Geometric Conceptual Spaces Ben Adams GEOG 288MR Spring 2008

Semantic Web One ontology, one language for all content on the web Integration of information facilitated by metadata Emphasis on the development of languages such as RDF (information representation) and OWL (web ontologies).

Semantic Web Problems Gärdenfors argues semantic web is not really “semantic”  Limited to first-order logical reasoning  Concepts are represented as sets of objects Many problems not solvable by deductive logic or using set theory Example: using similarity measurement as a tool for categorization  Tversky might disagree!

Semantic Web Problems Syllogistic reasoning important in many areas of AI, but only proven to be effective in domains already heavily dependent on deductive reasoning (e.g. law and medicine).  Rissland paper from week 1 Combinations of concepts  Intersections of sets too simple to work as a model for the meaning of combinations of concepts  e.g. tall squirrel, honey bee, stone lion, white Zinfandel

Symbol Grounding Ontologies in semantic web are free-floating  not grounded in the real world  cannot resolve conflicts between ontologies Human cognition includes non-symbolic representations Alternative proposed by Gärdenfors is to use conceptual spaces

Representation methodologies Symbolic  Symbol manipulation computations  Set-theoretic / feature based (Tversky, semantic web)‏ Associationist  Neural network models  fine-grained, complex, subconceptual representations Conceptual  Geometric conceptual models

Conceptual spaces Information represented by geometric structures  objects are points  properties and relations are regions Similarity is measured by the distance between points or regions in space Conceptual space defined by quality dimensions

Quality dimensions Can be tied to sensory input or can be abstract  e.g. hue, temperature, weight, spatial dimensions  e.g. functional concepts Integral or separable  Integral – an object's value on a given dimension requires a value on another dimension  Does not mean integral dimensions are non- orthogonal  Separable – dimensions that are not integral Dimension has a metric geometrical structure

Quality dimensions Represent qualities in various domains Domain - “a set of integral dimensions that are separable from all other dimensions”  Color – example of domain  Hue, saturation, brightness quality dimensions are integral with respect to each other, but they are separable from other dimensions. Together they are the set of quality dimensions that we define as the domain color.

Properties Definition based on geometrical structure of quality dimensions A property P is a convex region in some domain Objects with property P are represented as points within the convex region P betweenness holds due to convexity – if two points v1 and v2 are in region P then any point on the line between v1 and v2 is in region P  line can be curved or straight depending on metric (e.g. polar coordinates vs. Euclidean)‏

Convexity Convex – betweenness holdsConcave – betweenness does not hold

Polar Betweenness

Properties and Concepts Properties are distinct from concepts in conceptual model  Not covered by symbolic and connectionist representations  Property is a subset of concept Property - based on a single domain Concept - based on one or more domains Semantically, properties represent adjectives and concepts represent nouns

Concepts Concepts change depending the relevance of different domains for the given problem ex. “Apple” concept: Relative weightings of domains are determined by the context

Concepts More than just bundles of properties Also, correlations between associated region in one domain and associated region in another domain.  Quality dimensions are not necessarily orthogonal across domains

Concept definition “A concept is represented as a set of convex regions in a number of domains together with a prominence assignment to the domains and information about how the regions in different domains are correlated.”

Prototypes Prototypes are used to categorize objects Prototypical members are “most representative members of a category” In conceptual space, prototype can be seen as centroid of all objects of a category Voronoi tessellation of the space can be used to partition the conceptual space into convex categories (concepts)‏

Voronoi tessellation two dimensionalthree dimensional

Prototypes Concepts can be described as more or less similar to each other Objects can be described as more or less centrally representative of a concept

Concept hierarchies Because of geometric structure of concept spaces, concept hierarchies emerge from the shape of regions in space e.g. robin concept is a subregion contained in the bird concept region Don't need semantic information of relationships such as the kind used in OWL, because domain structure generates these relationships Question remains: how do we identify domains?

Concept combinations property-concept combinations  XY where X is property, Y is concept  compatible – intersection of concepts  incompatible – for incompatible regions, the region of X overrules the region of Y For example, pink elephant: X = pink, Y = elephant region of color domain for elephant (the gray area) is overruled by pink color domain

Concept combinations Shifts caused by X of a region in Y in one domain can cause shifts for regions in Y in other domains because of correlations Example: brown apple. Brown modifies color domain of apple and also the texture domain

Contrast Classes Accounts for changes in meaning of the concept based on context “The combination XY of two concepts X and Y is determined by letting the regions for the domains of X, confined to the contrast class defined by Y, replace the corresponding regions for Y.”

Formalizing Conceptual Spaces [Raubal, 2004] Gärdenfors' model formalized using linear algebra and statistics Use z-scores to make sure same relative unit of measurement is used by all variables. That way distance measurements make sense.

Conceptual Vector Space Conceptual vector space  C n = {(c 1, c 2,..., c n ) | c i ε C} Quality dimension can represent a domain  c j = D n = {(d 1, d 2,..., d n ) | d k ε D}

Example Conceptual Vector Space c1c2c5c4c3c6c7 facade area shape factor shape deviation identifiability by signs cultural importance visibilitycolor C conceptual space d1d2d3 redgreenblue color domain facade

Distances and weights Calculating semantic distances between concepts u and v with z scores  (u 1, u 2,..., u n ) → (z 1 u, z 2 u,..., z n u ), (v 1, v 2,..., v n ) → (z 1 v, z 2 v,..., z n v )‏  |d uv | 2 = (z 1 v – z 1 u ) 2 + (z 2 v – z 2 u ) (z n v - z n u ) 2 Weighting dimensions for context  C n = {(w 1 c 1, w 2 c 2,..., w n c n ) | c i ε C, w j ε W}

Facade example 'Facade' concept represented in two different conceptual spaces – system and user Different contexts, day and night, modeled using different sets of weight values Mapping between system and user concept spaces  projection to smaller number of dimensions  transformations  can result in loss of information

Spatial Relations for Semantic Similarity Measurement [Schwering & Raubal, 2005] Application of geometric conceptual space modeling Ordinance Survey MasterMap case study to find flooding areas in Great Britain Found that adding spatial relations to the quality dimensions improved the results of queries

Spatial relations Objects described as concepts with properties and relations between concepts Converted spatial relations to either boolean or ordinal values Addition of spatial relations gave better matches for all concepts that were compared Some confirmation that city-block metric works better with separable dimensions