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Instead of sums of multiplications you could for example use sum of squares of differences.

Means squared error instead of dot product, it's not cheaper but it's close

If you want to go cheaper you could use sum of abs of differences.

This is effectively "the same" as dot product.

For a lot of embeddings we have today, norm of any embedding vector is roughly of same size, so the angle between two vectors is roughly same size as length of difference that you are saying, and can be expressed in terms of 1 - dot product after scaling

I don't have an answer for this really outside of silly ones like "strict equality check", but I assert that no one else does either, at least today and right now, and its an inherent limitation due to the nature of embeddings and the space it desires to be (cheap, fast, good enough similarity for your use case).

You're probably best off using the commercial suggestion, and if its dot product, go for it. I am no expert in this area and my interest wanes every day.

Interested to know as well