scholarly journals Characteristic of partition-circuit matroid through approximation number

2012 ◽  
Author(s):  
Yanfang Liu ◽  
William Zhu
Keyword(s):  
Approximation Number

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals Function Values Are Enough for $$L_2$$-Approximation

2020 ◽  
Author(s):  
David Krieg ◽  
Mario Ullrich
Keyword(s):  
Hilbert Space ◽  
Fourier Coefficients ◽  
Approximation Number ◽  
Approximation Numbers ◽  
Approximation Of Functions ◽  
Worst Case ◽  
Linear Information ◽  
Grid Algorithm ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

AbstractWe study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

Quantitative analysis for covering-based rough sets through the upper approximation number

2013 ◽  
Vol 220 ◽  
pp. 483-491 ◽  
Author(s):  
Shiping Wang ◽  
Qingxin Zhu ◽  
William Zhu ◽  
Fan Min
Keyword(s):  
Quantitative Analysis ◽  
Rough Sets ◽  
Approximation Number ◽  
Upper Approximation

scholarly journals Equivalent Characterizations of Some Graph Problems by Covering-Based Rough Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shiping Wang ◽  
Qingxin Zhu ◽  
William Zhu ◽  
Fan Min
Keyword(s):  
Graph Theory ◽  
Rough Sets ◽  
Systematic Approach ◽  
Rough Set Theory ◽  
Approximation Number ◽  
Edge Cover ◽  
Upper Approximation ◽  
Graph Problems ◽  
New Concepts ◽  
Equivalent Characterization

Covering is a widely used form of data structures. Covering-based rough set theory provides a systematic approach to this data. In this paper, graphs are connected with covering-based rough sets. Specifically, we convert some important concepts in graph theory including vertex covers, independent sets, edge covers, and matchings to ones in covering-based rough sets. At the same time, corresponding problems in graphs are also transformed into ones in covering-based rough sets. For example, finding a minimal edge cover of a graph is translated into finding a minimal general reduct of a covering. The main contributions of this paper are threefold. First, any graph is converted to a covering. Two graphs induce the same covering if and only if they are isomorphic. Second, some new concepts are defined in covering-based rough sets to correspond with ones in graph theory. The upper approximation number is essential to describe these concepts. Finally, from a new viewpoint of covering-based rough sets, the general reduct is defined, and its equivalent characterization for the edge cover is presented. These results show the potential for the connection between covering-based rough sets and graphs.

A New Description of Transversal Matroids Through Rough Set Approach

2021 ◽  
Vol 179 (4) ◽  
pp. 399-416
Author(s):  
Zhaohao Wang
Keyword(s):  
Rough Set ◽  
Rough Set Theory ◽  
Sufficient Conditions ◽  
Mining Machine ◽  
Matroid Theory ◽  
Approximation Number ◽  
Upper Approximation ◽  
Transversal Matroid ◽  
Necessary And Sufficient ◽  
The Relationship

Matroid theory is a useful tool for the combinatorial optimization issue in data mining, machine learning and knowledge discovery. Recently, combining matroid theory with rough sets is becoming interesting. In this paper, rough set approaches are used to investigate an important class of matroids, transversal matroids. We first extend the concept of upper approximation number functions in rough set theory and propose the notion of generalized upper approximation number functions on a set system. By means of the new notion, we give some necessary and sufficient conditions for a subset to be a partial transversal of a set system. Furthermore, we obtain a new description of a transversal matroid by the generalized upper approximation number function. We show that a transversal matroid can be induced by the generalized upper approximation number function which can be decomposed into the sum of some elementary generalized upper approximation number functions. Conversely, we also prove that a generalized upper approximation number function can induce a transversal matroid. Finally, we apply the generalized upper approximation number function to study the relationship among transversal matroids.

The approximation number function and the characterization of covering approximation space

2015 ◽  
Vol 305 ◽  
pp. 196-207 ◽  
Author(s):  
Zhaohao Wang ◽  
Hong Wang ◽  
Qinrong Feng ◽  
Lan Shu
Keyword(s):  
Approximation Number ◽  
Approximation Space

scholarly journals Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Jianguo Tang ◽  
Kun She ◽  
William Zhu
Keyword(s):  
Graph Theory ◽  
Rough Sets ◽  
Mathematical Theory ◽  
Research Field ◽  
Rank Function ◽  
Complete Graphs ◽  
Important Research ◽  
Approximation Number ◽  
Lower Approximation ◽  
The Universe

Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. Some new properties, which have not been found in rough sets, are obtained. Furthermore, by defining the concept of lower approximation number, the rank function of some subset of the universe and the approximations of the subset are connected. Finally, the relationships between the two types of matroids are discussed, and the result shows that they are just dual matroids.

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