Rough sets and matroidal contraction

Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide well-established platforms for greedy algorithms. In this paper, we apply rough sets to matroids and study the contraction of the dual of the corresponding matroid. First, for an equivalence relation on a universe, a matroidal structure of the rough set is established through the lower approximation operator. Second, the dual of the matroid and its properties such as independent sets, bases and rank function are investigated. Finally, the relationships between the contraction of the dual matroid to the complement of a single point set and the contraction of the dual matroid to the complement of the equivalence class of this point are studied.Comment: 11 page

Some characteristics of matroids through rough sets

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.Comment: 13 page

Matroids, delta-matroids and embedded graphs

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal

Two Studies in Representation of Signals

The thesis consists of two parts. In the first part deals with a multi-scale approach to vector quantization. An algorithm, dubbed reconstruction trees, is proposed and analyzed. Here the goal is parsimonious reconstruction of unsupervised data; the algorithm leverages a family of given partitions, to quickly explore the data in a coarse-to-fine multi-scale fashion. The main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown probability measure. Both asymptotic and finite sample results are provided, under suitable regularity assumptions on the probability measure. Special attention is devoted to the case in which the probability measure is supported on a smooth sub-manifold of the ambient space, and is absolutely continuous with respect to the Riemannian measure of it; in this case asymptotic optimal quantization is well understood and a benchmark for understanding the results is offered. The second part of the thesis deals with a novel approach to Graph Signal Processing which is based on Matroid Theory. Graph Signal Processing is the study of complex functions of the vertex set of a graph, based on the combinatorial Graph Laplacian operator of the underlying graph. This naturally gives raise to a linear operator, that to many regards resembles a Fourier transform, mirroring the graph domain into a frequency domain. On the one hand this structure asymptotically tends to mimic analysis on locally compact groups or manifolds, but on the other hand its discrete nature triggers a whole new scenario of algebraic phenomena. Hints towards making sense of this scenario are objects that already embody a discrete nature in continuous setting, such as measures with discrete support in time and frequency, also called Dirac combs. While these measures are key towards formulating sampling theorems and constructing wavelet frames in time-frequency Analysis, in the graph-frequency setting these boil down to distinguished combinatorial objects, the so called Circuits of a matroid, corresponding to the Fourier transform operator. In a particularly symmetric case, corresponding to Cayley graphs of finite abelian groups, the Dirac combs are proven to completely describe the so called lattice of cyclic flats, exhibiting the property of being atomistic, among other properties. This is a strikingly concise description of the matroid, that opens many questions concerning how this highly regular structure relaxes into more general instances. Lastly, a related problem concerning the combinatorial interplay between Fourier operator and its Spectrum is described, provided with some ideas towards its future development

Tropical varieties, maps and gossip

Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group