Hausdorff Distance on Orbit Multisets

• Hausdorff Distance on Orbit Multisets is a framework for measuring dissimilarities between collections of orbits by extending the classical Hausdorff metric to account for element multiplicities.
• The approach generalizes five multiset metrics, ensuring invariance under group actions while penalizing mismatched multiplicities through a fixed parameter M.
• It finds practical application in fields such as mathematical chemistry, voting theory, and symmetry-aware geometric comparisons with rigorous metric properties.

The Hausdorff distance on orbit multisets is a framework for quantifying dissimilarity between collections of orbits (or equivalence classes) in a metric space under a group action, via a principled generalization of the classical Hausdorff metric to finite multisets. This construction is principally founded on the five multiset metrics formulated on bounded metric spaces, three of which directly generalize the Hausdorff metric and are particularly relevant for multisets of orbits. These metrics achieve invariance under group actions and account for both the identities and multiplicities of elements, facilitating robust analysis in settings such as mathematical chemistry, voting theory, and symmetry-aware geometric comparison.

1. Multiset Models and Metric Construction

The formalism introduces several models for multisets on a metric space $(X,d)$, where $d$ is assumed bounded. Multisets are encoded using the set $E$ (finite multisets of $X$), and more generally, $F'$, the finite subsets of

$A = (X \times \mathbb{N})/\{(x,0)\sim(x',0)\}$

where each element $(x,r)$ denotes $r$ copies of $x \in X$. The cardinality function $C(e)$ totals the multiplicities within a multiset, and the further quotient $G = F'/\!\sim$ identifies subsets of $A$ with the same multiplicity profile.

The five multiset distances $(d_1, \dots, d_5)$ are defined as follows:

Name	Domain	Defining Formula (summary)
$d_E$	$E \times E$	Minimal total matching cost between flattened multisets, plus $M$ for surplus elements
$d_{Em}$	$E \times E$	Normalized version of $d_E$ by maximal multiplicity
$d_F$	$F' \times F'$	Hausdorff-type metric using a modified distance $d_A$ on stacks
$d_{Fm}$	$F' \times F'$	Hausdorff-type metric with per-copy normalization
$d_G$	$G \times G$	Infimum of chains in $F'$ connecting representatives, with per-link cost $d_F$

Here $M>0$ is a fixed cost parameter, and $\theta = \sup_{x,y\in X}d(x,y)/M$ controls metric properties. The modified metric on "stacks" is

$$d_A(r\,e_x,\;t\,e_z) = M|t - r| + \min\{r, t\} \, d(x, z)$$

and its normalized cousin is

$$d_{Am}(r\,e_x,\;t\,e_z) = \frac{d_A(r\,e_x,\;t\,e_z)}{\max\{r, t\}}.$$

2. Generalization of the Classical Hausdorff Metric

The standard Hausdorff metric on finite subsets $A, B \subset X$ is

$$d_H(A, B) = \max \Bigl\{ \sup_{a \in A} \inf_{b \in B} d(a, b),\ \sup_{b \in B} \inf_{a \in A} d(a, b) \Bigr\}.$$

When all multiplicities $r$ are restricted to $\{0, 1\}$, $d_F$ and $d_{Fm}$ reduce to $d_H$, rendering the constructions direct generalizations. In general, the models operate by replacing each $x \in X$ with a stack of copies indexed by multiplicity, and measuring the cost of matching these stacks with the metric $d_A$. The supremum-infimum expressions in $d_F$ and $d_{Fm}$ replicate the Hausdorff sup–inf matching over multisets, with explicit penalties for mismatched multiplicities.

3. Metric Properties: Completeness and Discreteness Conditions

The metrics $d_A$ and $d_{Am}$ are genuine metrics on $A$ if and only if $\theta \le 2$ (i.e., the maximum original space diameter is no more than twice the penalty $M$), as shown by failure of the triangle inequality otherwise. Consequently, $d_F$ and $d_{Fm}$ are metrics on all finite subsets of $A$ when $\theta \le 2$, and they are complete exactly when $(X,d)$ is complete. For the quotient metric $d_G$ on $G$, completeness and separation of points hold if and only if $(X,d)$ is uniformly discrete; otherwise, $d_G$ is a pseudometric.

Notably, for any two equivalence classes $[U], [V]$ in $G$,

$d_G([U], [V]) \ge d_H(R([U]), R([V]))$

where $R([U])$ denotes the underlying support sets in $X$. This lower bound ensures that the generalized metric respects the classical Hausdorff distance at a coarse granularity.

4. Application to Multisets of Orbits

Given a group $G$ acting on a metric space $(X, d)$, comparison of multisets of orbits proceeds by defining an orbit pseudometric: $$d_{\mathrm{orb}}(Gx, Gy) = \inf_{g, h \in G} d(gx, hy).$$ This construction, under mild assumptions, yields a pseudometric on the orbit space $X/G$. The theory enables one to regard each orbit $Gx$ as a point in $(X/G, d_{\mathrm{orb}})$, thus fitting multisets of orbits into the finite-multiset frameworks above (models E, F, G). The invariance of the base metric under group actions ensures that all multiset distances inherit $G$-invariance, making the framework intrinsically suitable for symmetry-aware contexts.

Critical considerations include (i) verifying boundedness/discreteness to ensure meaningful metrics, and (ii) ensuring the penalty parameter $M$ is chosen relative to the action-induced orbit pseudometric so that $\theta \le 2$ or uniform discreteness holds as required.

5. Worked Example: Voting via Multiset Metrics

The theory finds direct application in voting theory. Let $X$ be the set of all linear orderings (ballots) of $n$ candidates, equipped with the Kendall-$\tau$ metric: $$d_\tau(\sigma, \pi) = \#\{\text{adjacent transpositions to turn } \sigma \text{ into } \pi\}.$$ Election outcomes are multisets over $X$, with multiplicity reflecting vote counts. Comparing two elections $e, f \in E$ involves: $$d_E(e, f) = \min_\gamma \Bigl\{ \sum_\text{votes} d_\tau(\cdot, \gamma(\cdot)) + M|C(e) - C(f)| \Bigr\}$$ where $\gamma$ ranges over permutations matching votes. When ballots counts differ, the surplus is penalized with $M$. The normalization in $d_{Em}$ and $d_{Fm}$ accounts for per-voter average deviation. This construction yields a robust metric quantifying both the diversity of ballot rankings and discrepancies in voter turnout, and admits adaptation to cases where ballots are identified up to group action (e.g., symmetries, relabelings).

6. Context, Significance, and Extensions

These metrics, particularly the Hausdorff-type generalizations, offer foundational tools for quantifying distances between structured collections with symmetry or multiplicity. Their applicability hinges on properties of the ground metric (boundedness, completeness, uniform discreteness) and the penalty parameter $M$. They capture both set-level and multiset-level distinctions through explicit treatment of multiplicities and are invariant under transformations induced by group actions.

A plausible implication is that, provided the hypotheses on the base metric and the group action are verified, these constructions can be extended to contexts such as isomorphism classes, symmetry-adapted clustering, and other domains where comparisons factor through equivalence relations imposed by group actions or symmetries.

The approach systematically generalizes the Hausdorff metric to multiset and orbit-multiset contexts, supporting rigorous analysis in applications where both identity and replication of elements are mathematically or practically significant (Turner, 2011).