Physical and Psychological Representations of Faces: Evidence FromMorphing

Models of face recognition and classification often adopt a framework in which faces are represented as points in a multi-dimensional space. This psychological face space organizes the faces according to similarity and makes predictions for representational theories of faces. A variety of image-processing techniques have been used to create novel stimuli in this space that represent the average of a population or make a face appear more distinctive. The current research examined the relation between the stimuli created by these image-processing techniques and the underlying psychological representation as measured by multidimensional scaling (MDS) procedures. Morphing procedures were used to create 16 faces that were embedded in a set of 84 other faces. Similarity ratings between all possible pairs of faces were collected, and the data were analyzed using MDS procedures. Dimensions that emerged from the MDS solution included age, race, adiposity, and facial hair. In the MDS space, the morphs appeared more typical than the parents, as predicted by the geometric model. A number of biases were examined, including the tendency of the morphs to be less typical than predicted, which may be attributed to the effects of density near the center of face space. In addition, age and facial-adiposity biases were found. The results support the use of the face-space framework for models of face recognition, although image-processing techniques that are designed to create novel stimuli in this space may introduce systematic biases.     

On the intricacies involved in the study of similarity judgments: comment on Ritov, Gati, and Tversky

The purpose of this comment is to point out some potential pitfalls in the study of similarity judgments, particularly with regard to comparison of similarity judgments between modalities. These are discussed in the context of the article by Ritov, Gati, and Tversky (1990).     

Generalization,similarity, and Bayesian inference

Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.     

Measures of similarity in models of categorization

This paper concerns the use of similarities based on geometric distance in models of categorization. Two problematic implications of such similarities are outlined. First, in a comparison between two stimuli, geometric distance implies that matching features are not taken into account. Second, missing features are assumed not to exist. Only nonmatching features enter into calculations of similarity. A new model is constructed that is based on the ALCOVE model (Kruschke, 1992), but it uses a feature-matching similarity measure (see, e.g., Tversky, 1977) rather than a geometric one. It is an on-line model in the sense that both dimensions and exemplars are constructed during the categorization process. The model accounts better than ALCOVE does for data with missing features (Experiments 1 and 2) and at least as well as ALCOVE for a data set without missing features (Nosofsky, Kruschke, & McKinley, 1992). This suggests that, at least for some stimulus materials, similarity in categorization is more akin to a feature-matching procedure than to geometric distance calculation.     

Similarity, feature discovery, and the size principle

In this paper we consider the “size principle” for featural similarity, which states that rare features should be weighted more heavily than common features in people’s evaluations of the similarity between two entities. Specifically, it predicts that if a feature is possessed by n objects, the expected weight scales according to a 1/n law. One justification of the size principle emerges from a Bayesian analysis of simple induction problems ( [Tenenbaum and Griffiths, 2001a] and [Tenenbaum and Griffiths, 2001b]), and is closely related to work by Shepard (1987) proposing universal laws for inductive generalization. In this article, we (1) show that the size principle can be more generally derived as an expression of a form of representational optimality, and (2) present analyses suggesting that across 11 different data sets in the domains of animals and artifacts, human judgments are in agreement with this law. A number of implications are discussed.     

Exploring children’s face-space: A multidimensional scaling analysis of the mental representation of facial identity

We explored differences in the mental representation of facial identity between 8-year-olds and adults. The 8-year-olds and adults made similarity judgments of a homogeneous set of faces (individual hair cues removed) using an “odd-man-out” paradigm. Multidimensional scaling (MDS) analyses were performed to represent perceived similarity of faces in a multidimensional space. Five dimensions accounted optimally for the judgments of both children and adults, with similar local clustering of faces. However, the fit of the MDS solutions was better for adults, in part because children’s responses were more variable. More children relied predominantly on a single dimension, namely eye color, whereas adults appeared to use multiple dimensions for each judgment. The pattern of findings suggests that children’s mental representation of faces has a structure similar to that of adults but that children’s judgments are influenced less consistently by that overall structure.     

Geometric analogue of holographic reduced representation

Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1, one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on ‘projected products’, a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.     

Shepard's universal law supported by honeybees in spatial generalization

An animal that is rewarded for a response in one situation (the S+) is likely to respond to similar but recognizably different stimuli, the ubiquitous phenomenon of stimulus generalization. On the basis of functional analyses of the probabilistic structure of the world, Shepard formulated a universal law of generalization, claiming that generalization gradients, as a function of the appropriately scaled distance of a stimulus from S+, should be exponential in shape. This law was tested in spatial generalization in honeybees. Based on theoretically derived scales, generalization along both the dimensions of the distance from a landmark and the direction to a landmark followed Shepard's law. Support in an invertebrate animal increases the scope of the law, and suggests that the ecological structure of the world may have driven the evolution of cognitive structures in diverse animals.     

Avoiding the dangers of averaging across subjects when using multidimensional scaling

Ashby, Maddox and Lee (Psychological Science, 5 (3) 144) argue that it can be inappropriate to fit multidimensional scaling (MDS) models to similarity or dissimilarity data that have been averaged across subjects. They demonstrate that the averaging process tends to make dissimilarity data more amenable to metric representations, and conduct a simulation study showing that noisy data generated using one distance metric, when averaged, may be better fit using a different distance metric. This paper argues that a Bayesian measure of MDS models has the potential to address these difficulties, because it takes into account data-fit, the number of dimensions used by an MDS representation, and the precision of the data. A method of analysis based on the Bayesian measure is demonstrated through two simulation studies with accompanying theoretical analysis. In the first study, it is shown that the Bayesian analysis rejects those MDS models showing better fit to averaged data using the incorrect distance metric, while accepting those that use the correct metric. In the second study, different groups of simulated ‘subjects’ are assumed to use different underlying configurations. In this case, the Bayesian analysis rejects MDS representations where a significant proportion of subjects use different configurations, or when their dissimilarity judgments contain significant amounts of noise. It is concluded that the Bayesian analysis provides a simple and principled means for systematically accepting and rejecting MDS models derived from averaged data.     

Stimulus dimensions or stimulus aspects

Summary The questionableness of geometric models of stimulus similarity has led to the development of an alternative approach by Tversky which makes no dimensional or metric assumptions. Rather, stimuli are described as sets of qualitative stimulus aspects and stimulus similarity as a function of common and non-common aspects. According to Restle's model of stimulus similarity, however, the perception of stimuli of a categorial nature can be organized along dimensions because stimulus aspects form dimensions under certain conditions. The present study supports the empirical validity of this assumption. Further it is suggested that contrary to the present opinion quantitative stimulus characteristics are probably not perceived as dimensions in the sense of the geometric models but only dimensions as described in the Restle model.     

Representation is representation of similarities

Advanced perceptual systems are faced with the problem of securing a principled (ideally, veridical) relationship between the world and its internal representation. I propose a unified approach to visual representation, addressing the need for superordinate and basic-level categorization and for the identification of specific instances of familiar categories. According to the proposed theory, a shape is represented internally by the responses of a small number of tuned modules, each broadly selective for some reference shape, whose similarity to the stimulus it measures. This amounts to embedding the stimulus in a low-dimensional proximal shape space spanned by the outputs of the active modules. This shape space supports representations of distal shape similarities that are veridical as Shepard's (1968) second-order isomorphisms (i.e., correspondence between distal and proximal similarities among shapes, rather than between distal shapes and their proximal representations). Representation in terms of similarities to reference shapes supports processing (e.g., discrimination) of shapes that are radically different from the reference ones, without the need for the computationally problematic decomposition into parts required by other theories. Furthermore, a general expression for similarity between two stimuli, based on comparisons to reference shapes, can be used to derive models of perceived similarity ranging from continuous, symmetric, and hierarchical ones, as in multidimensional scaling (Shepard 1980), to discrete and nonhierarchical ones, as in the general contrast models (Shepard & Arabie 1979; Tversky 1977).     

Similarity and proximity: When does close in space mean close in mind?

People often describe things that are similar as close and things that are dissimilar as far apart. Does the way people talk about similarity reveal something fundamental about the way they conceptualize it? Three experiments tested the relationship between similarity and spatial proximity that is encoded in metaphors in language. Similarity ratings for pairs of words or pictures varied as a function of how far apart the stimuli appeared on the computer screen, but the influence of distance on similarity differed depending on the type of judgments the participants made. Stimuli presented closer together were rated more similar during conceptual judgments of abstract entities or unseen object properties but were rated less similar during perceptual judgments of visual appearance. These contrasting results underscore the importance of testing predictions based on linguistic metaphors experimentally and suggest that our sense of similarity arises from our ability to combine available perceptual information with stored knowledge of experiential regularities.     

Finding similarity in a model of relational reasoning

Similarity plays a central role in the study of perception and cognition. Previous attempts to model similarity have captured effects of either featural or structural similarity, but typically not both. We simulated both by fitting similarity data with the LISA model of relational reasoning [Hummel, J. E., & Holyoak, K. J. (1997). Distributed representations of structure: A theory of analogical access and mapping. Psychological Review, 104, 427–466, Hummel, J. E., & Holyoak, K. J. (2003a). A symbolic-connectionist theory of relational inference and generalization. Psychological Review, 110, 220–264]. The same mechanisms LISA uses to simulate analogy also provide a natural account of feature-based similarity effects (e.g., violations of symmetry), structural effects (e.g., the advantage of alignable over non-alignable differences), and the combined effects of featural and structured information (i.e., MIPs and MOPs; “Matches In/Out of Place”) on similarity judgments. Our approach differs from most models of similarity in that LISA was not originally designed to simulate similarity judgments, but rather analogical reasoning. LISA’s incidental ability to simulate diverse similarity effects speaks to the plausibility of the model’s account of human knowledge representation.     

Exemplar-based accounts of "multiple-system" phenomena in perceptual categorization

We demonstrate that a wide variety of recently reported "rule-described" and "prototype-described" phenomena in perceptual classification, which have led to the development of a number of multiple-system models, can be given an alternative interpretation in terms of a single-system exemplar-similarity model. The phenomena include various rule- and prototype-described patterns of generalization, dissociations between categorization and similarity judgments, and dissociations between categorization and old-new recognition. The alternative exemplar-based interpretation relies on the idea that similarity is not an invariant relation but a context-dependent one. Similarity relations among exemplars change systematically because of selective attention to dimensions and because of changes in the level of sensitivity relating judged similarity to distance in psychological space. Adaptive learning principles may help explain the systematic influence of the selective attention process and of modulation in sensitivity settings on judged similarity.     

Recognition memory for realistic synthetic faces

A series of experiments examined short-term recognition memory for trios of briefly presented, synthetic human faces derived from three real human faces. The stimuli were a graded series of faces, which differed by varying known amounts from the face of the average female. Faces based on each of the three real faces were transformed so as to lie along orthogonal axes in a 3-D face space. Experiment 1 showed that the synthetic faces' perceptual similarity structure strongly influenced recognition memory. Results were fit by a noisy exemplar model (NEMO) of perceptual recognition memory. The fits revealed thatrecognition memory was influenced both by the similarity of the probe to the series items and by the similarities among the series items themselves. Nonmetric multidimensional scaling (MDS) showed that the faces' perceptual representations largely preserved the 3-D space in which the face stimuli were arrayed. NEMO gave a better account of the results when similarity was defined as perceptual MDS similarity, rather than as the physical proximity of one face to another. Experiment 2 confirmed the importance of within-list homogeneity directly, without mediation of a model. We discuss the affinities and differences between visual memory for synthetic faces and memory for simpler stimuli.     

The generalized universal law of generalization

It has been argued by Shepard that there is a robust psychological law that relates the distance between a pair of items in psychological space and the probability that they will be perceived as similar. Specifically, this probability is a negative exponential function of the distance between the pair of items. In experimental contexts, distance is typically defined in terms of a multidimensional space—but this assumption seems unlikely to hold for complex stimuli. We show that, nonetheless, the Universal Law of Generalization can be derived in the more complex setting of arbitrary stimuli, using a much more universal measure of distance. This universal distance is defined as the length of the shortest program that transforms the representations of the two items of interest into one another: The algorithmic information distance. It is universal in the sense that it minorizes every computable distance: It is the smallest computable distance. We show that the Universal Law of Generalization holds with probability going to one—provided the probabilities concerned are computable. We also give a mathematically more appealing form of the Universal Law.     

Interpersonal similarity as a social distance dimension: Implications for perception of others’ actions

Building on the assumption that interpersonal similarity is a form of social distance, the current research examines the manner in which similarity influences the representation and judgment of others’ actions. On the basis of a construal level approach, we hypothesized that greater levels of similarity would increase the relative weight of subordinate and secondary features of information in judgments of others’ actions. The results of four experiments showed that compared to corresponding judgments of a dissimilar target, participants exposed to a similar target person identified that person’s actions in relatively more subordinate means-related rather than superordinate ends-related terms (Experiment 1), perceived his or her actions to be determined more by feasibility and less by desirability concerns (Experiment 3), and gave more weight to secondary aspects in judgments of the target’s decisions (Experiment 2) and performance (Experiment 4). Implications for the study of interpersonal similarity, as well as social distance in general, are discussed.     

Stimulus hierarchy generalization in systematic desensitization

A group of snake phobic Ss were desensitized to the first 5 items of a standard 7 item snake fear hierarchy in which the items were ordered on the basis of distance from a snake. This group and a no-treatment control group (which did not receive the desensitization) were treated for fear of all the hierarchy stimuli in terms of ratings both before and after the desensitization. The Ss rated their fear responses to all the stimuli as presented to them in (1) real form (2) by slide, and (3) in imaginal form. The results verified all four experimental hypotheses: (1) The experimental Ss showed greater fear reduction than the control Ss to both the training stimuli (the stimuli on which desensitization done) and the generalization stimuli (the other two stimuli), (2) The systematic desensitization (SD) group showed (a) more fear reduction to the last training stimuli than to the lower generalization stimulus and (b) more to the latter than to the higher generalization stimulus, (3) The SD group showed less fear reduction to the generalization stimuli in their real form than in both the slide and imaginai modalities (in the desensitization each stimulus presentation was done first in slide form and then in imaginal form), and (4) There were significant overall individual differences in generalization of fear reduction to the generalization stimuli.