Tree-Structured Stick Breaking Processes for Hierarchical Data

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

Bayesian non-parametrics and the probabilistic approach to modelling.

Modelling is fundamental to many fields of science and engineering. A model can be thought of as a representation of possible data one could predict from a system. The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of probability theory in order to make predictions, compare alternative models, and learn model parameters and structure from data. This simple and elegant framework is most powerful when coupled with flexible probabilistic models. Flexibility is achieved through the use of Bayesian non-parametrics. This article provides an overview of probabilistic modelling and an accessible survey of some of the main tools in Bayesian non-parametrics. The survey covers the use of Bayesian non-parametrics for modelling unknown functions, density estimation, clustering, time-series modelling, and representing sparsity, hierarchies, and covariance structure. More specifically, it gives brief non-technical overviews of Gaussian processes, Dirichlet processes, infinite hidden Markov models, Indian buffet processes, Kingman's coalescent, Dirichlet diffusion trees and Wishart processes.

Simultaneous colour search renders other object properties less salient

A common approach to visualise multidimensional data sets is to map every data dimension to a separate visual feature. It is generally assumed that such visual features can be judged independently from each other. However, we have recently shown that interactions between features do exist [Hannus et al. 2004; van den Berg et al. 2005]. In those studies, we first determined individual colour and size contrast or colour and orientation contrast necessary to achieve a fixed level of discrimination performance in single feature search tasks. These contrasts were then used in a conjunction search task in which the target was defined by a combination of a colour and a size or a colour and an orientation. We found that in conjunction search, despite the matched feature discriminability, subjects significantly more often chose an item with the correct colour than one with correct size or orientation. This finding may have consequences for visualisation: the saliency of information coded by objects' size or orientation may change when there is a need to simultaneously search for colour that codes another aspect of the information. In the present experiment, we studied whether a colour bias can also be found in a more complex and continuous task, Subjects had to search for a target in a node-link diagram consisting of SO nodes, while their eye movements were being tracked, Each node was assigned a random colour and size (from a range of 10 possible values with fixed perceptual distances). We found that when we base the distances on the mean threshold contrasts that were determined in our previous experiments, the fixated nodes tend to resemble the target colour more than the target size (Figure 1a). This indicates that despite the perceptual matching, colour is judged with greater precision than size during conjunction search. We also found that when we double the size contrast (i.e. the distances between the 10 possible node sizes), this effect disappears (Figure 1b). Our findings confirm that the previously found decrease in salience of other features during colour conjunction search is also present in more complex (more 'visualisation- realistic') visual search tasks. The asymmetry in visual search behaviour can be compensated for by manipulating step sizes (perceptual distances) within feature dimensions. Our results therefore also imply that feature hierarchies are not completely fixed and may be adapted to the requirements of a particular visualisation. Copyright © 2005 by the Association for Computing Machinery, Inc.

Beyond 2D-grids: A dependence maximization view on image browsing

Ideally, one would like to perform image search using an intuitive and friendly approach. Many existing image search engines, however, present users with sets of images arranged in some default order on the screen, typically the relevance to a query, only. While this certainly has its advantages, arguably, a more flexible and intuitive way would be to sort images into arbitrary structures such as grids, hierarchies, or spheres so that images that are visually or semantically alike are placed together. This paper focuses on designing such a navigation system for image browsers. This is a challenging task because arbitrary layout structure makes it difficult - if not impossible - to compute cross-similarities between images and structure coordinates, the main ingredient of traditional layouting approaches. For this reason, we resort to a recently developed machine learning technique: kernelized sorting. It is a general technique for matching pairs of objects from different domains without requiring cross-domain similarity measures and hence elegantly allows sorting images into arbitrary structures. Moreover, we extend it so that some images can be preselected for instance forming the tip of the hierarchy allowing to subsequently navigate through the search results in the lower levels in an intuitive way. Copyright 2010 ACM.

Mechanical properties of hierarchical lattices

This paper focuses on the stiffness and strength of lattices with multiple hierarchical levels. We examine two-dimensional and three-dimensional lattices with up to three levels of structural hierarchy. At each level, the topology and the orientation of the lattice are prescribed, while the relative density is varied over a defined range. The properties of selected hierarchical lattices are obtained via a multiscale approach applied iteratively at each hierarchical level. The results help to quantify the effect that multiple orders of structural hierarchy produces on stretching and bending dominated lattices. Material charts for the macroscopic stiffness and strength illustrate how the property range of the lattices can expand as subsequent levels of hierarchy are added. The charts help to gain insight into the structural benefit that multiple hierarchies can impart to the macroscopic performance of a lattice. © 2013 Elsevier Ltd. All rights reserved.

Unified form language: A domain-specific language for weak formulations of partial differential equations

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies formultifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted. © 2014 ACM.