Scalar fields and the leietanic property
The pairwise interaction function, the homogeneous and heterogeneous fields, the isolation measure B, and Definition 1.
In plain words
To judge whether a region of the map is clean, every point is given a small "pull" on every other point. The pull is strongest, exactly 1, when two points coincide and fades smoothly as they move apart, a bit like gravity but without blowing up at zero distance. For each point, add up the pull from points of its own class (the more the better) and from points of other classes (the less the better).
Summing these over the whole dataset gives a single number, called \(B\), that runs from \(+1\) when everything is mixed together to \(-1\) when the classes are perfectly separated. A redrawing of the map earns the name leietanic when it lowers \(B\) compared with the original features. Rigid moves such as rotating or shifting the whole picture change nothing, so they do not count.
Pairwise interaction
When two instances \(\boldsymbol{x}_a\), \(\boldsymbol{x}_b\) are transformed into \(\boldsymbol{q}_a\), \(\boldsymbol{q}_b\), their relationship is described by
The function is inspired by the potential energies of classical mechanics, \(U(\boldsymbol{q}_a,\boldsymbol{q}_b)=k/|\boldsymbol{q}_a-\boldsymbol{q}_b|\) (5), where \(k\) depends on the masses or charges. That potential is infinite at \(\boldsymbol{q}_a=\boldsymbol{q}_b\), precisely the case that is desirable for two instances of the same class. Using \(\epsilon+|\boldsymbol{q}_a-\boldsymbol{q}_b|\) as denominator avoids the infinity but leaves the derivative discontinuous at coincidence; using \(\epsilon+(\boldsymbol{q}_a-\boldsymbol{q}_b)^2\) gives
which is equivalent to (4) with the convention \(\zeta(\boldsymbol{q},\boldsymbol{q})=1\), so that \(\epsilon/k=1\) and \(\rho^2=1/k\). Writing \(\rho^2\) emphasises that it is positive.
Homogeneous and heterogeneous fields
For a quon \(\boldsymbol{q}_c\) of class \(c\), its interactions with the rest of \(\Omega\) split into those with same-class instances and those with other classes:
\(W_{\text{hom}}\) measures the closeness of \(\boldsymbol{q}_c\) to the other instances of its class: the higher, the easier it is to isolate them in a closed area of low entropy. \(W_{\text{het}}\) measures how close and mixed \(\boldsymbol{q}_c\) is with instances of other classes: the lower, the easier it is to reduce the entropy around it. In the extreme cases,
where \(|\Omega|\) is the cardinality of the dataset. The limit (9) is reached only if all instances of the class are placed at the same point of \(Q\), a perfect generalisation of the class. Real values fall in between.
The isolation measure B
The two fields are combined into an evaluation of the uniformity of the distribution of all instances:
\(B\) is a LEI evaluation of the dataset, an estimate of the balance between the homogeneous and heterogeneous fields over all quons. It can be evaluated in any space, including the original \(X\). For a completely heterogeneous space with all instances mixed in close areas, \(B=1\); for a perfect isolation of classes, \(B=-1\), as follows from (9) and (10).
A transformation function \(L\colon X\subset\mathbb{R}^F\mapsto Q\subset\mathbb{R}^D\), \((F,D)\in(\mathbb{N}^+,\mathbb{N}^+)\), is leietanic for a deterministic classification problem with a dataset \(\Omega\) when \(B_Q\) in the space \(Q\) satisfies \(B_Q\lt B_X\), compared with \(B_X\) in the original space of features \(X\).
A leietanic transformation cannot be rigid: it explicitly excludes any linear transformation that preserves the Euclidean distance (rotations, translations, reflections and their combinations), since these leave \(B\) unchanged.