#StackBounty: #distance #similarities Intended interpretation of one-mode, three-way (dis)similarities?

Bounty: 50

I have what I think is a very simple question, the answer has just eluded me so far. A two-way similarity, $s_{ij}$ (for objects $i$ and $j$) can be interpreted fairly straightforwardly as the degree to which $j$ resembles $i$. When I’ve seen discussion of three-way similarities, however, the various authors I have read have never given a clear interpretation. They’ll say something like "a three-way [similarity] is defined as the resemblance between objects taken three at a time" (slightly paraphrased from de Rooij & Heiser (2000)).

In most presentations the triadic distances are a function of the dyadic ones, e.g., $d_{ijk} = max ( d_{ij}, d_{ik}, d_{jk} ) $, where $d_{ij}$ is the Euclidean distance between the points representing $i$ and $j$. Triadic distances can also be treated axiomatically, however, and obey generalizations of the usual metric axioms. From Heiser and Bennani (1997), the triadic distance function $d : O times O times O to mathbb{R}{geq 0}$ is such that for all $i,j,k in O$, and for all permutations $pi$ on ${i,j,k}$, we have:
{ijk} geq 0,
d_{ijk} = d_{pi(i) pi(j) pi(k)},

d_{iji} = d_{ijj},

$$ d_{ijk} = 0 text{ only if } i = j = k, and$$
$$ 2d_{ijk} leq d_{ikl} + d_{jkl} + d_{ijl}.$$

This last one is called the tetrahedral inequality and generalizes the triangle inequality.

I think I get the math behind the notion, unfortunately I can’t make sense of "resemblance between objects taken three at a time". Is there a natural language gloss on this ternary similarity relation that might help me to get an intuitive grasp on it?

de Rooij & Heiser, 2000, "Triadic distance models for the analysis of asymmetric three-way proximity data"

Heiser & Bennani, 1997, "Triadic distance models: axiomatization and least squares represenation"

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