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Following this paper and this paper, I'm trying to implement the formula for the Lorentzian distance between 2 points (aka the distance between 2 points in Lorentzian space). I'll use this a the distance metric for a KNN classifier.

According to the papers, the distance looks something like this:

Lorentz Distance

where n is the number of dimensions (features), and X and Y are 2 points represented by vectors of features [X1, X2, ... , Xn] and [Y1, Y2, ... , Yn]

If I'm reading the formula correctly, the right hand term inside the square root is always going to be the absolute difference value between the 2 points for the last feature n, then squared. However, this right term is subtracted from the other squared term on the left, so at first it seems to be possible to end up with a negative value inside the square root. And indeed, when I tried this metric against my dataset, some terms errored out because of a negative value inside the square root. My dataset features were all normalized via minmax from 0-1.

It doesn't appear to be anywhere in the paper, at least to me, something to suggest the Lorentzian distance could be a complex number, unless I missed something.

Has anyone come across this metric for distance and know better than I? Or maybe I misread the papers or the formula somehow.

Any input would be much appreciated.

Thanks.

Ricardo Chavarria
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  • Not an expert of Lorentzian distance, but notice that you mention "some terms errored out...", did you remember to sum all the terms before taking square root? There is a summation inside so there should only be 1 scalar when taking the sqrare root. – lpounng May 03 '23 at 08:00
  • Thanks @lpounng . Indeed I added all the terms from the summation. It's just that some of the individual terms can be negative, and as it happens, the total sum can also be negative if the negative individual terms dominate. – Ricardo Chavarria May 03 '23 at 13:59

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the Lorentzian metric is just a generalization of the spacetime interval, with a signature (n,0,1).

the imaginary result for the metric corresponds to negative real distance. elapsed time is a negative distance in space. you subtract it from all 3 space directions equally.

People don't want to believe that because they've defined distance as always being positive or an absolute value. except that's not how reality works.

===[ time is negative space.

Jess Fuckett
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