arXiv subjects

Pavel Rytíř

Publications and source records attributed to Pavel Rytíř.

5 recordsLinked to original sources

arXiv2026

ExMAG: Learning of Maximally Ancestral Graphs

In mixed graphs, there are both directed and bidirected edges. An extension of acyclicity to this mixed-graph setting is known as maximally ancestral graphs. This extension is of considerable interest in causal learning in the presence of confounders. There, directed edges represent a clear direction of causality, while bidirected edges represent confounding. We propose a branch-and-cut algorithm for learning maximally ancestral graphs using a formulation as a mixed-integer quadratic program. Empirically, our method achieves comparable or improved reconstruction quality while requiring an order of magnitude fewer samples than state-of-the-art approaches.

cs.LG↗

arXiv2016

Binary Linear Codes, Dimers and Hypermatrices

We show that the weight enumerator of any binary linear code is equal to the permanent of a 3-dimensional hypermatrix (3-matrix). We also show that each permanent is a determinant of a 3-matrix. As an application we write the dimer partition function of a finite 3-dimensional cubic lattice as the determinant of the vertex-adjacency 3-matrix of a 2-dimensional simplicial complex which preserves the natural embedding of the cubic lattice.

math.CO↗

arXiv2012

Geometric representations of binary codes embeddable in three dimensions

We say that a binary linear code C has a geometric representation if there exists a two dimensional simplicial complex D such that C is a punctured code of the kernel ker D of the incidence matrix of D and dim C = dim ker D. We show that every binary linear code has a geometric representation that can be embedded into R^4. Moreover, we show that a binary linear code C has a geometric representation in R^3 if and only if there exists a graph G such that C equals the cut space of G. This is a polynomially testable property and hence we can conclude that there is a polynomial algorithm that decides the minimal dimension of a geometric representation of a binary linear code.

math.CO↗

arXiv2011

Geometric representations of linear codes

We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex $Δ$ such that C is a punctured code of the kernel ker $Δ$ of the incidence matrix of $Δ$ over F and there is a linear mapping between C and ker $Δ$ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.

math.CO↗

arXiv2009

Geometric representation of binary codes and computation of weight enumerators

For every linear binary code $C$, we construct a geometric triangular configuration $Δ$ so that the weight enumerator of $C$ is obtained by a simple formula from the weight enumerator of the cycle space of $Δ$. The triangular configuration $Δ$ thus provides a geometric representation of $C$ which carries its weight enumerator. This is the initial step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration $Δ$, a triangular configuration $Δ'$ and a bijection between the cycle space of $Δ$ and the set of the perfect matchings of $Δ'$.

math.CO↗