CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH

Transcription

1 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH Yaron Azrieli and Ehud Lehrer Tel-Aviv University CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 1/3

2 Outline Categorization and Prototypes - short general background. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

3 Outline Categorization and Prototypes - short general background. Categorization system the model. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

4 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

5 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. Voronoi diagrams. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

6 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. Voronoi diagrams. Proof sketch. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

7 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. Voronoi diagrams. Proof sketch. Examples. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

8 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. Voronoi diagrams. Proof sketch. Examples. Discussion. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

9 Outline Categorization and Prototypes - short general background. Categorization system the model. Categorization systems generated by prototypes Axiomatization. Voronoi diagrams. Proof sketch. Examples. Discussion. Remarks, open questions,... CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 2/3

10 General background Why do we categorize? CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 3/3

11 General background Why do we categorize? If you have never seen this particular creature before, why are you afraid of it? CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 3/3

12 General background It seems that for decision under stress we use a prototype-oriented decision process. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 4/3

13 General background It seems that for decision under stress we use a prototype-oriented decision process. A team trained to recognize a prototype from a set of prototypical scenarios and react accordingly, do not need a coordinator to act in harmony. This is crucial when the team might need to react instantaneously with no time for coordination. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 4/3

14 General background The classical view: A category as a list of features If man has one meaning, let this be two-footed animal. By has one meaning I mean this: if X means man, then if anything is a man, it s humanity will consist in being X. Aristotle, Metaphysics. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 5/3

15 General background The classical view: Categories are defined in terms of a conjunction of necessary and sufficient features. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 6/3

16 General background The classical view: Categories are defined in terms of a conjunction of necessary and sufficient features. Features are binary. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 6/3

17 General background The classical view: Categories are defined in terms of a conjunction of necessary and sufficient features. Features are binary. All members of a category have equal status. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 6/3

18 General background The prototypes approach: Another way to achieve separateness and clarity of actually continuous categories is by conceiving of each category in terms of its clear cases rather than its boundaries. As Wittgenstein (1953) has pointed out, categorical judgments become a problem only if one is concerned with boundaries in the normal course of life, two neighbors know on whose property they are standing without exact demarcation of the boundary line. E. Rosch, Principles of Categorization (1978). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 7/3

19 General background The prototypes approach: Certain members of a category are more typical than others. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 8/3

20 General background The prototypes approach: Certain members of a category are more typical than others. Categories have radial structure. They form around the prototypical cases. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 8/3

21 General background The prototypes approach: Certain members of a category are more typical than others. Categories have radial structure. They form around the prototypical cases. Category membership is a matter of degree. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 8/3

22 General background The prototypes approach: Certain members of a category are more typical than others. Categories have radial structure. They form around the prototypical cases. Category membership is a matter of degree. People categorize more typical members faster than less typical ones. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 8/3

23 Entities and Categories A finite set of categories L = {1, 2,...,l} CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 9/3

24 Entities and Categories A finite set of categories L = {1, 2,...,l} The set of entities to be categorized is R d + (d 2). That is, an entity is defined by the intensity of d attributes, and the intensity is unbounded. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 9/3

25 Entities and Categories A finite set of categories L = {1, 2,...,l} The set of entities to be categorized is R d + (d 2). That is, an entity is defined by the intensity of d attributes, and the intensity is unbounded. Typically, the d attributes represent just a partial list of attributes of the entity under consideration. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 9/3

26 Entities and Categories A finite set of categories L = {1, 2,...,l} The set of entities to be categorized is R d + (d 2). That is, an entity is defined by the intensity of d attributes, and the intensity is unbounded. Typically, the d attributes represent just a partial list of attributes of the entity under consideration. The set R d + can be replaced by the entire Euclidean space, or the simplex. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 9/3

27 Categorization system Definition: An open partition of R d + is a collection of non-empty, pairwise disjoint open sets, say, A 1,A 2,... such that cl A i = R d +. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 10/3

28 Categorization system Definition: An open partition of R d + is a collection of non-empty, pairwise disjoint open sets, say, A 1,A 2,... such that cl A i = R d +. Definition: 1. A categorization system is a collection of open partitions P A = (P A (i)) i A of R d +, A L ( A 2). 2. When x P A (i) we say that x is categorized as i when A is considered. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 10/3

29 Axioms Convexity: For every A L and for each i A, P A (i) is a convex set. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 11/3

30 Axioms Convexity: For every A L and for each i A, P A (i) is a convex set. Hierarchic Consistency: For every A L and for each i A, P A (i) = B A,i B P B(i). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 11/3

31 Axioms Convexity: For every A L and for each i A, P A (i) is a convex set. Hierarchic Consistency: For every A L and for each i A, P A (i) = B A,i B P B(i). Non-Redundancy: For every three categories {i,j,k} L, P {i,j} (i) P {i,k} (i). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 11/3

32 Axioms Convexity: For every A L and for each i A, P A (i) is a convex set. Hierarchic Consistency: For every A L and for each i A, P A (i) = B A,i B P B(i). Non-Redundancy: For every three categories {i,j,k} L, P {i,j} (i) P {i,k} (i). Variety: For every four distinct categories {i,j,k,m} L, cl(p {i,j,k} (i)) cl(p {i,j,k} (j)) cl(p {i,j,k} (k)) cl(p {m,j,k} (m)) cl(p {m,j,k} (j)) cl(p {m,j,k} (k)) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 11/3

33 Extended prototypes Definition: A categorization system, P A = (P A (i)) i A, A L, is generated by extended prototypes, if there exist l points x 1,x 2,...,x l in R d+1 +, such that for any A L and any i A, P A (i) = {y R d + : d i (y) < d j (y) for every j A,j i}, where d i (y) = (y, 0) x i 2 (i = 1,...,l) and (y, 0) is the vector in R d+1 + whose first d coordinates coincide with y and the last coincides with 0. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 12/3

34 Extended prototypes Definition: A categorization system, P A = (P A (i)) i A, A L, is generated by extended prototypes, if there exist l points x 1,x 2,...,x l in R d+1 +, such that for any A L and any i A, P A (i) = {y R d + : d i (y) < d j (y) for every j A,j i}, where d i (y) = (y, 0) x i 2 (i = 1,...,l) and (y, 0) is the vector in R d+1 + whose first d coordinates coincide with y and the last coincides with 0. Interpretation: The extended prototype is in R d+1 +. Recall that the d attribute may represent only a partial information about the entity. The hidden attribute may me interpreted as an imaginary completion of the unknown attributes. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 12/3

35 Partial information Suppose that you encounter this: CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 13/3

36 Partial information Suppose that you encounter this: CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 13/3

37 Partial information Suppose that you encounter this: What do you do? CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 13/3

38 Axiomatization The main result: If {P A } A L is a categorization system generated by extended prototypes, then it satisfies Convexity and Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 14/3

39 Axiomatization The main result: If {P A } A L is a categorization system generated by extended prototypes, then it satisfies Convexity and Hierarchic Consistency. If a categorization system {P A } A L satisfies Convexity and Hierarchic Consistency, and in addition satisfies Non-Redundancy and Variety, then it is generated by extended prototypes. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 14/3

40 Remarks Hierarchic Consistency is a necessary condition for a categorization system to be generated by prototypes regardless of the metric (e.g., maximum norm, l 1 etc.). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 15/3

41 Remarks Hierarchic Consistency is a necessary condition for a categorization system to be generated by prototypes regardless of the metric (e.g., maximum norm, l 1 etc.). The Euclidean metric is the result of the Convexity axiom. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 15/3

42 Voronoi Diagrams Let x 1,x 2,...,x n be some n distinct points in R d. For every i = 1, 2,...,n, let A i = {y R d : y x i < y x j for every j = 1, 2,...,n j i} where stands for the Euclidian norm. Definition: The Voronoi diagram induced by x 1,x 2,...,x n is the collection of sets A 1,A 2,...,A n. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 16/3

43 Voronoi Diagrams CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 17/3

44 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal distance from x i and x j is the hyperplane which is the perpendicular bisector of the segment between x i and x j. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 18/3

45 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal distance from x i and x j is the hyperplane which is the perpendicular bisector of the segment between x i and x j. For every i = 1,...,n, A i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 18/3

46 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal distance from x i and x j is the hyperplane which is the perpendicular bisector of the segment between x i and x j. For every i = 1,...,n, A i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. x i A i. Therefore A i is not empty. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 18/3

47 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal distance from x i and x j is the hyperplane which is the perpendicular bisector of the segment between x i and x j. For every i = 1,...,n, A i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. x i A i. Therefore A i is not empty. The closure of the union of the sets A i (1 i n) is the entire space. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 18/3

48 Voronoi Diagrams Generalization: Power diagrams Every point x i has a non-negative weight w i. For a point y R d, let d i (y) = y x i 2 + w i. Similarly to the former case, define for i = 1, 2,...,n B i = {y R d : d i (y) < d j (y) for every j = 1, 2,...,n j i} Definition: The Power diagram induced by (x 1,w 1 ),..., (x n,w n ) is the collection of sets B 1,B 2,...,B n. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 19/3

49 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal weighted distance from x i and x j is a hyperplane perpendicular to the segment between x i and x j (not necessarily the perpendicular bisector). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 20/3

50 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal weighted distance from x i and x j is a hyperplane perpendicular to the segment between x i and x j (not necessarily the perpendicular bisector). For every i = 1,...,n, B i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 20/3

51 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal weighted distance from x i and x j is a hyperplane perpendicular to the segment between x i and x j (not necessarily the perpendicular bisector). For every i = 1,...,n, B i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. B i can be empty. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 20/3

52 Voronoi Diagrams Simple observations: For every two points x i,x j, the set of points in R d which are in equal weighted distance from x i and x j is a hyperplane perpendicular to the segment between x i and x j (not necessarily the perpendicular bisector). For every i = 1,...,n, B i is the intersection of a (finite) number of half spaces. Therefore, it is a convex polyhedral set. B i can be empty. The closure of the union of the sets B i (1 i n) is the entire space. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 20/3

53 Proof sketch Part 1 of the main theorem: If {P A } A L is a categorization system generated by extended prototypes, then it satisfies Convexity and Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 21/3

54 Proof sketch Part 1 of the main theorem: If {P A } A L is a categorization system generated by extended prototypes, then it satisfies Convexity and Hierarchic Consistency. Lemma: Let {P A } A L be a categorization system. Then {P A } A L satisfies Hierarchic Consistency if and only if for every A L ( A 2) and for each i A, P A (i) = j A\{i} P {i,j}(i). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 21/3

55 Proof sketch Part 2 of the main theorem: If a categorization system {P A } A L satisfies Convexity and Hierarchic Consistency, and in addition satisfies Non-Redundancy and Variety, then it is generated by extended prototypes. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 22/3

56 Proof sketch Lemma: If {P A } A L satisfies Convexity then for every two categories i,j L there exists a hyperplane H i,j, such that P {i,j} (i) is the open half space on one side of it and P {i,j} (j) is the complementary open half. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 23/3

57 Proof sketch Lemma: If {P A } A L satisfies Convexity then for every two categories i,j L there exists a hyperplane H i,j, such that P {i,j} (i) is the open half space on one side of it and P {i,j} (j) is the complementary open half. Lemma: Non-Redundancy implies that for any three distinct categories i,j,k the hyperplanes H i,j and H i,k are not parallel. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 23/3

58 Proof sketch Lemma: If {P A } A L satisfies Convexity then for every two categories i,j L there exists a hyperplane H i,j, such that P {i,j} (i) is the open half space on one side of it and P {i,j} (j) is the complementary open half. Lemma: Non-Redundancy implies that for any three distinct categories i,j,k the hyperplanes H i,j and H i,k are not parallel. Lemma: Let {P A } A L be a categorization system which satisfies Convexity, Hierarchic Consistency and Non-Redundancy. For any three distinct categories i, j and k H i,j H i,k = H i,j H j,k = H j,k H i,k CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 23/3

59 Proof sketch Proposition: If a categorization system {P A } A L satisfies Convexity, Hierarchic Consistency, Non-Redundancy and Variety, then there are l points x 1,...,x l Rd + such that: 1. x i x j is perpendicular to the hyperplane H i,j, for every i,j L; and 2. The direction from x i to x j is the same as the direction from P {i,j} (i) to P {i,j} (j) (we call such points well oriented ). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 24/3

60 Proof sketch P L (2) P L (4) 3 P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

61 Proof sketch P L (2) P L (4) x P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

62 Proof sketch x 2 P L (2) P L (4) x P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

63 Proof sketch x 2 P L (2) P L (4) x 1 3 x 3 4 P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

64 Proof sketch x 2 P L (2) x 4 P L (4) x 1 3 x 3 4 P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

65 Proof sketch x 2 P L (2) x 4 P L (4) x 1 3 x 3 4 P L (1) P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

66 Proof sketch x 2 P L (2) x 4 P L (4) P L (1) x 1 3 x P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

67 Proof sketch x 2 P L (2) x 4 P L (4) P L (1) x 1 3 x P L (3) CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

68 Proof sketch x 2 P L (2) x 4 P L (4) P L (1) x 1 3 x P L (3) 1 4 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

69 Proof sketch x 2 P L (2) x 4 P L (4) P L (1) x 1 3 x P L (3) 1 4 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 25/3

70 Proof sketch Proposition: Let {x 1,...,x l } Rd + be points that satisfy the previous proposition. There exist a set of numbers w 1,...,w l, such that P {i,j} (i) = {y R d +; d i (y) < d j (y)} for every i,j L, where d i (y) = (y, 0) (x i,w i) 2. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 26/3

71 Proof sketch Conclusion of the main result s proof: It is possible to find points x 1 = (x 1,w 1),...,x l = (x l,w l) in such that for every two categories i,j L, P {i,j} (i) = {y R d +; (y, 0) x i 2 < (y, 0) x j 2 }. Therefore, for any A L and any i A, R d+1 + P A (i) = j A\{i} {y R d +; (y, 0) x i 2 < (y, 0) x j 2 } = {y R d +; (y, 0) x i 2 < (y, 0) x j 2 for every j A,j i} Thus, the categorization system is generated by extended prototypes. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 27/3

72 Examples Categorization which is not generated by extended prototypes L = {i,j,k} (3 categories), d = 2 (2 attributes). P {i,j} (i) = {(x 1,x 2 ) R 2 +; x 1 < 1} P {i,k} (i) = {(x 1,x 2 ) R 2 +; x 2 < 1} P {j,k} (j) = {(x 1,x 2 ) R 2 +; x 2 < 2 x 1 } By Hierarchic Consistency, when L is considered: P L (i) = {(x 1,x 2 ) R 2 +; x 1 < 1, x 2 < 1} P L (j) = {(x 1,x 2 ) R 2 +; x 1 > 1, x 2 < 2 x 1 } P L (k) = {(x 1,x 2 ) R 2 +; x 2 > 1, x 2 > 2 x 1 } CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 28/3

73 Examples Categorization which is not generated by extended prototypes x 2 i j k j P L (k) k i P L (i) P L (j) x 1 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 29/3

74 Examples The hidden attribute L = {i,j,k,m} (4 categories), d = 2 (2 attributes). P {i,j} (i) = {(x 1,x 2 ) R 2 +; 2x 2 > 4 x 1 } P {i,k} (i) = {(x 1,x 2 ) R 2 +; x 2 > 1} P {i,m} (i) = {(x 1,x 2 ) R 2 +; 2x 2 > x 1 1} P {j,k} (j) = {(x 1,x 2 ) R 2 +; 2x 2 > x 1 } P {j,m} (j) = {(x 1,x 2 ) R 2 +; 2x 1 < 5} P {k,m} (k) = {(x 1,x 2 ) R 2 +; 2x 2 < 5 x 1 } When more than 2 categories are considered, the partitions are determined by Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 30/3

75 Examples The hidden attribute x 2 j m P L (i) P L (j) P L (m) P L (k) x 1 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 31/3

76 Examples Convexity and Hierarchic Consistency are insufficient L = {i,j,k} (3 categories), d = 2 (2 attributes). P {i,j} (i) = {(x 1,x 2 ) R 2 +; x 1 < 1} P {i,k} (i) = {(x 1,x 2 ) R 2 +; x 1 < 1} P {j,k} (j) = {(x 1,x 2 ) R 2 +; x 2 > x 1 } By Hierarchic Consistency, when L is considered: P L (i) = {(x 1,x 2 ) R 2 +; x 1 < 1} P L (j) = {(x 1,x 2 ) R 2 +; x 1 > 1, x 2 > x 1 } P L (k) = {(x 1,x 2 ) R 2 +; x 1 > 1, x 2 < x 1 } CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 32/3

77 Examples Convexity and Hierarchic Consistency are insufficient x 2 i i j k P L (j) j k P L (i) P L (k) x 1 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 33/3

78 Examples Variety is not satisfied L = {i,j,k,m} (4 categories), d = 2 (2 attributes). P {i,j} (i) = {(x 1,x 2 ) R 2 +; x 1 < 1} P {i,k} (i) = {(x 1,x 2 ) R 2 +; x 2 > 1} P {i,m} (i) = {(x 1,x 2 ) R 2 +; x 2 > 2x 1 1} P {j,k} (j) = {(x 1,x 2 ) R 2 +; 2x 2 > 3 x 1 } P {j,m} (j) = {(x 1,x 2 ) R 2 +; x 2 > x 1 } P {k,m} (k) = {(x 1,x 2 ) R 2 +; x 2 < 2 x 1 } When more than 2 categories are considered, the partitions are determined by Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 34/3

79 Examples Variety is not satisfied x 2 i j x 2 P {i,j,k} (i) j P {m,j,k} (j) j m i k P {i,j,k} (j) k P {m,j,k} (m) P {i,j,k} (k) j P {m,j,k} (k) k k m x 1 x 1 CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 35/3

80 Discussion Limitations of the axioms: More than one prototypical example resulting in non-convex categories. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 36/3

81 Discussion Limitations of the axioms: More than one prototypical example resulting in non-convex categories. The effect of the set of categories under consideration on the prototypes Violation of Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 36/3

82 Discussion Decision theory and categorization Categorization of decision problems according to their best response satisfies Hierarchic Consistency. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 37/3

83 Discussion Decision theory and categorization Categorization of decision problems according to their best response satisfies Hierarchic Consistency. If a decision problem is defined by a distribution P over some state space (say Ω), and u(p,a) is the expected utility when choosing the action a, then the categorization of decision problems (distributions) according to their best response forms convex categories. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 37/3

84 Remarks & Open questions Categorizations generated by (non-extended) prototypes. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3

85 Remarks & Open questions Categorizations generated by (non-extended) prototypes. Other domains. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3

86 Remarks & Open questions Categorizations generated by (non-extended) prototypes. Other domains. Other metrics. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3

87 Remarks & Open questions Categorizations generated by (non-extended) prototypes. Other domains. Other metrics. Prototypical sets. CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3

88 Remarks & Open questions Categorizations generated by (non-extended) prototypes. Other domains. Other metrics. Prototypical sets. Another version of the Non-redundancy axiom which is also necessary: Non-Redundancy : For every three categories {i,j,k} L, if P {i,j} (i) = P {i,k} (i) then either P {i,j,k} (i) P {j,k} (j) or P {i,j,k} (i) P {j,k} (k). CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3

89 Remarks & Open questions Categorizations generated by (non-extended) prototypes. Other domains. Other metrics. Prototypical sets. Another version of the Non-redundancy axiom which is also necessary: Non-Redundancy : For every three categories {i,j,k} L, if P {i,j} (i) = P {i,k} (i) then either P {i,j,k} (i) P {j,k} (j) or P {i,j,k} (i) P {j,k} (k). Is the theorem still true without assuming Variety? CATEGORIZATION GENERATED BY PROTOTYPES AN AXIOMATIC APPROACH p. 38/3