Xarray: Support alignment/broadcasting with unlabeled dimensions of size 1

[shoyer]

Sometimes, it's convenient to include placeholder dimensions of size 1, which allows for removing any ambiguity related to the order of output dimensions.

Currently, this is not supported with xarray:

>>> xr.DataArray([1], dims='x') + xr.DataArray([1, 2, 3], dims='x')
ValueError: arguments without labels along dimension 'x' cannot be aligned because they have different dimension sizes: {1, 3}

>>> xr.Variable(('x',), [1]) + xr.Variable(('x',), [1, 2, 3])
ValueError: operands cannot be broadcast together with mismatched lengths for dimension 'x': (1, 3)

However, these operations aren't really ambiguous. With size 1 dimensions, we could logically do broadcasting like NumPy arrays, e.g.,

>>> np.array([1]) + np.array([1, 2, 3])
array([2, 3, 4])

This would be particularly convenient if we add keepdims=True to xarray operations (#2170).

[ambrosejcarr]

We'd really like to be able to do this for broadcastable matrix multiplication. See e.g. https://github.com/spacetx/starfish/pull/983

Do you have a sense of how this might be implemented, or a suggested work around in the mean time?

[shoyer]

I think there are three places where you would need to fix this:

Index based alignment in align():
https://github.com/pydata/xarray/blob/6d2076688d4f5466cf77ace2b196e910c1c0fbb8/xarray/core/alignment.py#L143

Unified dimensions for broadcasting variables, which is defined in two places:
https://github.com/pydata/xarray/blob/6d2076688d4f5466cf77ace2b196e910c1c0fbb8/xarray/core/variable.py#L1978
https://github.com/pydata/xarray/blob/6d2076688d4f5466cf77ace2b196e910c1c0fbb8/xarray/core/computation.py#L441

The easiest work around is probably to call .squeeze() on any objects with size 1 dimensions -- then xarray's broadcasting/alignment should all work fine.