arXiv subjects

Harry Crane

Publications and source records attributed to Harry Crane.

26 records · Page 2Linked to original sources

arXiv2015

Lipschitz partition processes

We introduce a family of Markov processes on set partitions with a bounded number of blocks, called Lipschitz partition processes. We construct these processes explicitly by a Poisson point process on the space of Lipschitz continuous maps on partitions. By this construction, the Markovian consistency property is readily satisfied; that is, the finite restrictions of any Lipschitz partition process comprise a compatible collection of finite state space Markov chains. We further characterize the class of exchangeable Lipschitz partition processes by a novel set-valued matrix operation.

math.ST↗

arXiv2014

The cut-and-paste process

We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $σ$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a Lévy-Itô representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.

math.PR↗

arXiv2013

Exchangeable Markov Processes on $[k]^{\zz{N}}$ with Cadlag Sample Paths

Any exchangeable Markov processes on $[k]^{\mathbb{N}}$ with cadlag sample paths projects to a Markov process on the simplex whose sample paths are cadlag and of locally bounded variation. Furthermore, any such process has a de Finetti-type description as a mixture of i.i.d. copies of time-inhomogeneous Markov processes on $[k]$. In the Feller case, these time-inhomogeneous Markov processes have a relatively simple structure; however, in the non-Feller case a greater variety of behaviors is possible since the transition law of the underlying Markov process on $[k]^{\zz{N}}$ can depend in a non-trivial way on the exchangeable $σ$-algebra of the process.

math.PR↗

arXiv2013

Some algebraic identities for the alpha-permanent

We show that the permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers αand β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computational complexity of the α-permanent and provide some numerical illustrations.

math.CO↗

arXiv2012

Convergence Rates of Markov Chains on Spaces of Partitions

We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic matrices, which describe the chain induced on the simplex by taking asymptotic frequencies. Using this representation, we establish upper bounds for the mixing times of ergodic cut-and-paste chains, and under certain conditions on the distribution of the governing random matrices we show that the "cutoff phenomenon" holds.

math.PR↗

arXiv2011

Ancestral branching, cut-and-paste algorithms and associated tree and partition-valued processes

We introduce an algorithm for generating a random sequence of fragmentation trees, which we call the ancestral branching algorithm. This algorithm builds on the recursive partitioning structure of a tree and gives rise to an associated family of Markovian transition kernels whose finite-dimensional transition probabilities can be written in closed-form as the product over partition-valued Markov kernels. The associated tree-valued Markov process is infinitely exchangeable provided its associated partition-valued kernel is infinitely exchangeable. We also identify a transition procedure on partitions, called the cut-and-paste algorithm, which corresponds to a previously studied partition-valued Markov process on partitions with a bounded number of blocks. Specifically, we discuss the corresponding family of tree-valued Markov kernels generated by the combination of both the ancestral branching and cut-and-paste transition probabilities and show results for the equilibrium measure of this process, as well as its associated mass fragmentation-valued and weighted tree-valued processes.

math.PR↗

arXiv2011

Infinitely exchangeable random graphs generated from a Poisson point process on monotone sets and applications to cluster analysis for networks

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $Λ$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, and every monotone subset maps to an undirected graph $G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an infinitely exchangeable measure on the projective system $\catg^{\rest}$ of graphs $\catg$ under permutation and restriction mappings given an infinitely exchangeable family of measures on the projective system $\cate^{\rest}$ of subsets with permutation and restriction maps. We show potential connections of this process to applications in cluster analysis, machine learning, classification and Bayesian inference.

math.ST↗

arXiv2011

A consistent Markov partition process generated from the paintbox process

We study a family of Markov processes on $\mathcal{P}^{(k)}$, the space of partitions of the natural numbers with at most $k$ blocks. The process can be constructed from a Poisson point process on $\mathbb{R}^+\times\prod_{i=1}^k\mathcal{P}^{(k)}$ with intensity $dt\otimes\varrho_ν^{(k)}$, where $\varrho_ν$ is the distribution of the paintbox based on the probability measure $ν$ on $\masspartition$, the set of ranked-mass partitions of 1, and $\varrho_ν^{(k)}$ is the product measure on $\prod_{i=1}^k\mathcal{P}^{(k)}$. We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

math.PR↗