Measuring the performance of a clustering algorithm with "true" clusters available Suppose I have observational units $x_1, \dots, x_n$. Each of these units is in a known cluster $C_1, \dots, C_m$, $m < n$. A ML algorithm generates based on a metric new clusters $C^{*}_1, \dots, C^{*}_p$, $p$ not necessarily equal to $m$.
Is there a standard way to measure the performance of the ML algorithm?
Best-case scenario is the following: $m = p$ and for each $C^{*}_i$, there exists some $C_j$ such that $C^{*}_i = C_j$, where equality is defined by having the same observational units.
Edit as clarification: The reason I'm doing this in the first place is because I have $N > n$ units available, with the remaining $N - n$ units not having known clusters. I'd like to see how well the algorithm performs with the $n$ units before I scale it up to the remaining $N-n$ units.

A great discussion of clustering in a Bayesian framework appears in:
Hanson, Robin, John Stutz, and Peter Cheeseman. "Bayesian Classification Theory." http://robinhanson.com/ijcai91.pdf

... the AutoClass system searches for the most probable classifications, automatically choosing the number of classes and complexity of class descriptions ...


... evaluation function optimally trades off the complexity of the model with its fit to the data, and is used
to guide an open-ended search for the best classification ...

Note AutoClass does not use "true" clusters, and has a history of illuminating new meaningful classes to subject matter experts. https://ti.arc.nasa.gov/tech/rse/synthesis-projects-applications/autoclass/
