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Is there a transformation of a set of float vectors to a set of binary integer vectors that preserves the dot product. I found conformal transformations but I'm interested in large vectors (size 300) while they are mainly for the complex domain.

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No. The dot product of float vectors can be fractional, but the dot product of integers is always an integer. You can certainly just scale up the floats to just before you overflow ints and round to get pretty close.

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