Topic
Unique negative dimension
About: Unique negative dimension is a research topic. Over the lifetime, 6 publications have been published within this topic receiving 113 citations.
Papers
1 Dec 1989
TL;DR: This paper explores algorithms that learn a concept from a concept class of Vapnik-Chervonenkis (VC) dimension d by saving at most d examples at a time, the model of learning introduced by Valiant.
Abstract: This paper explores algorithms that learn a concept from a concept class of Vapnik-Chervonenkis (VC) dimension d by saving at most d examples at a time. The framework is the model of learning introduced by Valiant [V84]. A maximum concept class of VC dimension d is defined. For a maximum class C of VC dimension d , we give an algorithm for representing a finite set of positive and negative examples of a concept by a subset of d labeled examples of that set. This data compression scheme of size d is used to construct a space-bounded algorithm that learns a concept from the class C by saving at most d examples at a time. These d examples represent the current hypothesis of the learning algorithm. A space-bounded learning algorithm is called acyclic if a hypothesis that has been rejected as incorrect is never reinstated. We give a sufficient condition for this algorithm to be acyclic on a maximum class C. Classes for which this algorithm is acyclic include positive half-spaces in Euclidean space E n , balls in E n , and arbitrary rectangles in the plane. This algorithm is also acyclic on positive sets in the plane where each positive set is defined by a polynomial of degree at most n. The algorithm can be thought of as learning a boundary between the positive and the negative examples.
82 citations
TL;DR: The notion of dynamic sampling is introduced, wherein the number of examples examined may increase with the complexity of the target concept, and this method is used to establish the learnability of various concept classes with an infinite Vapnik-Chervonenkis dimension.
Abstract: We consider the problem of learning a concept from examples in the distribution-free model by Valiant. (An essentially equivalent model, if one ignores issues of computational difficulty, was studied by Vapnik and Chervonenkis.) We introduce the notion of dynamic sampling, wherein the number of examples examined may increase with the complexity of the target concept. This method is used to establish the learnability of various concept classes with an infinite Vapnik-Chervonenkis dimension. We also discuss an important variation on the problem of learning from examples, called approximating from examples. Here we do not assume that the target concept T is a member of the concept class C from which approximations are chosen. This problem takes on particular interest when the VC dimension of C is infinite. Finally, we discuss the problem of computing the VC dimension of a finite concept set defined on a finite domain and consider the structure of classes of a fixed small dimension.
46 citations
TL;DR: In this article, the authors introduced the notion of negative dimensions through several examples of random fractal constructions, and extended Minkowski's definition of the dimension by ∊ -neighborhoods to ∊-pseudo-neighbors.
Abstract: Applied blindly, the formula for the dimension of the intersection can give negative results. Extending Minkowski's definition of the dimension by ∊-neighborhoods to ∊-pseudo-neighborhoods, that is, replacing (A ∩ B)∊ with A∊ ∩ B∊, we introduce the notion of negative dimensions through several examples of random fractal constructions.
9 citations
TL;DR: The connection between resource-bounded dimension and the online mistake-bound model of learning is used to show that the following classes have polynomial-time dimension zero.
Abstract: We use the connection between resource-bounded dimension and the online mistake-bound model of learning to show that the following classes have polynomial-time dimension zero.
The class of problems which reduce to nondense sets via a majority reduction.
The class of problems which reduce to nondense sets via an iterated reduction that composes a bounded-query truth-table reduction with a conjunctive reduction.
Intuitively, polynomial-time dimension is a means of quantifying the size and complexity of classes within the exponential time complexity class E. The class P has dimension 0, E itself has dimension 1, and any class with dimension less than 1 cannot contain E. As a corollary, it follows that all sets which are hard for E under these types of reductions are exponentially dense. The first item subsumes two previous results and the second item answers a question of Lutz and Mayordomo. Our proofs use Littlestone’s Winnow2 algorithm for learning r-of-k threshold functions and Maass and Turan’s algorithm for learning halfspaces.
6 citations
Journal Article•
TL;DR: After Analyzing the signification of structure and definition about dimension to aggregating operation, it is confirmed that some dimension classes maybe produce analysis demands which are similar to dimension hierarchies, and CDCDH method is provided for conversion from dimension classes to dimension hierarchy.
Abstract: After Analyzing the signification of structure and definition about dimension to aggregating operation,confirms that some dimension classes maybe produce analysis demands which are similar to dimension hierarchies,provides CDCDH method for conversion from dimension classes to dimension hierarchies.Gives definition on dimension class and dimension hierarchy,furthermore,gives definition on visible dimension hierarchy and hide dimension hierarchy.About hide dimension hierarchy which shows dimension class,confirms the condition and steps for conversion to visible dimension hierarchy.In the meantime,produces the realizing procedure about CDCDH method through a practical instance.
1 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2015 | 1 |
| 2011 | 1 |
| 2009 | 1 |
| 2008 | 1 |
| 1991 | 1 |
| 1989 | 1 |