A Computational Level Theory of Similarity

A Computational Level Theory of Similarity Bradley C. Love Department of Psychology The University of Texas at Austin MEZ 330 Austin, TX 78712 USA love@psy.utexas.edu Abstract Why are some pairs of objects (or events) perceived to be more similar to each other than other pairs? A computational level theory of perceived similarity is presented that extends previ- ous geometric and set-theoretic formulations. Like previous approaches, the current account posits that the similarity of two objects is a function of the common and distinctive fea- tures of the two objects. Unlike previous approaches, simi- larity is also a function of higher-order compatibility relations among features (as it is in models of analogy). Objects (or con- cepts) are represented as directed feature graphs as opposed to feature vectors or sets. Like current accounts of human analogical processing, the approach presented here holds that representational elements are put into correspondence during the comparison processes. Correspondences are chosen in or- der to maximize an objective function. The function contains four terms that are motivated by theories of human compari- son. The maximum of the function is monotonically related to perceived similarity. Thus, similarity is characterized as the byproduct of comparison and structural alignment. The objec- tive function serves as a quantitative computational level the- ory of human comparison. Introduction Since William James (1890/1950), psychologists have held that detecting the “sameness” or similarity of objects is at the backbone of cognition. Clearly, detecting similarities be- tween novel events and previous experiences is crucial in rea- soning, analogy, and object recognition. Many theories of category learning hold that similarity is the basis for catego- rization (see [7]). A fundamental question then is what makes two objects similar? Almost all accounts of perceived similarity hold that sim- ilarity increases as the number of feature matches increases and decreases as the number of feature mismatches increases. In geometric models of similarity, such as multidimensional scaling (MDS) models of similarity, concepts or objects are represented as points in a multidimensional space and sim- ilarity is inversely related to the distance between points in the space [20]. Objects that match on many features will be closer together in the space than objects that mismatch on a number of features. Unfortunately, the axioms of metric spaces (e.g., minimality, symmetry, and the triangle inequal- ity) appear to be violated by human similarity judgments (see [22]). More recently, Medin, Goldstone, and Gentner (1993) have demonstrated that an object can be rated as both more similar and more dissimilar to the same object in an object pair, which seems problematic for distance models. Tversky’s (1977) contrast model is a non-metric set- theoretic account of perceived similarity that aims to address some of the shortcoming of the distance models. Tversky’s model is based on evaluating sets of matching and mismatch- ing features: sim ( x; y ) = 1 F ( X \ Y ) sim 2 F ( X Y ) 3 F ( Y X ) where 1 ; 2 ; 3 0 where ( x; y ) reads “the similarity of x to y ,” X is the set of features that represents x , Y is the set of features that represents y , X \ Y is the set of features common to x and y , X Y is the set of features uniquely possessed by x , Y X is the set of features uniquely possessed by y , 1 , 2 , and 3 are free parameters, and F is a function over sets of features related to the features’ saliency. For simplicity and without loss of generalization, we assume here that all features are equally salient: ( X ) = j X j F where j X j denotes the cardinality of the set X . Of course, in many situations certain features will weigh more heavily on the evaluation of similarity than other features. Tversky’s contrast model can account for asymmetries that occur in similarity judgments. For example, “North Korea” is rated as being more similar to “China” than vice versa. The contrast model can explain such asymmetries by setting 2 > 3 . Ostensibly, when comparing x and y , the focus is on first term x , which I will refer to as the target, and not on the second term, which I will refer to as the base. Both x and y will be referred to as analogs. In the example above, most people know more about China than North Korea. Accord- ingly, when evaluating how similar China is to North Korea j X Y j will be larger than j Y X j . Another comparison related explanation for asymmetries is that subjects prefer the base to be the object out of an object pair that allows for more analogical inferences to be projected [1]. Such asymmetries may be attributable to the similarity predicate in particular. Another alternative is that asymmetries arise from general principles related to sentence interpretation such as the fig- ure/ground relationship between the target and base [21] or from general syntactic properties [6]. Although the contrast model can address a wide range of data, it cannot account for judgments of similarity that are re- lational or analogical in nature. Two analogs can be similar