What's the difference between a Permutation and a Perturbation? I frequently come across the terms Permutation and Perturbation within the field of explainable AI.
I understand that both terms refer to methods that make changes to a sample's features. However, I'm not certain of the difference between the two.
From mathematics, a permutation is a rearrangement of elements in a set. This makes me think permutation methods draw their changes from other samples in the dataset.
From physics, a perturbation is a minor change to a model that disturbs its usual output. This makes me think perturbation methods can make any change to the sample regardless of what values are held in other samples.
If that's true then permutation methods would be special cases of perturbation methods. Have I got this right, or is there more to the distinction?
Example situation
To avoid confusion, consider the example of a single sample, which is part of a highly localised dataset.
The high locality of the dataset implies there are many other samples nearby. Permutation will therefore produce samples close to the single sample. Similarly, perturbing the single sample will produce multiple samples close to the dataset. In this situation it's possible the two methods would produce overlapping samples, the only difference being the permutation samples are more restricted than the perturbation samples.
In the above example, wouldn't that make permutation a special case of perturbation?
Related

*

*What is perturbation in Machine Learning?

*Permutation Feature Importance

There may be a deeper connection between the two, I'll speak from a perspective of graduate level applied math/physics.

If that's true then permutation methods would be special cases of perturbation methods. Have I got this right, or is there more to the distinction?

No, I don't think so.  Permutations are mappings from a finite set onto itself.  In that way, a permutation can be like a function.
In english, to perturb something is "to an influence tending to alter its normal or regular state or path" or maybe "to unsettle something". This aligns fairly closely to applications in physics/applied math where we take a dynamical system and perturb the solution to understand the dynamics near fixed points.  At least in physics, perturbations can be conceived as functions I suppose, but its more to understand  how some other function or system behaves near some point of interest.
