A dependent partition-valued process for multitask clustering and time evolving network modelling

The fundamental aim of clustering algorithms is to partition data points. We consider tasks where the discovered partition is allowed to vary with some covariate such as space or time. One approach would be to use fragmentation-coagulation processes, but these, being Markov processes, are restricted to linear or tree structured covariate spaces. We define a partition-valued process on an arbitrary covariate space using Gaussian processes. We use the process to construct a multitask clustering model which partitions datapoints in a similar way across multiple data sources, and a time series model of network data which allows cluster assignments to vary over time. We describe sampling algorithms for inference and apply our method to defining cancer subtypes based on different types of cellular characteristics, finding regulatory modules from gene expression data from multiple human populations, and discovering time varying community structure in a social network.

Efficient sampling of atomic configurational spaces.

We describe a method to explore the configurational phase space of chemical systems. It is based on the nested sampling algorithm recently proposed by Skilling (AIP Conf. Proc. 2004, 395; J. Bayesian Anal. 2006, 1, 833) and allows us to explore the entire potential energy surface (PES) efficiently in an unbiased way. The algorithm has two parameters which directly control the trade-off between the resolution with which the space is explored and the computational cost. We demonstrate the use of nested sampling on Lennard-Jones (LJ) clusters. Nested sampling provides a straightforward approximation for the partition function; thus, evaluating expectation values of arbitrary smooth operators at arbitrary temperatures becomes a simple postprocessing step. Access to absolute free energies allows us to determine the temperature-density phase diagram for LJ cluster stability. Even for relatively small clusters, the efficiency gain over parallel tempering in calculating the heat capacity is an order of magnitude or more. Furthermore, by analyzing the topology of the resulting samples, we are able to visualize the PES in a new and illuminating way. We identify a discretely valued order parameter with basins and suprabasins of the PES, allowing a straightforward and unambiguous definition of macroscopic states of an atomistic system and the evaluation of the associated free energies.

Network structure preserving model reduction with weak a priori structural information

This paper extends a state projection method for structure preserving model reduction to situations where only a weaker notion of system structure is available. This weaker notion of structure, identifying the causal relationship between manifest variables of the system, is especially relevant is settings such as systems biology, where a clear partition of state variables into distinct subsystems may be unknown, or not even exist. The resulting technique, like similar approaches, does not provide theoretical performance guarantees, so an extensive computational study is conducted, and it is observed to work fairly well in practice. Moreover, conditions characterizing structurally minimal realizations and sufficient conditions characterizing edge loss resulting from the reduction process, are presented. ©2009 IEEE.

Nested sampling for materials: the case of hard spheres

The recently introduced nested sampling algorithm allows the direct and efficient calculation of the partition function of atomistic systems. We demonstrate its applicability to condensed phase systems with periodic boundary conditions by studying the three dimensional hard sphere model. Having obtained the partition function, we show how easy it is to calculate the compressibility and the free energy as functions of the packing fraction and local order, verifying that the transition to crystallinity has a very small barrier, and that the entropic contribution of jammed states to the free energy is negligible for packing fractions above the phase transition. We quantify the previously proposed schematic phase diagram and estimate the extent of the region of jammed states. We find that within our samples, the maximally random jammed configuration is surprisingly disordered.

Decentralised minimum-time consensus

We consider the discrete-time dynamics of a network of agents that exchange information according to a nearest-neighbour protocol under which all agents are guaranteed to reach consensus asymptotically. We present a fully decentralised algorithm that allows any agent to compute the final consensus value of the whole network in finite time using the minimum number of successive values of its own state history. We show that the minimum number of steps is related to a Jordan block decomposition of the network dynamics, and present an algorithm to compute the final consensus value in the minimum number of steps by checking a rank condition of a Hankel matrix of local observations. Furthermore, we prove that the minimum number of steps is related to graph theoretical notions that can be directly computed from the Laplacian matrix of the graph and from the minimum external equitable partition. © 2013 Elsevier Ltd. All rights reserved.

The Random Forest Kernel and other kernels for big data from random partitions

We present Random Partition Kernels, a new class of kernels derived by demonstrating a natural connection between random partitions of objects and kernels between those objects. We show how the construction can be used to create kernels from methods that would not normally be viewed as random partitions, such as Random Forest. To demonstrate the potential of this method, we propose two new kernels, the Random Forest Kernel and the Fast Cluster Kernel, and show that these kernels consistently outperform standard kernels on problems involving real-world datasets. Finally, we show how the form of these kernels lend themselves to a natural approximation that is appropriate for certain big data problems, allowing $O(N)$ inference in methods such as Gaussian Processes, Support Vector Machines and Kernel PCA.