If you are just asking **is this a valid metric** then the answer is **almost**, it is a valid **pseudometric** if only `.computeCost`

is deterministic.

For simplicity i denote `f(A) := model.computeCost(A)`

and `d(A, B) := |f(A)-f(B)|`

Short proof: `d`

is a L1 applied to an image of some function, thus is a pseudometric itself, and a metric if `f`

is **injective** (in general, yours is **not**).

Long(er) proof:

`d(A,B) >= 0`

**yes**, since `|f(A) - f(B)| >= 0`

`d(A,B) = d(B,A)`

**yes**, since `|f(A) - f(B)| = |f(B) - f(A)|`

`d(A,B) = 0`

iff `A=B`

, **no**, this is why it is **pseudo**metric, since you can have many `A != B`

such that `f(A) = f(B)`

`d(A,B) + d(B,C) <= d(A,C)`

, **yes**, directly from the same inequality for absolute values.

If you are asking **will it work** for your **problem**, then the answer is **it might, depends on the problem**. There is no way to answer this without analysis of your problem and data. As shown above this is a valid pseudometric, thus it will measure something **decently behaving** from mathematical perspective. Will it work for your particular case is completely different story. The good thing is most of the algorithms which work for metrics will work with pseudometrics as well. The only difference is that you simply "glue together" points which have the same image (`f(A)=f(B)`

), if this is not the issue for your problem - then you can apply this kind of pseudometric in any metric-based reasoning without any problems. In practise, that means that if your `f`

is

computes the sum of squared distances between the input point and the corresponding cluster center

this means that this is actually a distance to closest center (there is no summation involved when you consider **a single point**). This would mean, that 2 points in two separate clusters are considered **identical** when they are equally far away from their own clusters centers. Consequently your measure captures "how different are relations of points and their respective clusters". This is a well defined, indirect dissimilarity computation, however you have to be fully aware what is happening before applying it (since it will have specific consequences).