arXiv subjects

Daniel Horsley

Publications and source records attributed to Daniel Horsley.

43 records · Page 3Linked to original sources

arXiv2015

Compressed sensing with combinatorial designs: theory and simulations

In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given.

cs.IT↗

arXiv2014

On factorisations of complete graphs into circulant graphs and the Oberwolfach Problem

Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p = 5 (mod 8) is prime.

math.CO↗

arXiv2014

Hierarchical Recovery in Compressive Sensing

A combinatorial approach to compressive sensing based on a deterministic column replacement technique is proposed. Informally, it takes as input a pattern matrix and ingredient measurement matrices, and results in a larger measurement matrix by replacing elements of the pattern matrix with columns from the ingredient matrices. This hierarchical technique yields great flexibility in sparse signal recovery. Specifically, recovery for the resulting measurement matrix does not depend on any fixed algorithm but rather on the recovery scheme of each ingredient matrix. In this paper, we investigate certain trade-offs for signal recovery, considering the computational investment required. Coping with noise in signal recovery requires additional conditions, both on the pattern matrix and on the ingredient measurement matrices.

cs.IT↗

arXiv2014

Embedding partial Steiner triple systems with few triples

It was proved in 2009 that any partial Steiner triple system of order $u$ has an embedding of order $v$ for each admissible integer $v\geq 2u+1$. This result is best-possible in the sense that, for each $u\geq 9$, there exists a partial Steiner triple system of order $u$ that does not have an embedding of order $v$ for any $v<2u+1$. Many partial Steiner triple systems do have embeddings of orders smaller than $2u+1$, but little has been proved about when these embeddings exist. In this paper we construct embeddings of orders less than $2u+1$ for partial Steiner triple systems with few triples. In particular, we show that a partial Steiner triple system of order $u \geq 62$ with at most $\frac{u^2}{50}-\frac{11u}{100}-\frac{116}{75}$ triples has an embedding of order $v$ for each admissible integer $v \geq \frac{8u+17}{5}$.

math.CO↗

arXiv2013

Decompositions of complete graphs into cycles of arbitrary lengths

We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the complete graph on $n$ vertices can be decomposed into a perfect matching and $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is even, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\ldots+m_t=\binom n2-\frac n2$.

math.CO↗

arXiv2013

On balanced incomplete block designs with specified weak chromatic number

A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly $c$-chromatic if $c$ is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all $c \geq 2$ and $k \geq 3$ with $(c,k) \neq (2,3)$, the obvious necessary conditions for the existence of a $(v,k,λ)$-BIBD are asymptotically sufficient for the existence of a weakly $c$-chromatic $(v,k,λ)$-BIBD.

math.CO↗

arXiv2012

Decomposing various graphs into short even-length cycles

We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the second longest. We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length.

math.CO↗