arXiv subjects

Matthew Johnson

Publications and source records attributed to Matthew Johnson.

64 records · Page 4Linked to original sources

arXiv2016

The Segmented iHMM: A Simple, Efficient Hierarchical Infinite HMM

We propose the segmented iHMM (siHMM), a hierarchical infinite hidden Markov model (iHMM) that supports a simple, efficient inference scheme. The siHMM is well suited to segmentation problems, where the goal is to identify points at which a time series transitions from one relatively stable regime to a new regime. Conventional iHMMs often struggle with such problems, since they have no mechanism for distinguishing between high- and low-level dynamics. Hierarchical HMMs (HHMMs) can do better, but they require much more complex and expensive inference algorithms. The siHMM retains the simplicity and efficiency of the iHMM, but outperforms it on a variety of segmentation problems, achieving performance that matches or exceeds that of a more complicated HHMM.

stat.ML↗

arXiv2016

A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs

For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.

cs.CC↗

arXiv2016

A Serial Multilevel Hypergraph Partitioning Algorithm

The graph partitioning problem has many applications in scientific computing such as computer aided design, data mining, image compression and other applications with sparse-matrix vector multiplications as a kernel operation. In many cases it is advantageous to use hypergraphs as they, compared to graphs, have a more general structure and can be used to model more complex relationships between groups of objects. This motivates our focus on the less-studied hypergraph partitioning problem. In this paper, we propose a serial multi-level bipartitioning algorithm. One important step in current heuristics for hypergraph partitioning is clustering during which similar vertices must be recognized. This can be particularly difficult in irregular hypergraphs with high variation of vertex degree and hyperedge size; heuristics that rely on local vertex clustering decisions often give poor partitioning quality. A novel feature of the proposed algorithm is to use the techniques of rough set clustering to address this problem. We show that our proposed algorithm gives on average between 18.8 per cent and 71.1 per cent better quality on these irregular hypergraphs by comparing it to state-of-the-art hypergraph partitioning algorithms on benchmarks taken from real applications.

cs.DS↗

arXiv2015

Efficient non-greedy optimization of decision trees

Decision trees and randomized forests are widely used in computer vision and machine learning. Standard algorithms for decision tree induction optimize the split functions one node at a time according to some splitting criteria. This greedy procedure often leads to suboptimal trees. In this paper, we present an algorithm for optimizing the split functions at all levels of the tree jointly with the leaf parameters, based on a global objective. We show that the problem of finding optimal linear-combination (oblique) splits for decision trees is related to structured prediction with latent variables, and we formulate a convex-concave upper bound on the tree's empirical loss. The run-time of computing the gradient of the proposed surrogate objective with respect to each training exemplar is quadratic in the the tree depth, and thus training deep trees is feasible. The use of stochastic gradient descent for optimization enables effective training with large datasets. Experiments on several classification benchmarks demonstrate that the resulting non-greedy decision trees outperform greedy decision tree baselines.

cs.LG↗

arXiv2015

Kempe Equivalence of Colourings of Cubic Graphs

Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain is a subset of $V$ that induces a maximal connected subgraph of $G$ in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$, all $k$-colourings of $k$-regular graphs that are not complete are Kempe equivalent. We address the case $k=3$ by showing that all $3$-colourings of a cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism.

cs.DM↗

arXiv2015

Tensor Modes in Bigravity: Primordial to Present

Massive bigravity, a theoretically consistent modification of general relativity with an additional dynamical rank two tensor, successfully describes the observed accelerated expansion of the Universe without a cosmological constant. Recent analyses of perturbations around a cosmological background have revealed power law instabilities in both scalar and tensor perturbations, motivating an analysis of the initial conditions, evolution, and cosmological observables to determine the viability of these theories. In this paper we focus on the tensor sector, and study a primordial stochastic gravitational wave background in massive bigravity. The phenomenology can differ from standard General Relativity due to non-trivial mixing between the two linearized tensor fluctuations in the theory, only one of which couples to matter. We study perturbations about two classes of cosmological solutions in bigravity, computing the tensor contribution to the temperature anisotropies in the Cosmic Microwave Background radiation and the present stochastic gravitational wave background. The result is strongly dependent on the choice of cosmological background and initial conditions. One class of background solution generically displaying tremendous growth in the amplitude of large-wavelength gravitational waves, while the other remains observationally indistinguishable from standard General Relativity for a wide variety of initial conditions. We analyze the initial conditions for tensor modes expected in an inflationary cosmology, finding again that there is a strong dependence on the assumed background. For one choice of background, the semi-classical theory is beyond the perturbative regime. For the other choice, inflation generically yields initial conditions that, when evolved, give rise to a stochastic background observationally indistinguishable from standard General Relativity.

astro-ph.CO↗

arXiv2015

A Reconfigurations Analogue of Brooks' Theorem and its Consequences

Let $G$ be a simple undirected graph on $n$ vertices with maximum degree~$Δ$. Brooks' Theorem states that $G$ has a $Δ$-colouring unless~$G$ is a complete graph, or a cycle with an odd number of vertices. To recolour $G$ is to obtain a new proper colouring by changing the colour of one vertex. We show an analogue of Brooks' Theorem by proving that from any $k$-colouring, $k>Δ$, a $Δ$-colouring of $G$ can be obtained by a sequence of $O(n^2)$ recolourings using only the original $k$ colours unless $G$ is a complete graph or a cycle with an odd number of vertices, or $k=Δ+1$, $G$ is $Δ$-regular and, for each vertex $v$ in $G$, no two neighbours of $v$ are coloured alike. We use this result to study the reconfiguration graph $R_k(G)$ of the $k$-colourings of $G$. The vertex set of $R_k(G)$ is the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. We prove that for $Δ\geq 3$, $R_{Δ+1}(G)$ consists of isolated vertices and at most one further component which has diameter $O(n^2)$. This result enables us to complete both a structural classification and an algorithmic classification for reconfigurations of colourings of graphs of bounded maximum degree.

cs.CC↗

arXiv2014

Finding Shortest Paths between Graph Colourings

The $k$-colouring reconfiguration problem asks whether, for a given graph $G$, two proper $k$-colourings $α$ and $β$ of $G$, and a positive integer $\ell$, there exists a sequence of at most $\ell+1$ proper $k$-colourings of $G$ which starts with $α$ and ends with $β$ and where successive colourings in the sequence differ on exactly one vertex of $G$. We give a complete picture of the parameterized complexity of the $k$-colouring reconfiguration problem for each fixed $k$ when parameterized by $\ell$. First we show that the $k$-colouring reconfiguration problem is polynomial-time solvable for $k=3$, settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all $k \geq 4$, we show that the $k$-colouring reconfiguration problem, when parameterized by $\ell$, is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.

cs.CC↗

arXiv2014

The nonlinear dynamical stability of infrared modifications of gravity

Scalar forces "screened" by the Vainshtein mechanism may hold the key to understanding the cosmological expansion of our universe, while predicting new and exciting features in the interaction between massive bodies. Here we explore the dynamics of the Vainshtein screening mechanism, focusing on the decoupling limit of the DGP braneworld scenario and dRGT massive gravity. We show that there is a vast set of initial conditions whose evolution is well defined and which are driven to the static, screening solutions of these theories. Screening solutions are stable and behave coherently under small fluctuations: they oscillate and eventually settle to an equilibrium configuration, the timescale for the oscillations and damping being dictated by the Vainshtein radius of the screening solutions. At very late times, a power-law decay ensues, in agreement with known analytical results. However, we also conjecture that physically interesting processes such as gravitational collapse of compact stars may not possess a well-posed initial value problem. Finally, we construct solutions with nontrivial multipolar structure describing the screening field of deformed, asymmetric bodies and show that higher multipoles are screened more efficiently than the monopole component.

hep-th↗

arXiv2014

Narrowing the Complexity Gap for Colouring ($C_s$,$P_t$)-Free Graphs

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a $k$-colouring of $G$ exists. The $k$-Precolouring Extension problem is to decide, for a given $G=(V,E)$, whether a colouring of a subset of $V$ can be extended to a $k$-colouring of $G$. A $k$-list assignment of a graph is an allocation of a list -a subset of $\{1,\ldots,k\}$- to each vertex, and the List $k$-Colouring problem is to decide, for a given $G$, whether $G$ has a $k$-colouring in which each vertex is coloured with a colour from its list. We continued the study of the computational complexity of these three decision problems when restricted to graphs that contain neither a cycle on $s$ vertices nor a path on $t$ vertices as induced subgraphs (for fixed positive integers $s$ and~$t$).

cs.CC↗