Related papers: Matroidal structure of generalized rough sets base…
Unsupervised representation learning methods are widely used for gaining insight into high-dimensional, unstructured, or structured data. In some cases, users may have prior topological knowledge about the data, such as a known cluster…
It is described how one comes to the Wigner-Dyson random matrix theory (RMT) starting from a model of a disordered metal. The lectures start with a historical introduction where basic ideas of the RMT and theory of disordered metals are…
A coreset is a small set that can approximately preserve the structure of the original input data set. Therefore we can run our algorithm on a coreset so as to reduce the total computational complexity. Conventional coreset techniques…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…
This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the…
We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between…
Conventional machine learning algorithms have traditionally been designed under the assumption that input data follows a vector-based format, with an emphasis on vector-centric paradigms. However, as the demand for tasks involving set-based…
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of…
In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal…
We examine non-dual relational extensions of rough set approximations and find an extension which satisfies surprisingly many of the usual rough set properties. We then use this definition to give an explanation for an observation made by…
Symmetries of geometric structures such as hyperplane arrangements, point configurations and polytopes have been studied extensively for a long time. However, symmetries of oriented matroids, a common combinatorial abstraction of them, are…
A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented…
Traversals are commonly seen in tree data structures, and performance-enhancing transformations between tree traversals are critical for many applications. Existing approaches to reasoning about tree traversals and their transformations are…
Compositional generalization is one of the main properties which differentiates lexical learning in humans from state-of-art neural networks. We propose a general framework for building models that can generalize compositionally using the…
Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations,…
A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of…
Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework permits the formal definition of random vectors (and random time series) whose desired joint distributions are a priori prescribed. Its key…
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…