Theory and applications of freedom in matroids

To each cell e in a matroid M we can associate a non-negative integer ǁ e ǁ called the freedom of e. Geometrically the value ǁ e ǁ indicates how freely placed the cell is in the matroid. We see that ǁ e ǁ is equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e. The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and the applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of M*. If ζ(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M ǁ e ǁ= \frac{max\ f \ (e)}{f\in\zeta} We look at freedom in binary matroids and show that for a connected binary matroid M, ǁ e ǁ is the number of connected components of M/e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M/e disconnected

Abstract 3-Rigidity and Bivariate C21C_2^1-Splines II: Combinatorial Characterization

We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function) and Lov\'{a}sz and Yemini (which suggested a sufficient connectivity condition for rigidity) hold for this matroid

Abstract 3-Rigidity and Bivariate C½-Splines II: Combinatorial Characterization

We showed in the first paper of this series that the generic C1-cofactor matroid is the unique maximal abstract 3-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function) and Lovász and Yemini (which suggested a sufficient connectivity condition for rigidity) hold for the C1-cofactor matroid

The Category of Matroids

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial.Comment: 31 pages, 10 diagrams, 28 reference

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

Graduate School: Course Decriptions, 1972-73

Official publication of Cornell University V.64 1972/7

Matroid Erection and Duality

Las Vergnas [6] and Nguyen [7] independently described the free erection of a matroid M and gave an algorithm for the determination of the hyperplanes of the free erection of M. The purpose of this paper is to look at the connection between erections of M and one-point extensions of the dual M, and show that the Las Vergnas—Nguyen algorithm is the dual of an algorithm to construct the modular cut of M generated by the cyclic flats of M