(
            "The last argument passed to `einops.einsum` must be a string, representing the einsum pattern."
        )
    tensors = tensors_and_pattern[:-1]
    pattern = _compactify_pattern_for_einsum(pattern)
    return get_backend(tensors[0]).einsum(pattern, *tensors)