Abstract
The Representational Distortion (RD) approach to similarity (e.g., Hahn, Chater, & Richardson, 2003) proposes that similarity is computed using the transformation distance between two entities. We argue that researchers who adopt this approach need to be concerned with how representational transformations can be determined a priori. We discuss several roadblocks to using this approach. Specifically, we demonstrate the difficulties inherent in determining what transformations are psychologically salient and the importance of considering the directionality of transformations.
Keywords: Similarity, Representation, Transformation, Judgment, Distance
Similarity, the psychological likeness of a pair of items, is central to cognition and is often incorporated as a construct in models of cognitive processing. Most theoretical approaches to similarity are comparison-based models that make some assumption about the nature of mental representations and the processes that act over them. One alternative is to identify similarity as the distance between items in an n-dimensional representational space (Shepard, 1957). Another is to determine the features items have in common and those that are distinct, and to allow similarity to increase with the number of common features and decrease with the number of distinct features (Tversky, 1977). These approaches are both intuitive, but they fail to capture the rich structure inherent in representation and the kinds of commonalities and differences that result from this structure (Gentner & Markman, 1997; Markman & Gentner, 1993). A third type of comparison model is the structural-alignment approach, which assumes that similarity depends on the degree of correspondence between the relational structures that represent the items.
One prominent structural theory is the structure mapping account, which likens similarity computation to analogical reasoning (Gentner, 1983; Gentner & Markman, 1997). Under this account, items are represented as structured systems of relational predicates, predicate arguments or roles, objects that fill those roles, and object properties. Similarity is computed by comparing these structures. Two items are similar to the extent that they share objects that play same relational roles. One case that reveals the importance of this kind of structure in similarity is the distinction between identicality and cross-mapping. Consider Item 1 and Item 2 in Figure 1. Although the two items have the same objects with the same properties, those objects play different roles within the items' relational structure; that is, there is cross-mapping. One would therefore judge them to be similar, but not as similar as they would be if the objects in Item 2 were reversed.
Figure 1.
Stimuli like those used in Larkey & Markman (2005) and Hodgetts, Hahn, and Chater (2009).
An alternative to comparison theories of similarity is the representational distortion approach (RD: Hahn, Chater, & Richardson, 2003). On this view, the similarity between two items is determined by the degree to which one representation would have to be distorted for it to be fully transformed into the other (Chater & Hahn, 1997; Hahn et al., 2003). In some respects, RD requires some comparison (i.e., a mapping of matching components that don't require transformation) but the means of calculating similarity is different from that used by comparison models. Transformational complexity is intuitively defined as the length of the shortest computer program that could perform the transformation. This account also has an explanation for the distinction between identicality and cross-mapping: Item 2 must undergo some form of transformation to look like Item 1, while Item 1 need not be transformed at all to look like Item 1.
Comparison of Structural Alignment and RD
In the first direct comparison of structural alignment and representational distortion theories, Larkey and Markman (2005) had participants rate the similarity of object pairs like those in Figure 1. SIAM (Goldstone, 1994), a structural alignment model, provided the best qualitative fit to observed behavior. Hodgetts, Hahn, and Chater (2009), however, argued that Larkey and Markman's implementation of representational distortion was not an accurate reflection of the theory. Instead of treating each observable change in the stimuli as a transformation, as Larkey and Markman did, they averred that an appropriate distortion model must operate on changes in the representation of the stimuli.
Hodgetts et al. (2009), in turn, offered a more nuanced representational distortion account, based on three types of psychologically plausible transformations between base (i.e., the item selected to be transformed into the second item in the pair) and target (i.e., the second item):
Create feature – This introduces a new feature (i.e. not present in base item) into the representation set.
Apply feature – This takes a feature in the representation set, newly-created or in the base, and applies it to one or both of the existing features in the target.
Swap – This reverses individual features or whole objects in an item.
Consider Items 1 and 3 in Figure 1 with Item 1 treated as the base and Item 3 as the target. Transforming the former into the latter requires all three operations: swapping the two objects, creating a triangle feature, and applying it to the circle feature. In contrast, transforming Item 1 into Item 2 requires a single swap, reflecting the greater similarity between 1 and 2 than between 1 and 3.
To test the predictions of this model, Hodgetts et al. performed two straightforward experiments, similar to those of Larkey and Markman. In Experiment 1, there were 14 unique combinations of items. In this set, shape was varied but color was held constant, as in Items 1-3 in Figure 1. On each trial, participants compared one pair of items (e.g., 1 and 2) to another pair (e.g., 1 and 3), selecting one pair as more similar than the other. The relative similarity between items in a pair was defined as the percentage of trials in which that pair was selected as more similar. As expected, items that required more transformations were perceived as being less similar than items that required fewer transformations.
Assessing Representational Changes a Priori
The disagreement between Hodgetts et al. and Larkey and Markman hinges on the extent to which researchers are capable of determining the psychological transformations thought to occur during the process of comparison. In the remainder of this manuscript, we suggest that there are several major roadblocks to researchers who wish to define similarity using representational distortion, as the theory is currently formulated. First, we demonstrate that it is difficult in practice for RD to constrain the notion of a transformation a priori even using something as fundamental as Gestalt processing to identify which transformations are seen by participants to be psychologically salient. Second, we re-analyze the Hodgetts et al. data and present some new study data to argue that researchers need to account for which item participants are using as the base item in a pair. Lastly, we conclude that RD is not currently a viable theory of similarity and discuss a computational investigation by Müller, van Rooij, and Wareham (2009) that considers the computational tractability of RD.
Determining the Psychologically Salient Transformations
Early work on perceptual organization by Gestalt psychologists suggests that the way points are configured can influence how these points are perceived. Using simple rows of black dots, Goldmeier (1972) demonstrated that a row of dots that is closer together is more readily perceived as a line than is a row of dots spaced further apart. Furthermore, with more complex configurations of points, the Gestalt notion of “pattern goodness” (Garner & Clement, 1963) suggests that some configurations of points are more readily perceived as patterns than others. Similarly, theories in object perception argue that objects are perceived and processed with respect to the relations among object parts (Biederman, 1987; Hummel & Stankiewicz, 1996).
Experiment 1a
Based on these Gestalt notions, we used two different methods to test how participants perceive transformations. In Experiment 1a, we transform configurations of points that have a high- or low-degree of structure with two different manipulations. In the configuration-conserved condition, the overall configural structure is preserved by the combination of two transformations that independently do not preserve the configuration. In contrast, the other manipulation is not configuration-preserving when a pair of transformations is combined or when the transformations are presented independently. We call these sets configuration-conserved and configuration-broken, respectively. Importantly, our configuration-conserved and configuration-broken stimuli were constructed so that the same physical transformations occur in both types of manipulations within a stimulus set.
For example, as shown in Figure 2, we expect that similarity comparisons of the configuration-broken stimuli, in which the configurations are not preserved, will replicate the findings typical in RD studies (e.g., Hahn et al., 2003). The similarity between the base and the target stimuli should decrease as the number of transformations needed to transform the base into the target increases, making Target 3 the least similar item. In contrast, for the configuration-conserved stimuli, the overall configural match is destroyed by either of the transformations alone, but preserved when the transformations are combined, making Target 3 the most similar item. Based on the work in perception, the overall configuration should be encoded and used in the similarity comparison. Thus, for configuration-conserved stimuli, the similarity will be lower for one transformation stimuli than for the two transformation stimuli. RD would not predict this difference between the configuration-conserved and configuration-broken stimuli a priori. The stimuli were constructed using exactly the same transformations within each set, thereby receiving the same number and type of counted transformations.
Figure 2.
Sample Stimulus Set: Configuration-conserved and configuration-broken bases were transformed one time to yield targets 1 and 2. Target 3 was created by combining the transformations.1
As shown in Figure 2, the two-transformation configuration-conserved items are critical items. Despite being constructed with two transformations, it is possible that subjects will represent the change as one transformation, say expansion. Empirically, then, each stimulus would have one transformation producing identical similarity judgments. Theoretically, lower similarity judgments would suggest two transformations and higher similarity judgments would suggest one transformation. Using rated similarity to determine the number of transformations creates a circular argument. One needs to know the similarity judgments to determine the number of transformations represented, but the number of transformations needs to be determined a priori for RD to provide an explanation for similarity judgments. Our hypothesis for the configuration-conserved items relies on Gestalt notions, which avoids the problem of determining a priori how subjects will respond to particular transformations and what counts as a transformation. Experiment 1a will demonstrate that it is important to understand the influence of transformations of a stimulus on people's representation of that stimulus and vice versa. The influence of the number of transformations on rated similarity depends on the structure of the stimuli.
Method
Participants
The 40 participants in Experiments 1a were undergraduates at the University of Texas at Austin. They participated to fulfill a research requirement.
Materials
The stimuli consisted of line drawings constructed in Microsoft Word using the autoshapes function and formed 8 unique stimulus sets (see Appendix for stimuli). In Experiment 1a, within a stimulus set, there was a configuration-conserved and a configuration-broken stimulus base. As shown in the sample set in Figure 2, these bases were transformed one or two times to create the target stimuli. Each configuration-conserved stimulus had a corresponding configuration-broken stimulus that used the same one or two transformations. For example, the same transformation altered the configuration-conserved base and the configuration-broken base to form Target 1 in each set. Furthermore, following the procedure of Hahn et al. (2003), the pairs representing two transformations were constructed by combining both types of the individual transformations in one stimulus.
Procedure
Participants were presented with a booklet containing 48 pairs of stimuli. The pairs of stimuli represent the 6 unique base/target pairs of stimuli for the 8 different stimulus sets. These pairs were randomly ordered across sets and types of pairs, subject to the constraint that members of the same set could not be presented consecutively. For each pair, the participant rated the similarity of the stimuli on a 1 to 7 scale, with 1 indicating that the stimuli were very dissimilar and 7 indicating that the stimuli were very similar.
Results
Participants rated the similarity of base/target pairs with two types of sets: configuration-conserved and configuration-broken. In addition, within the configuration-conserved and configuration-broken sets, the pairs differed according to the number of transformations needed to transform the base into the target stimulus. The data were analyzed using a 2 (Match Type: configuration-conserved, configuration-broken) × 2 (Transformation: 1, 2) within-subjects ANOVA. All post-hoc contrasts were Bonferroni corrected. The mean similarity ratings for pairs in each condition are shown in Figure 3.
Figure 3.
Mean similarity ratings for pairs of configuration-conserved and configuration-broken stimuli post one and two transformations. Error bars indicate standard errors.
There was a main effect for Match Type, F(1,39) = 260.65, MSE = .32, p < .001, partial η2 = .87, such that the configuration-conserved pairs (M = 4.7) received higher mean similarity ratings than did the configuration-broken pairs (M = 3.2). In addition, there was a main effect for Transformation, F(1,39) = 4.44, MSE = .34, p < .05, partial η2 = .10. The mean similarity rating for the one transformation pairs (M = 4.0) was greater than the mean rating for the two transformation pairs (M = 3.8).
Consistent with our predictions, there was a significant interaction between Match Type and Transformation, F(1,39) = 92.45, MSE = .18, p < .001, partial η2 = .70. This interaction reflects that for the configuration-broken stimuli, rated similarity decreased significantly from one transformation (M = 3.6) to two transformations (M = 2.8), F(1,39) = 69.57, MSE = .2, p < .001, partial η2 = .64. In contrast, for the configuration-conserved stimuli, rated similarity increased significantly from one transformation (M = 4.4) to two transformations (M = 4.9), F(1,39) = 12.67, MSE = .31, p < .01, partial η2 = .25.
This finding is supported by both participant and stimulus set counts. Of the 40 subjects, 35 participants rated the configuration-broken pairs as being less similar after two transformations than after one. For the configuration-conserved pairs of stimuli, a majority of participants (26 out of 40) rated the two-transformation pairs as being more similar than the one-transformation pairs, and as can be seen in the effect size for the statistical test, the increase in ratings is large. The stimulus set counts revealed that participants rated the configuration-broken stimuli in 7 of the 8 sets as being less similar after two transformations as compared to after one. In addition, participants rated the configuration-conserved stimuli in 5 of the 8 stimulus sets as being more similar after two transformations as compared to after one transformation.
Experiment 1b
We present Experiment 1b to assess whether perceptual Gestalts could be integrated into RD to identify transformations a priori. One might argue that in Experiment 1a there is a distinct difference between transforming more structured stimulus bases in the configuration-conserved set and less structured bases in the configuration-broken set. According to this argument, the transformations of the configuration-conserved bases and the configuration-broken bases may actually be perceived and represented as different transformations.
In Experiment 1b, the configuration-conserved stimuli from Experiment 1a were augmented to form more complex stimuli, which we call configuration-buried stimuli. For example, as shown in Figure 4, the configuration-conserved base is embedded in the center of the configuration-buried base. Using this method, the configuration-conserved bases and the configuration-buried bases had an identical set of elements that was transformed. Therefore, in each case, the configuration of the identical structure was conserved after two transformations of the base but not after one transformation.
Figure 4.
Sample Stimulus Set: Configuration-conserved and configuration-buried bases were transformed one time to yield targets 1 and 2. Target 3 was created by combining the transformations.2
We argue that for the configuration-conserved stimuli, the overall configuration of points has been preserved between the base and two-transformation target. However, this preserved configuration is not present in a comparison of the base and the one-transformation targets. Therefore, as in Experiment 1a, we expect that as the number of transformations increase, the similarity of the items will increase.
The configuration-buried stimuli also preserve the same configuration of points as the configuration-conserved stimuli. However, based on Goldmeier (1972), we expect that this matching configuration will have a reduced likelihood of being used in comparisons with these more complex stimuli, because the matching configuration does not represent a matching configuration of points for the overall figure. Instead, we expect that people will focus on the changes between each component part of the configuration. As the number of transformations increases, there will be more changes among the components. Therefore, we expect that the configuration-buried stimuli to be rated as less similar after two transformations than after one transformation.
Method
Participants
The 40 participants in Experiment 1b were undergraduates at the University of Texas at Austin. They participated to fulfill a research requirement, and had not previously completed Experiment 1a.
Materials and Procedure
The stimuli consisted of line drawings constructed in Microsoft Word using the autoshapes function and formed 8 unique stimulus sets (see Appendix for stimuli). In Experiment 1b, within a stimulus set, there was a configuration-conserved and a configuration-buried stimulus base. The configuration-conserved stimuli are the same as the configuration-conserved stimuli in Experiment 1a. The configuration-buried stimuli were created by doubling the number of shapes that appeared in the configuration-conserved stimuli. Furthermore, the configuration-conserved stimuli remained intact when the additional shapes were added to form the configuration-buried stimuli. As shown in the sample set in Figure 4, these bases were transformed one or two times to create the target stimuli. The exact same transformations were completed within target types. For example, the same transformation altered the configuration-conserved base and the configuration-buried base to form Target 1 in each set. Furthermore, following the procedure of Hahn et al. (2003), the pairs representing two transformations were constructed by combining both types of the individual transformations in one stimulus.
The procedure was identical to Experiment 1a.
Results
In Experiment 1b, the participants rated the similarity between pairs of stimuli with two types of representations: configuration-conserved and configuration-buried. In addition, the number of transformations needed to change one stimulus into the other varied. The data were analyzed using a 2 (Match Type: configuration-conserved, configuration-buried) × 2 (Transformation: 1, 2) within-subjects ANOVA. All post-hoc contrasts were Bonferroni corrected. The mean similarity ratings for pairs in each condition are shown in Figure 5.
Figure 5.
Mean similarity ratings for pairs of configuration-conserved and configuration-buried stimuli post one and two transformations. Error bars indicate standard errors.
There was a main effect for Match Type, such that mean similarity ratings of the configuration-conserved pairs (M = 4.3) were significantly greater than for the configuration-buried pairs (M = 4.0) F(1,39) = 6.58, MSE = .28, p < .05, partial η2 = .14. There was not a main effect of Transformation. The mean similarity ratings of the one transformation pairs (M = 4.2) did not differ significantly from the mean similarity ratings of the two transformation pairs (M = 4.1) F(1,39) = .49, MSE = .21, p > .05. However, there was a significant interaction between Match Type and Transformation F(1,39) = 46.87, MSE = .22, p < .001, partial η2 = .55. This interaction reflects that for the configuration-conserved stimuli, rated similarity increased from one transformation (M = 4.1) to two transformations (M = 4.5), F(1,39) = 9.1, MSE = .43, p < .01, partial η2 = .19. In contrast, for the configuration-buried stimuli, rated similarity decreased from one transformation (M = 4.4) to two transformations (M = 3.8), F(1,39) = 31.52, MSE = .22, p < .001, partial η2 = .45.
These findings are supported by both participant and stimulus set counts. Of the 40 participants, a majority (26 of 40 participants), rated the configuration-conserved stimuli as being more similar after two transformations. As in Experiment 1a, this is a relatively small majority but the effect size associated with the ratings is of reasonable size. In contrast, 33 rated the configuration-buried pairs of stimuli to be less similar after two transformations than after one transformation. With respect to the stimulus sets, participants rated the similarity of the configuration-conserved pairs as being higher for 5 of the 8 sets after two transformations. In contrast, the rated similarity of the configuration-buried pairs was higher for 6 of the 8 sets after one transformation.
Discussion
The rated similarity of the configuration-broken and configuration-buried pairs decreased as the number of transformations between the base and target stimuli increased. In contrast, the rated similarity of the configuration-conserved pairs was higher for the two transformation items than for the one transformation items. These results suggest that the number of physical transformations in the stimulus domain does not necessarily correspond to the transformation distance in the mental representation domain. Moreover, it suggests that the higher similarity did not result simply from a calculation of RD using one transformation; if so, each stimulus would have been rated identically by our participants. Thus, it is crucial to understand how stimulus transformations affect the underlying mental representations of the items and to determine what counts as a meaningful transformation. Our studies suggest that it is difficult to predict a priori what would count as one or two transformations. This problem worsens as the number of possible transformations increases. As such, RD may not be able to rely on perceptual Gestalts to gain traction on determining the number of transformations a priori.
Considering Which Item to Use as the Base Item for Transformation
In this section, we consider the importance of determining which item is psychologically treated as the base item during a similarity judgment (Tversky, 1977). Using RD, the assumption is that the number of transformations required to turn the “base” into the target is used. In this approach, the item considered to be the base is the item that produces the fewest transformations. Table 1 describes the stimuli and results from Experiment 1 in Hodgetts et al. (2009). Schematic descriptions of the items appear in the first and second column. Each item is described with two letters, corresponding to unique shape features. For example, Item 1 in Figure 1 would correspond to AB, while Item 3 would be BC.
Table 1.
Hodgetts et al. (2009) Experiment 1
| # of | # of | |||
|---|---|---|---|---|
| Percentage | transformations | transformations | ||
| Pair 1 | Pair 2 | selected | (Hodgetts et al.) | (Pair 1 base) |
| AA | AA | 91.56 | 0 | 0 |
| AB | AB | 73.67 | 0 | 0 |
| AB | BA | 69.39 | 1 | 1 |
| AA | AB | 51.12 | 2 | 2 |
| AA | BA | 49.65 | 2 | 2 |
| AB | AA | 47.63 | 2 | 1 |
| AB | BB | 49.64 | 2 | 1 |
| AA | BB | 61.99 | 2 | 2 |
| AB | AC | 51.09 | 2 | 2 |
| AB | CB | 41.82 | 2 | 2 |
| AB | BC | 36.7 | 3 | 3 |
| AB | CA | 34.38 | 3 | 3 |
| AA | BC | 29.45 | 4 | 4 |
| AB | CC | 23.57 | 4 | 2 |
| r = -.95 | r = -.80 |
The selection percentages (i.e., relative similarity) and the number of transformations between items in a pair, as defined by Hodgetts et al., appear in the third and fourth columns of Table 1, respectively. Overall, there was a strong negative correlation between similarity and number of transformations (r = -.95, p < .01). This correlation was significantly stronger than the -.76 correlation for SIAM (z = 1.88, p < .05) and the -.74 correlation for Falkenhainer, Forbus, and Gentner's (1989) Structure Mapping Engine (SME) (z = 1.98, p < .05). Hodgetts et al. emphasize that this superior fit is particularly impressive because their transformation model—unlike SIAM and SME—has no free parameters.
This impressive correlation for the transformation model is largely due to assumptions about which pair is considered to be the base pair. In Hodgetts et al.'s formulation, the first step is to calculate the number of transformations required to convert each item into the other. The item requiring the greater number of transformations is then selected as the base. Although it is plausible that participants compare items in this manner, it is more likely that people use the item that is presented first as the base. The items highlighted in gray in Table 1 are those items when Pair 2 instead of Pair 1 was used as the base pair by Hodgetts et al. for their calculations. When the number of transformations is calculated by using Pair 1 as the base for all items, the correlation drops to -.80, indistinguishable from SIAM (z = .23, ns) and SME (z = .34, ns).
Experiment 2
To consider this possibility, we performed a simple experiment in which we varied the position of the items on the screen. On each trial, participants saw two items and were asked to make a similarity judgment. Each critical trial contained a red circle and a second item created by transforming the red circle. On half of the trials the red circle appeared on the left side of the screen, and on the other half the red circle appeared on the right. We manipulated screen position as a way to influence the item treated as the base during the similarity comparison. Our participants were native English speakers and reading from left to right automatically could cause participants to select the leftmost item as the base. To increase this possibility we included words in our stimuli: [Shape A] is like [Shape B]. We used simple transformations: expansion, contraction, indention, and protrusion. Our selection of transformations was inspired by Leyton (1989), who argued that people infer the causal history of an object based on the presence of two features: curvature and symmetry. When encountering pairs of objects that appear to be the same object but at different stages, he further argues that people can infer the intervening-process history. RD would predict no difference in similarity judgments based on screen position for items with identical transformations because it is the number of transformations that determines similarity.
Method
Participants
The 50 participants in Experiment 2 were undergraduates at the University of Texas at Austin. They participated to fulfill a research requirement.
Materials
The stimuli consisted of shapes constructed in Serif DrawPlus. There were three transformations of each type: expansion, contraction, protrusion, and indention. For example, for expansion, the transformations referred to below as 1, 2, and 3 were the base circle expanded by ¼ inch, expanded by ½ inch, and expanded by ¾ inch. The expansion and contraction stimuli were created using the same method, and the protrusion and indention mirrored each other. For example, as shown in Figure 6, the 1 indention item was created by inverting the 1 protrusion item. As such, the difference in transformation between a 1 indention item and the base and the 1 protrusion item and the base is identical.
Figure 6. Two example trials from Experiment 2.
Procedure
Participants completed 262 similarity judgments in the Experiment for pairs of items presented on a computer screen. Of these, 24 are considered critical trials with the remainder being filler items. Item presentation was randomized for each participant. The critical stimuli represent the 6 unique base/target pairs of stimuli for the 4 different stimulus sets. That is, each transformation type has 3 forms and the circle was presented on both the right and the left. For each pair, the participant rated the similarity of the stimuli on a 1 to 9 scale, with 1 indicating that the stimuli were very dissimilar and 9 indicating that the stimuli were very similar.
Results
The data were analyzed using a 2 (Circle Position: left, right) × 4 (Transformation Type: expansion, contraction, indention, protrusion) × 3 (Transformation: 1, 2, 3) within-subjects ANOVA. There was a main effect of Transformation Type, F(3,66) = 94.34, MSE = 3.32, p < .001, partial η2 = .81, and a main effect of Transformation, F(2,44) = 62.89, MSE = .22, p < .001, partial η2 = .74. There was not a main effect of Circle Position. As the number of transformations increased, the rated similarity decreased (M = 6.4, 5.6, and 5.0 for 1, 2, and 3 transformations, respectively) showing a strong linear trend, F(1,22) = 88.69, MSE = 2.04, p < .001, partial η2 = .80, and participants' ratings for the protrusion and indention items did not differ (M = 4.4 and 4.6, respectively), t(45) = 1.33, p = .19. However, participants rated the expansion items as having greater similarity with the base (M = 7.4) as compared to the contraction items (M = 6.6), t(45) = 5.36, p < .001, d = .80.
There was also a Transformation Type × Circle Position interaction, F(3,66) = 3.89, MSE = 1.12, p = 0.013, partial η2 = .15. There were no other significant interaction effects. For Expansion, items were rated as more similar when the circle appeared on the right side of the screen (M = 7.5) than the left side of the screen (M = 7.2), F(1,24) = 8.02, MSE = .48, p = .009, partial η2 = .25, while there was no difference for Contraction (M = 6.6 and 6.6), F = .16. For Protrusion, items were rated as more similar when the circle appeared on the left side of the screen (M = 4.8) than the right side of the screen (M = 4.2), F(1,24) = 6.19, MSE = 1.7, p = .021, partial η2 = .21, while there was no difference for Indention (M = 4.3 and 4.4), F = .21.
Discussion
By recoding the stimulus transformation in Hodgetts et al. and performing Experiment 2, we demonstrate that it is critical to realize that the base item used by participants fundamentally affects similarity judgments. The expand and contract items had exactly the same transformations (e.g., ¼ inch smaller than the base or ¼ inch larger than the base in the 1 expand and 1 contract stimuli). However, participants' treatment of this transformation differed. Participants rated expansion-transformed items as being more similar to the circle than contraction transformed items. Moreover, the position of the circle on the screen, which could influence the item selected by participants as the base (as native English speakers our participants automatically read from left to right), also interacted with the type of transformation to alter similarity judgments. Our results suggest that adding and subtracting a feature from the base are also treated differently. RD cannot account for these effects because the number of transformations alone determines similarity. As suggested by Hahn et al. (2003), RD does allow for different transformation weights and therefore could theoretically apply higher weights to transformations using a specific item as a base. Based on our data, however, it is not clear how this unequal weighting could reasonably be applied a priori. Similarity judgments varied with the type of transformation and position of the circle base. Furthermore, allowing unequal weights to a transformation and its inverse seems very unnatural, and RD would no longer be void of free parameters. It will be critical for RD to account for base item selection in accounting for similarity judgments.
Conclusion
The Representation Distortion approach is not a viable theory of similarity in its current formulation. We do believe that RD has made a conceptual contribution to the study of similarity because it has challenged researchers to think about similarity processing in new ways. However, we believe that theories of similarity and tests of those theories must make meaningful a priori claims about representation. It is important that theories of similarity can identify psychologically meaningful transformations and account for the item selected as the base item that is transformed. In support of this view, participants in Experiment 1a rated the similarity of pairs of configuration-conserved and configuration-broken stimuli after one or two transformations. Overall, participants rated the configuration-conserved pairs to be more highly similar than the configuration-broken pairs. In addition, as compared to the similarity ratings after one transformation, participants rated the configuration-broken stimuli as being less similar and rated the configuration-conserved stimuli as being more similar after two transformations. In Experiment 1b, the similarity ratings of the configuration-conserved stimulus pairs again increased as the number of transformations increased. In contrast, like the configuration-broken stimuli in Experiment 1a, the similarity ratings of the configuration-buried stimulus pairs decreased as the number of transformations increased. The results from these experiments highlight the potential importance of transformation distance in the similarity comparison process, while recognizing that it is crucial to understand and predict the influence of transformations on item representations.
In Experiment 2, we demonstrated that researchers do need to account for which item is considered to be the base item during the process of computing similarity. Items with identical transformations were not rated as having identical similarity with the circle item and the placement of the circle item on the screen influenced judgments for half of the transformations. One avenue for future research could be examining when and why base selection matters.
Lastly, while the Hodgetts et al. model has no numerical free parameters, the idea of transformations is sufficiently vague to allow a great deal of flexibility. All of the possible transformation operations are essentially hidden degrees of freedom. These hidden degrees of freedom can lead to apparent statistical anomalies (McKay, Bar-Natan, Bar-Hillel, & Kalal, 1999). Model flexibility per se is not necessarily a problem, but that flexibility should be explicitly built in to the model. Implicit flexibility can lead to inappropriate comparisons between models, domain rigidity, and coding errors in some cases. Furthermore, it is not necessarily true that the computational processes required to implement RD in one domain (e.g., computers) are the same processes required to implement RD in human cognition.
Müller et al. (2009) explicitly tested the computational tractability of RD. They formulated 4 different computational versions of RD and discovered that computing similarity using each formulation required super-polynomial time algorithms. As explained by Müller et al., super-polynomial algorithms are considered to be intractable because they take a very long time unless there are only a small number of inputs. To address this intractability, Müller et al. examined specific parameters to determine the source of the intractability. Some of these parameters included the length of the shortest sequence of transformations, the size of the set of possible transformations, the maximum lengths of the representations of both stimuli, the maximum lengths of all of the intervening representations required to turn one representation into another, and the size of the set of possible transformations most likely to be relevant in the current context. Müller et al. find that RD is tractable only when researchers assume that similarity is only calculated for items that are somewhat similar, the set of transformations used in a specific context is relatively small, and that the number of total transformations cannot be too large. RD theorists could potentially allow for a non-computational process to select the set of transformations but this would reduce the explanatory power of RD (van Rooij, I., 2008). RD theorists recognize that the set of transformations needs to be rather large to account for the data (Hahn et al., 2003), because there needs to be a large enough set of possible transformations from which subsets of relevant transformations can be generated. However, the requirement of such a large set of possible transformations confronts the issues raised by Müller et al. that RD needs a smaller set of possible transformations to be computationally tractable. Thus, on the one hand RD needs a large set of possible transformations to provide sufficient degrees of freedom to fit the data, but on the other hand the theory is computationally intractable with large numbers of transformations.
We suggest that RD theorists should focus on considering how to constrain the notion of a transformation and to recognize that people likely select a particular item to use as a base during the similarity computation process. The current operationalization does not provide enough specificity to mirror similarity processing. As an example, we consider some hypotheses suggested by structural alignment theory. Derived from structure mapping theory (Gentner, 1983), structural alignment assumes that the similarity between two items results from both the commonalities and the differences that emerge from comparisons (Gentner & Markman, 1997) and this approach to similarity has been implemented by a variety of symbolic and connectionist models (Falkenhainer, Forbus, & Gentner, 1989; Holyoak & Thagard, 1989; Hummel & Holyoak, 1997; Keane, Ledgeway, & Duff, 1994; Love, 2000). The theory assumes that items are represented by structured hierarchical representations consisting of object attributes, which describe objects, and relations, which relate 2 or more object attributes or relations (Gentner, 1983; Gentner & Markman, 1997; Markman, 2001). During a similarity comparison, items are placed into correspondence that conserves their commonalities (Markman, 1999).
By applying the predictions of structural alignment to Representational Distortion, the notion of a transformation may be constrained. For example, transformations of attributes may be perceived to be qualitatively different and independent from perceptions of transformations of relations (Goldstone, Medin, & Gentner, 1991). Furthermore, dimensions like size and color may be quantified over entities defined as shapes so transformations on these attributes may be treated differently than transformations on shapes (Love & Markman, 2003).
A hybrid model combining structural alignment and RD could potentially use RD principles to select the best mapping of items when multiple mappings result from the comparison process. The best mapping would be the one that involves the lowest complexity to turn the representation of one item into the representation of the other. That said, if RD theorists are interested in creating such a model, it will be important to demonstrate empirically that the mappings created and the final mapping selected by a structural alignment model, such as SME, are not the mappings generated and selected by people, and therefore that RD principles could contribute to the process. We hope that RD theorists further constrain the transformation process to allow this framework to influence the body of work on similarity judgment.
Acknowledgments
Research supported by National Institutes of Health Grant 1 R21 DA015211-01A1 to Arthur B. Markman. Data collected at the University of Texas. The authors thank Iris van Rooij, 2 anonymous reviewers, and the AE for very helpful comments on the manuscript.
Appendix
Stimuli from Experiment 1a.
| Stimulus Set | Type | Base | Target 1 | Target 2 | Target 3 |
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| 1 | Configuration-conserved |
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| 3 | Configuration-conserved |
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| 4 | Configuration-conserved |
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| 5 | Configuration-conserved |
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| 6 | Configuration-conserved |
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| 7 | Configuration-conserved |
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Stimuli from Experiment 1b.
| Stimulus Set | Type | Base | Target 1 | Target 2 | Target 3 |
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| 1 | Configuration-conserved |
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| 2 | Configuration-conserved |
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| 4 | Configuration-conserved |
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| 5 | Configuration-conserved |
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| 6 | Configuration-conserved |
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| 7 | Configuration-conserved |
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| 8 | Configuration-conserved |
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Footnotes
Note: Participants did not see the grey boxes or arrows; they have been added to help clarify the transformations for the reader. Furthermore, the configuration-conserved stimuli retained the same overall configural structure between the base and target 3 after two transformations whereas configuration-broken stimuli did not. Also, the same transformations were performed within targets 1, 2, and 3 for the configuration-conserved and configuration-broken bases.
Note: The shading of the blocks in the configuration-buried stimuli depict where the configuration-conserved stimuli were embedded. Participants did not see the stimuli with this shading.
References
- Biederman I. Recognition-by-Components - a Theory of Human Image Understanding. Psychological Review. 1987;94:115–147. doi: 10.1037/0033-295X.94.2.115. [DOI] [PubMed] [Google Scholar]
- Chater N, Hahn U. Representational distortion, similarity, and the universal law of generalization; Paper presented at the Interdisciplinary Workshop on Similarity and Categorization, SimCat97; Edinburgh. Department of Artificial Intelligence, University of Edinburgh; 1997. [Google Scholar]
- Falkenhainer B, Forbus K, Gentner D. The structure-mapping engine: Algorithm and examples. Artificial Intelligence. 1989;41:1–63. [Google Scholar]
- Garner WR, Clement DE. Goodness of Pattern and Pattern Uncertainty. Journal of Verbal Learning and Verbal Behavior. 1963;2:446–452. [Google Scholar]
- Gentner D. Structure-mapping: A theoretical framework for analogy. Cognitive Science. 1983;7:155–170. [Google Scholar]
- Gentner D, Markman AB. Structure mapping in analogy and similarity. American Psychologist. 1997;52:45–56. [Google Scholar]
- Goldmeier E. Similarity in visually perceived forms. Psychological Issues. 1972;1 Monograph 29. [PubMed] [Google Scholar]
- Goldstone RL. Similarity, interactive activation, and mapping. Journal of Experimental Psychology: Learning, Memory, and Cognition. 1994;20:3–28. [Google Scholar]
- Goldstone RL, Medin DL, Gentner D. Relational similarity and the nonindependence of features in similarity judgments. Cognitive Psychology. 1991;23:222–262. doi: 10.1016/0010-0285(91)90010-l. [DOI] [PubMed] [Google Scholar]
- Hahn U, Chater N, Richardson LB. Similarity as transformation. Cognition. 2003;87:1–32. doi: 10.1016/s0010-0277(02)00184-1. [DOI] [PubMed] [Google Scholar]
- Hodgetts CJ, Hahn U, Chater N. Transformation and alignment in similarity. Cognition. 2009;113:62–79. doi: 10.1016/j.cognition.2009.07.010. [DOI] [PubMed] [Google Scholar]
- Holyoak KJ, Thagard P. Analogical mapping by constraint satisfaction. Cognitive Science. 1989;13:295–355. [Google Scholar]
- Hummel JE, Holyoak KJ. Distributed Representations of Structure: A Theory of Analogical Access and Mapping. Psychological Review. 1997;104:427–466. [Google Scholar]
- Hummel JE, Stankiewicz BJ. Categorical relations in shape perception. Spatial Vision. 1996;10:201–236. doi: 10.1163/156856896x00141. [DOI] [PubMed] [Google Scholar]
- Keane MT, Ledgeway T, Duff S. Constraints on Analogical Mapping: A comparison of three models. Cognitive Science. 1994;18:387–438. [Google Scholar]
- Larkey LB, Markman AB. Processes of similarity judgment. Cognitive Science. 2005;29:1061–1076. doi: 10.1207/s15516709cog0000_30. [DOI] [PubMed] [Google Scholar]
- Leyton M. Inferring causal history from shape. Cognitive Science. 1989;13:357–387. [Google Scholar]
- Love BC. A computational level theory of similarity. Proceedings of the Annual Meeting of the Cognitive Science Society; Philadelphia, PA. 2000. pp. 316–321. [Google Scholar]
- Love BC, Markman AB. The nonindependence of simulus properties in human category learning. Memory & Cognition. 2003;31:790–799. doi: 10.3758/bf03196117. [DOI] [PubMed] [Google Scholar]
- Markman AB. Knowledge Representation. Mahwah, NJ: Lawrence Erlbaum Associates; 1999. [Google Scholar]
- Markman AB. Structural alignment, similarity, and the internal structure of category representations. In: Hahn U, Ramscar M, editors. Similarity and Categorization. Oxford, UK: Oxford University Press; 2001. pp. 109–130. [Google Scholar]
- Markman AB, Gentner D. Structural alignment during similarity comparisons. Cognitive Psychology. 1993;25:431–467. [Google Scholar]
- McKay B, Bar-Natan D, Bar-Hillel M, Kalal G. Solving the Bible code puzzle. Statistical Science. 1999;14:150–173. [Google Scholar]
- Müller M, van Rooij I, Wareham T. Similarity as tractable transformation. Proceedings of the Annual Meeting of the Cognitive Science Society; Amsterdam, the Netherlands. 2009. pp. 50–55. [Google Scholar]
- Shepard RN. Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space. Psychometrika. 1957;22:325–345. doi: 10.1037/h0042354. [DOI] [PubMed] [Google Scholar]
- Tversky A. Features of similarity. Psychological Review. 1977;84:327–352. [Google Scholar]
- van Rooij I. The tractable cognition thesis. Cognitive Science. 2008;32:939–984. doi: 10.1080/03640210801897856. [DOI] [PubMed] [Google Scholar]