[Numpy-discussion] ENH: compute many inner products quickly

[Scott Sievert]

I recently ran into an application where I had to compute many inner products quickly (roughy 50k inner products in less than a second). I wanted a vector of inner products over the 50k vectors, or `[x1.T @ A @ x1, …, xn.T @ A @ xn]` with A.shape = (1k, 1k).

My first instinct was to look for a NumPy function to quickly compute this, such as np.inner. However, it looks like np.inner has some other behavior and I couldn’t get tensordot/einsum to work for me.

Then a labmate pointed out that I can just do some slick matrix multiplication to compute the same quantity, `(X.T * A @ X.T).sum(axis=0)`. I opened [a PR] with this, and proposed that we define a new function called `inner_prods` for this.

However, in the PR, @shoyer pointed out 

> The main challenge is to figure out how to transition the behavior of all these operations, while preserving backwards compatibility. Quite likely, we need to pick new names for these functions, though we should try to pick something that doesn't suggest that they are second class alternatives.

Do we choose new function names? Do we add a keyword arg that changes what np.inner returns?

[a PR]:https://github.com/numpy/numpy/pull/7690



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