A proposal for a Riemannian face space and application to atypical vs. typical face similarities

James T. Townsend, Hao Lun Fu, Cheng Ju Hsieh, Cheng Ta Yang

Research output: Contribution to journalReview articlepeer-review

Abstract

Two intriguing papers of the late 1990’s and early 2000s by J. Tanaka and colleagues put forth the hypothesis that a repository of face memories can be viewed as a vector space where points in the space represent faces and each of these is surrounded by an attractor field. This hypothesis broadens the thesis of T. Valentine that face space is constituted of feature vectors in a finite dimensional vector space (e.g., Valentine, 2001). The attractor fields in the atypical part of face space are broader and stronger than those in typical face regions. This notion makes the substantiated prediction that a morphed midway face between a typical and atypical parent will be perceptually more similar to the atypical face. We propose an alternative interpretation that takes a more standard geometrical approach but also departs from the popular types of metrics assumed in almost all multidimensional scaling studies. Rather we propose a theoretical structure based on our earlier investigations of non-Euclidean and especially, Riemannian Face Manifolds (e.g., Townsend, Solomon, & Spencer-Smith, 2001). We assert that this approach avoids some of the issues involved in the gradient theme by working directly with the type of metric inherently associated with the face space. Our approach emphasizes a shift towards a greater emphasis on non-Euclidean geometries, especially Riemannian manifolds, integrating these geometric concepts with processing-oriented modeling. We note that while fields like probability theory, stochastic process theory, and mathematical statistics are commonly studied in mathematical psychology, there is less focus on areas like topology, non-Euclidean geometry, and functional analysis. Therefore, both to elevate comprehension as well as to propagate the latter topics as critical for our present and future enterprises, our exposition moves forward in a highly tutorial fashion, and we embed the material in its proper historical context.

Original languageEnglish
Article number102870
JournalJournal of Mathematical Psychology
Volume122
DOIs
Publication statusPublished - Sept 2024

Keywords

  • Atypicality bias
  • Face perception
  • Face recognition
  • Face space
  • Non-Euclidean geometry
  • Riemannian face manifolds
  • Topology

ASJC Scopus subject areas

  • General Psychology
  • Applied Mathematics

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