The triangle inequality constraint in similarity judgments

Area of Science:

• Cognitive psychology
• Mathematical psychology
• Human perception

Background:

• The triangle inequality is a fundamental constraint in similarity judgments, alongside symmetry and minimality.
• While symmetry and minimality are well-understood, the triangle inequality's application to similarity has been complex.
• Previous models assumed linear relationships between dissimilarity and distance, leading to a specific form of the triangle inequality.

Purpose of the Study:

• To translate the dissimilarity triangle inequality into a multiplicative form for similarity.
• To investigate whether human similarity judgments adhere to this derived multiplicative triangle inequality.
• To empirically test for violations of the multiplicative triangle inequality in human perception.

Main Methods:

• Derivation of the multiplicative triangle inequality for similarity using Shepard's generalization law.
• Translating the linear dissimilarity constraint (Dissimilarity(A,B) + Dissimilarity(B,C) ≥ Dissimilarity(A,C)) into a similarity constraint (Sim(A,C) ≥ Sim(A,B)⋅Sim(B,C)).
• Empirical investigation to observe human similarity judgments.

Main Results:

• The multiplicative triangle inequality for similarity was mathematically derived.
• Empirical evidence demonstrated that human similarity judgments can violate this multiplicative triangle inequality.
• Human behavior is not always consistent with the derived mathematical constraint.

Conclusions:

• The multiplicative triangle inequality is not a universally followed constraint in human similarity judgments.
• Human perception of similarity can exhibit non-linear or context-dependent properties not captured by simple multiplicative rules.
• Further research is needed to understand the conditions under which these violations occur.