Geometric Conceptual Spaces Ben Adams GEOG 288MR Spring 2008.
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Transcript of Geometric Conceptual Spaces Ben Adams GEOG 288MR Spring 2008.
Geometric Conceptual Spaces
Ben AdamsGEOG 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 holds Concave – betweenness doesnot 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
Domain RegionColor Red-yellow-greenShape Roundish (cycloid)Texture SmoothTaste Regions of the sweet and sour dimensionsFruit Region defined by specification of various characteristicsNutrition Values of sugar content, vitamins, fiber, etc.
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 dimensional three 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 Cn = {(c
1, c
2, ..., c
n) | c
i ε C}
Quality dimension can represent a domain cj = Dn = {(d1, d2, ..., dn) | dk ε D}
Example Conceptual Vector Space
c1 c2 c5c4c3 c6 c7
facadearea
shapefactor
shapedeviation
identifiabilityby signs
culturalimportance
visibilitycolor
C
conceptualspace
d1 d2 d3
red green blue
color domain
facade
Distances and weights
Calculating semantic distances between concepts u and v with z scores (u1, u2, ..., un) → (z1
u, z2u, ..., zn
u), (v1, v2, ..., vn) → (z1v, z2
v, ..., zn
v)
|duv|2 = (z1v – z1
u)2 + (z2v – z2
u)2 + ... + (znv - zn
u)2
Weighting dimensions for context Cn = {(w1c1, w2c2, ..., wncn) | ci ε C, wj 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