=Paper=
{{Paper
|id=None
|storemode=property
|title=Structure, Similarity and Spaces
|pdfUrl=https://ceur-ws.org/Vol-1007/paper4.pdf
|volume=Vol-1007
|dblpUrl=https://dblp.org/rec/conf/shapes/FioriniAG13
}}
==Structure, Similarity and Spaces==
https://ceur-ws.org/Vol-1007/paper4.pdf
Structure, Similarity and Spaces
Sandro Rama FIORINI a,1 , Mara ABEL a and Peter GÄRDENFORS b
a
Institute of Informatics, UFRGS, Brazil
b
Lund University Cognitive Science, Lund University, Sweden
Abstract. Much of the discussion about shape representation during the
last two decades was fundamentally related to questions about the rep-
resentation of parts. Inspired by the cognitive processes governing how
people represent and think about parts, we give a brief summary of
our framework for representing part structures. In particular, we are
interested in the role of similarity and prototype e↵ects in this context.
Keywords. Part-whole relations, similarity, conceptual spaces
Introduction
Humans seem to be prone to divide the complex shape of objects into parts.
In seeing a cat, we divide its overall shape in some more-or-less well defined
parts, such as the head, trunk, tail and legs. We can then use this information
to recognize and think about cats. It seems that structure is intrinsically related
to our everyday notion of shape. That leads us to a broader question: what are
the cognitive phenomena that allow us to represent and think about parts of
object as a whole, and not just parts of their shapes? In this paper, we introduce
a novel way to represent the relation between parts and wholes that takes into
consideration some of these phenomena.
In an influential work, Farah [4] carried out a meta-study about di↵erent
kinds of agnosia in humans and proposed that two processes participate in object
recognition. Object recognition can be structural, where recognition is achieved
by the identification of parts of the object and its internal structure. On the
other side, recognition can be holistic, based on the global characteristics of the
object, such as overall shape. According to Farah, both processes work together in
recognitions of broad categories of entities (e.g. faces, objects and written words).
It is important to note that, as pointed by Peissig and Tarr [7], the structural
versus holism problem is correlated, but independent of the recurrent question
whether object (shape) recognition is model- or view-based (i.e. [2,3]).
If we assume that holistic and structural processes are necessary for object
recognition, therefore it is reasonable to expect that both require an underly-
ing conceptual structure conveying holistic and structural information. When it
1 Corresponding Author: Sandro Fiorini, Institute of Informatics, UFRGS, P.O. Box 15064,
CEP 91501-970, Porto Alegre-RS, Brazil; E-mail: srfiorini@inf.ufrgs.br. We thank CAPES and
Petrobras project PB PRH-17 for the support in this work and Joel Carbonera for the comments.
71
comes to conceptual structures for object recognition, similarity e↵ects seem to
play a fundamental role (c.f. [3,1]). The general notion is that recognition could
be reduced to judgements of similarity between perceptual input and internal
representations. The question we are trying to answer is how a holistic/structural
representation framework that supports similarity judgements should look like.
We base our e↵orts on the Theory of Conceptual Spaces [5], a representation
framework that embeds the notions of concept similarity and prototypes. Our
general approach is to extend conceptual spaces so that it becomes more suitable
for the representing holistic and, specially, structural information about concepts
and objects. It should provide the grounds for novel computational approaches
to concept representation based on holistic and structural similarity.
1. Conceptual Spaces
Gärdenfors’ Theory of Conceptual Spaces [5] puts forward a new way for repre-
senting concepts using geometrical and topological structures, which complements
symbolic and connexionist approaches. Given the available space, we present just
a brief introduction, but it is enough to say that the Theory is based on the
notions of concept similarity and prototypes. Put it simple, a conceptual space
is a space in the mathematical sense, where objects are points and concepts are
regions or sets of regions. If this space has a well-defined metric, then it is possible
to tell the similarity between objects (and concepts) by measuring the distance
between them: further objects are apart, less similar they are. The dimensions of
a conceptual space have a special meaning: they denote the features — or qual-
ities — through which entities can be compared and are frequently grounded in
perception. Good examples are hue, mass, height, etc. Certain quality dimensions
always co-occur, forming subspaces called quality domains. Examples of quality
domains are colour, shape, taste, etc. Complex concepts are defined as set of re-
gions in many quality domains. For instance, the concept of apple can be defined
as a combination of the regions green and red in the colour domain, plus the
cycloid regions in the shape domain, plus the sweet and acid regions in the taste
domain and so on. An individual apple is represented by a single point (or vector)
in the multi-dimensional space formed by all quality domains of apple and that
is close (similar) enough to the regions that form the concept of apple. A type of
apple is formed by subregions of apple.
2. Structure Spaces
We are interested in using conceptual spaces to represent holistic and structural
information about concepts and objects. The holistic portion of an entity can be
seen as its usual features: colour, shape, weight, etc. These can be readily repre-
sented as regions in quality domains. However, representing structural informa-
tion is far from trivial. In doing so, we are fundamentally interested in describing
the partonomic relations between parts and wholes. The Theory of Conceptual
Spaces does not provide a complete solution for representing relations in general;
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it simply suggests that they could be represented by a (Cartesian) product be-
tween the relata. We take this basic notion and develop it further to account for
structural information.
Structural information can be represented in conceptual spaces through what
we call structure spaces. A concept are represented as regions in a conceptual
space; a structure space is a subspace of it, where part structures are described.
A vector in a structure space denotes a particular configuration of parts of an
individual. That is, a single vector encode the information about what parts
compose a whole and also about how these parts are related to whole. Similar
configurations of parts are close together in this space. For a single vector to
convey all this information, much of it is naturally allocated to the dimensions.
Given a concept C and a set of concepts P1 , . . . , Pn denoting parts of C, then we
can generally define the structure space containing C as the product space of the
quality domains of P1 , . . . , Pn and n quality domains denoting specific structure
information about each part Pi . We call these domains structure domains; they
represent information such as the displacement of the part in the whole, part
quantification and so on. The actual structure space of C is the product space
of the regions that define P1 , . . . , Pn , plus regions in the structure domains. For
instance, the structure space of Apple could be formed by the product of Core,
Flesh, Seed and Stein, plus regions denoting the general positioning each part in
an object-centred coordinate system. A vector in the structure space of C denotes
a particular apple-structure: a combination of individual parts, each with a specific
value for colour, shape, taste and so on. Close points in this space represent
similar apple-structures. The combination of regions of each part in the product
restricts what are the valid individual components of an apple. More importantly,
the structure space can be further divided into specific regions defining types of
apple-structures; e.g., the concept of an apple with acid flesh and short stein.
This basic formulation of our framework might raise questions, such as prob-
lem of co-determination between holistic and part qualities; the role of parts in
taxonomies (c.f. Tversky [8]), the representation of the many kinds of partonomic
relations; the question about dependent and essential parts; computational fea-
sibility and so on. To all of those we have at least partial answers, but due to
the limited space, we shall touch upon the issue we consider most critical: the
problem of transitivity.
Wholes have parts, which can also have parts and so on. This might become
an issue when we define a whole as a product of parts: given parts also can
have parts, complex wholes could soon become multidimensional monsters; the
structure space of the concept Universe would be impossible to describe fully.
In more formal terms, we could say that the structure space of a whole might
become the transitive closure of its parts. In order to solve this problem, we first
assume that the part relation is not essentially transitive. This position contrast
with formal theories of parts, such as Classical Mereology, but it has becoming
increasingly common in recent years (c.f. [6]). Instead, we take a more cognitive
stance; experience and perception are the sole determinants of which parts directly
compose a whole. For instance, what determines that the person’s heart is not
part of the company where she works in is the fact that there is no use for
such conceptualization in the actual context. However, if there is a change in
73
context (e.g. parts of employee body becoming property of the company), we
can easily adapt our conceptualization. We do not assume any hard a priori
ontological distinction on parts and wholes, for it might hinder the plasticity of
the representation. Instead, we provide a way in such plasticity could operate in
our framework, by improving the definition of structure spaces. We introduce the
notion of dimensional filter. A dimensional filter is a conceptual operator that
projects a subset of the quality domains of a concept onto a smaller subset; it
“selects” relevant domains of another concept. We can then redefine a concept
as a product space of filtered parts, were just the relevant domains are selected
to compose the structure space of the whole. The filtering (i.e. projection) is
controlled by processes like attention and context. For instance, a combustion
engine can be part of a car or part of an electricity generator. The quality domains
of the engine that are relevant for car are related to its characteristics as a car
mover. So, the projection of combustion engine into the structure spaces of car
carries only some of its more relevant domains. This scheme solves the dimensional
explosion by providing a way in such the transitive closure can be avoided; parts
of parts that are not relevant for the whole can be filtered out.
3. Final Remarks
We are now developing a mathematical formulation of structure spaces based on
metric spaces. This should pave the way for computational applications. We are
also investigating the use of structure spaces in robotics and geology. Some auto-
localization algorithms for robots employ similarity reasoning to compare its sur-
roundings with its internal map. This comparison could benefit from a represen-
tation that allows structural similarity matching. In petroleum geology, an anal-
ogous problem of structural similarity exists in the task of matching geological
structures in di↵erent exploration wells, to which no satisfactory computational
solution yet exists. In the same way, structure spaces could help solve this problem
by providing a principled way in which sequences of features can be compared.
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