[opt_transport] Replace np.sum(a * b) with np.vdot(a, b)

[suda-yuga]

Replace np.sum(a * b) with np.vdot(a, b)
#463

[HumphreyYang]

Hi @suda-yuga,

Thanks so much for your PR! This is a nice application of vdot. I am curious would np.sum(a*b) be more transparent than np.vdot(a, b) in this case as we are summing over a 2-D matrix?

[suda-yuga]

Hi @suda-yuga,

Thanks so much for your PR! This is a nice application of vdot. I am curious would np.sum(a*b) be more transparent than np.vdot(a, b) in this case as we are summing over a 2-D matrix?

Thank you for your comment.

You're absolutely right — in this case, np.sum(a * b) is arguably more transparent and easier to read when working with 2-D matrices.

The motivation behind this change was based on prior discussions (#463) suggesting that a @ b or np.vdot(a, b) can be more efficient than np.sum(a * b) due to avoiding intermediate arrays and leveraging BLAS-level optimizations. I applied the same reasoning to the Frobenius inner product of matrices here.

That said, I agree with you — clarity matters, and np.sum(a * b) may better communicate the intent in this specific context. I'm happy to revert or revise the change depending on what's preferred for readability.

[HumphreyYang]

Thanks @suda-yuga! I'm happy either way. I just feel that it's less obvious to readers what we're trying to compute here, unlike the more straightforward cases where we have two 1-D arrays. For the sake of presentation, I'm inclined to close this PR—though I really like what you've done!

Hi @mmcky, I will leave it to you to make the call!

[oyamad]

Maybe write something like "Here we use np.vdot (with a link to https://numpy.org/doc/stable/reference/generated/numpy.vdot.html) (instead of np.dot or @), where X and C are 2-D arrays"?

[oyamad]

Or "Here we use np.vdot for the trace inner product of X and C".
(The notation "tr (C' X)" is already introduced (with no explanation).)

[mmcky]

Thanks @oyamad for the suggestion.

I think being explicit around the use of vdot is a good idea, specifying what it achieves. I have added your second suggestion with a link to the vdot documentation.

[mmcky]

thanks @HumphreyYang and @oyamad for your valuable feedback and review.

Thanks @suda-yuga for the PR.