Python / NumPy 中的多维广播 - 或“numpy.squeeze()”的逆

Question

当对np.broadcast_to() 的简单调用失败时，将两个数组一起广播的最佳方式是什么？

考虑以下示例：

import numpy as np

arr1 = np.arange(2 * 3 * 4 * 5 * 6).reshape((2, 3, 4, 5, 6))
arr2 = np.arange(3 * 5).reshape((3, 5))

arr1 + arr2
# ValueError: operands could not be broadcast together with shapes (2,3,4,5,6) (3,5)

arr2_ = np.broadcast_to(arr2, arr1.shape)
# ValueError: operands could not be broadcast together with remapped shapes

arr2_ = arr2.reshape((1, 3, 1, 5, 1))
arr1 + arr2
# now this works because the singletons trigger the automatic broadcast

这仅在我手动选择要自动广播的形状时才有效。 自动执行此操作的最有效方法是什么？ 除了在巧妙构造的 broadcastable 形状上重塑之外，还有其他方法吗？

注意与np.squeeze() 的关系：这将通过删除单例来执行逆运算。所以我需要的是某种np.squeeze() 逆。 官方 documentation（截至 NumPy 1.13.0 表明 np.squeeze() 的倒数是 np.expand_dim()，但这并不像我需要的那样灵活，实际上np.expand_dim() 大致相当于np.reshape(array, shape + (1,)) 或array[:, None]。

此问题也与例如接受的 keepdims 关键字有关。 sum:

import numpy as np

arr1 = np.arange(2 * 3 * 4 * 5 * 6).reshape((2, 3, 4, 5, 6))

# not using `keepdims`
arr2 = np.sum(arr1, (0, 2, 4))
arr2.shape
# : (3, 5)

arr1 + arr2
# ValueError: operands could not be broadcast together with shapes (2,3,4,5,6) (3,5)


# now using `keepdims`
arr2 = np.sum(arr1, (0, 2, 4), keepdims=True)
arr2.shape
# : (1, 3, 1, 5, 1)

arr1 + arr2
# now this works because it has the correct shape

编辑：显然，如果np.newaxis 或keepdims 机制是合适的选择，则不需要unsqueeze() 函数。

然而，在某些用例中，这些都不是一个选项。

例如，考虑在numpy.average() 中对axis 指定的任意数量的维度进行加权平均的情况。 现在weights 参数必须与输入具有相同的形状。 但是，weights 不需要指定非缩减维度的权重，因为它们只是重复，NumPy 的广播机制会适当地处理它们。

因此，如果我们想拥有这样的功能，我们需要编写类似的代码（为了简单起见，省略了一些一致性检查）：

def weighted_average(arr, weights=None, axis=None):
    if weights is not None and weights.shape != arr.shape:
        weights = unsqueeze(weights, ...)
        weights = np.zeros_like(arr) + weights
    result = np.sum(arr * weights, axis=axis)
    result /= np.sum(weights, axis=axis)
    return result

或者，等效地：

def weighted_average(arr, weights=None, axis=None):
    if weights is not None and weights.shape != arr.shape:
        weights = unsqueeze(weights, ...)
        weights = np.zeros_like(arr) + weights
    return np.average(arr, weights, axis)

在两者中的任何一个中，都不能用类似weights[:, np.newaxis] 的语句替换unsqueeze()，因为我们事先不知道哪里需要新轴，也不能使用@ 的keepdims 功能987654347@ 因为代码将在arr * weights 失败。

如果np.expand_dims() 支持其axis 参数的整数迭代，则这种情况可以相对较好地处理，但从NumPy 1.13.0 开始不支持。

Answer 1

我实现这一点的方法是定义以下 unsqueezing() 函数来处理可以自动完成的情况，并在输入可能不明确时发出警告（例如，当源形状的某些源元素可能匹配多个元素时目标形状）：

def unsqueezing(
        source_shape,
        target_shape):
    """
    Generate a broadcasting-compatible shape.

    The resulting shape contains *singletons* (i.e. `1`) for non-matching dims.
    Assumes all elements of the source shape are contained in the target shape
    (excepts for singletons) in the correct order.

    Warning! The generated shape may not be unique if some of the elements
    from the source shape are present multiple timesin the target shape.

    Args:
        source_shape (Sequence): The source shape.
        target_shape (Sequence): The target shape.

    Returns:
        shape (tuple): The broadcast-safe shape.

    Raises:
        ValueError: if elements of `source_shape` are not in `target_shape`.

    Examples:
        For non-repeating elements, `unsqueezing()` is always well-defined:

        >>> unsqueezing((2, 3), (2, 3, 4))
        (2, 3, 1)
        >>> unsqueezing((3, 4), (2, 3, 4))
        (1, 3, 4)
        >>> unsqueezing((3, 5), (2, 3, 4, 5, 6))
        (1, 3, 1, 5, 1)
        >>> unsqueezing((1, 3, 5, 1), (2, 3, 4, 5, 6))
        (1, 3, 1, 5, 1)

        If there is nothing to unsqueeze, the `source_shape` is returned:

        >>> unsqueezing((1, 3, 1, 5, 1), (2, 3, 4, 5, 6))
        (1, 3, 1, 5, 1)
        >>> unsqueezing((2, 3), (2, 3))
        (2, 3)

        If some elements in `source_shape` are repeating in `target_shape`,
        a user warning will be issued:

        >>> unsqueezing((2, 2), (2, 2, 2, 2, 2))
        (2, 2, 1, 1, 1)
        >>> unsqueezing((2, 2), (2, 3, 2, 2, 2))
        (2, 1, 2, 1, 1)

        If some elements of `source_shape` are not presente in `target_shape`,
        an error is raised.

        >>> unsqueezing((2, 3), (2, 2, 2, 2, 2))
        Traceback (most recent call last):
          ...
        ValueError: Target shape must contain all source shape elements\
 (in correct order). (2, 3) -> (2, 2, 2, 2, 2)
        >>> unsqueezing((5, 3), (2, 3, 4, 5, 6))
        Traceback (most recent call last):
          ...
        ValueError: Target shape must contain all source shape elements\
 (in correct order). (5, 3) -> (2, 3, 4, 5, 6)

    """
    shape = []
    j = 0
    for i, dim in enumerate(target_shape):
        if j < len(source_shape):
            shape.append(dim if dim == source_shape[j] else 1)
            if i + 1 < len(target_shape) and dim == source_shape[j] \
                    and dim != 1 and dim in target_shape[i + 1:]:
                text = ('Multiple positions (e.g. {} and {})'
                        ' for source shape element {}.'.format(
                    i, target_shape[i + 1:].index(dim) + (i + 1), dim))
                warnings.warn(text)
            if dim == source_shape[j] or source_shape[j] == 1:
                j += 1
        else:
            shape.append(1)
    if j < len(source_shape):
        raise ValueError(
            'Target shape must contain all source shape elements'
            ' (in correct order). {} -> {}'.format(source_shape, target_shape))
    return tuple(shape)

这可用于将unsqueeze() 定义为np.squeeze() 的更灵活的逆函数，而np.expand_dims() 一次只能附加一个单例：

def unsqueeze(
        arr,
        axis=None,
        shape=None,
        reverse=False):
    """
    Add singletons to the shape of an array to broadcast-match a given shape.

    In some sense, this function implements the inverse of `numpy.squeeze()`.


    Args:
        arr (np.ndarray): The input array.
        axis (int|Iterable|None): Axis or axes in which to operate.
            If None, a valid set axis is generated from `shape` when this is
            defined and the shape can be matched by `unsqueezing()`.
            If int or Iterable, specified how singletons are added.
            This depends on the value of `reverse`.
            If `shape` is not None, the `axis` and `shape` parameters must be
            consistent.
            Values must be in the range [-(ndim+1), ndim+1]
            At least one of `axis` and `shape` must be specified.
        shape (int|Iterable|None): The target shape.
            If None, no safety checks are performed.
            If int, this is interpreted as the number of dimensions of the
            output array.
            If Iterable, the result must be broadcastable to an array with the
            specified shape.
            If `axis` is not None, the `axis` and `shape` parameters must be
            consistent.
            At least one of `axis` and `shape` must be specified.
        reverse (bool): Interpret `axis` parameter as its complementary.
            If True, the dims of the input array are placed at the positions
            indicated by `axis`, and singletons are placed everywherelse and
            the `axis` length must be equal to the number of dimensions of the
            input array; the `shape` parameter cannot be `None`.
            If False, the singletons are added at the position(s) specified by
            `axis`.
            If `axis` is None, `reverse` has no effect.

    Returns:
        arr (np.ndarray): The reshaped array.

    Raises:
        ValueError: if the `arr` shape cannot be reshaped correctly.

    Examples:
        Let's define some input array `arr`:

        >>> arr = np.arange(2 * 3 * 4).reshape((2, 3, 4))
        >>> arr.shape
        (2, 3, 4)

        A call to `unsqueeze()` can be reversed by `np.squeeze()`:

        >>> arr_ = unsqueeze(arr, (0, 2, 4))
        >>> arr_.shape
        (1, 2, 1, 3, 1, 4)
        >>> arr = np.squeeze(arr_, (0, 2, 4))
        >>> arr.shape
        (2, 3, 4)

        The order of the axes does not matter:

        >>> arr_ = unsqueeze(arr, (0, 4, 2))
        >>> arr_.shape
        (1, 2, 1, 3, 1, 4)

        If `shape` is an int, `axis` must be consistent with it:

        >>> arr_ = unsqueeze(arr, (0, 2, 4), 6)
        >>> arr_.shape
        (1, 2, 1, 3, 1, 4)
        >>> arr_ = unsqueeze(arr, (0, 2, 4), 7)
        Traceback (most recent call last):
          ...
        ValueError: Incompatible `[0, 2, 4]` axis and `7` shape for array of\
 shape (2, 3, 4)

        It is possible to reverse the meaning to `axis` to add singletons
        everywhere except where specified (but requires `shape` to be defined
        and the length of `axis` must match the array dims):

        >>> arr_ = unsqueeze(arr, (0, 2, 4), 10, True)
        >>> arr_.shape
        (2, 1, 3, 1, 4, 1, 1, 1, 1, 1)
        >>> arr_ = unsqueeze(arr, (0, 2, 4), reverse=True)
        Traceback (most recent call last):
          ...
        ValueError: When `reverse` is True, `shape` cannot be None.
        >>> arr_ = unsqueeze(arr, (0, 2), 10, True)
        Traceback (most recent call last):
          ...
        ValueError: When `reverse` is True, the length of axis (2) must match\
 the num of dims of array (3).

        Axes values must be valid:

        >>> arr_ = unsqueeze(arr, 0)
        >>> arr_.shape
        (1, 2, 3, 4)
        >>> arr_ = unsqueeze(arr, 3)
        >>> arr_.shape
        (2, 3, 4, 1)
        >>> arr_ = unsqueeze(arr, -1)
        >>> arr_.shape
        (2, 3, 4, 1)
        >>> arr_ = unsqueeze(arr, -4)
        >>> arr_.shape
        (1, 2, 3, 4)
        >>> arr_ = unsqueeze(arr, 10)
        Traceback (most recent call last):
          ...
        ValueError: Axis (10,) out of range.

        If `shape` is specified, `axis` can be omitted (USE WITH CARE!) or its
        value is used for addiotional safety checks:

        >>> arr_ = unsqueeze(arr, shape=(2, 3, 4, 5, 6))
        >>> arr_.shape
        (2, 3, 4, 1, 1)
        >>> arr_ = unsqueeze(
        ...     arr, (3, 6, 8), (2, 5, 3, 2, 7, 2, 3, 2, 4, 5, 6), True)
        >>> arr_.shape
        (1, 1, 1, 2, 1, 1, 3, 1, 4, 1, 1)
        >>> arr_ = unsqueeze(
        ...     arr, (3, 7, 8), (2, 5, 3, 2, 7, 2, 3, 2, 4, 5, 6), True)
        Traceback (most recent call last):
          ...
        ValueError: New shape [1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1] cannot be\
 broadcasted to shape (2, 5, 3, 2, 7, 2, 3, 2, 4, 5, 6)
        >>> arr = unsqueeze(arr, shape=(2, 5, 3, 7, 2, 4, 5, 6))
        >>> arr.shape
        (2, 1, 3, 1, 1, 4, 1, 1)
        >>> arr = np.squeeze(arr)
        >>> arr.shape
        (2, 3, 4)
        >>> arr = unsqueeze(arr, shape=(5, 3, 7, 2, 4, 5, 6))
        Traceback (most recent call last):
          ...
        ValueError: Target shape must contain all source shape elements\
 (in correct order). (2, 3, 4) -> (5, 3, 7, 2, 4, 5, 6)

        The behavior is consistent with other NumPy functions and the
        `keepdims` mechanism:

        >>> axis = (0, 2, 4)
        >>> arr1 = np.arange(2 * 3 * 4 * 5 * 6).reshape((2, 3, 4, 5, 6))
        >>> arr2 = np.sum(arr1, axis, keepdims=True)
        >>> arr2.shape
        (1, 3, 1, 5, 1)
        >>> arr3 = np.sum(arr1, axis)
        >>> arr3.shape
        (3, 5)
        >>> arr3 = unsqueeze(arr3, axis)
        >>> arr3.shape
        (1, 3, 1, 5, 1)
        >>> np.all(arr2 == arr3)
        True
    """
    # calculate `new_shape`
    if axis is None and shape is None:
        raise ValueError(
            'At least one of `axis` and `shape` parameters must be specified.')
    elif axis is None and shape is not None:
        new_shape = unsqueezing(arr.shape, shape)
    elif axis is not None:
        if isinstance(axis, int):
            axis = (axis,)
        # calculate the dim of the result
        if shape is not None:
            if isinstance(shape, int):
                ndim = shape
            else:  # shape is a sequence
                ndim = len(shape)
        elif not reverse:
            ndim = len(axis) + arr.ndim
        else:
            raise ValueError('When `reverse` is True, `shape` cannot be None.')
        # check that axis is properly constructed
        if any([ax < -ndim - 1 or ax > ndim + 1 for ax in axis]):
            raise ValueError('Axis {} out of range.'.format(axis))
        # normalize axis using `ndim`
        axis = sorted([ax % ndim for ax in axis])
        # manage reverse mode
        if reverse:
            if len(axis) == arr.ndim:
                axis = [i for i in range(ndim) if i not in axis]
            else:
                raise ValueError(
                    'When `reverse` is True, the length of axis ({})'
                    ' must match the num of dims of array ({}).'.format(
                        len(axis), arr.ndim))
        elif len(axis) + arr.ndim != ndim:
            raise ValueError(
                'Incompatible `{}` axis and `{}` shape'
                ' for array of shape {}'.format(axis, shape, arr.shape))
        # generate the new shape from axis, ndim and shape
        new_shape = []
        i, j = 0, 0
        for l in range(ndim):
            if i < len(axis) and l == axis[i] or j >= arr.ndim:
                new_shape.append(1)
                i += 1
            else:
                new_shape.append(arr.shape[j])
                j += 1

    # check that `new_shape` is consistent with `shape`
    if shape is not None:
        if isinstance(shape, int):
            if len(new_shape) != ndim:
                raise ValueError(
                    'Length of new shape {} does not match '
                    'expected length ({}).'.format(len(new_shape), ndim))
        else:
            if not all([new_dim == 1 or new_dim == dim
                        for new_dim, dim in zip(new_shape, shape)]):
                raise ValueError(
                    'New shape {} cannot be broadcasted to shape {}'.format(
                        new_shape, shape))

    return arr.reshape(new_shape)

使用这些，可以编写：

import numpy as np

arr1 = np.arange(2 * 3 * 4 * 5 * 6).reshape((2, 3, 4, 5, 6))
arr2 = np.arange(3 * 5).reshape((3, 5))

arr3 = unsqueeze(arr2, (0, 2, 4))

arr1 + arr3
# now this works because it has the correct shape

arr3 = unsqueeze(arr2, shape=arr1.shape)
arr1 + arr3
# this also works because the shape can be expanded unambiguously

所以现在可以进行动态广播了，这和keepdims的行为是一致的：

import numpy as np

axis = (0, 2, 4)
arr1 = np.arange(2 * 3 * 4 * 5 * 6).reshape((2, 3, 4, 5, 6))
arr2 = np.sum(arr1, axis, keepdims=True)
arr3 = np.sum(arr1, axis)
arr3 = unsqueeze(arr3, axis)
np.all(arr2 == arr3)
# : True

实际上，这扩展了np.expand_dims() 以处理更复杂的场景。

对这段代码的改进显然是非常受欢迎的。