# #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:
$$d$$
{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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