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