Asymptotic independence for unimodal densities

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (dfs). Dfs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, one can obtain a good geometric image of the asymptotic shape of the level sets of the density. This paper establishes a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.Comment: 33 pages, 4 figure

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

Improving "bag-of-keypoints" image categorisation: Generative Models and PDF-Kernels

In this paper we propose two distinct enhancements to the basic ''bag-of-keypoints" image categorisation scheme proposed in [4]. In this approach images are represented as a variable sized set of local image features (keypoints). Thus, we require machine learning tools which can operate on sets of vectors. In [4] this is achieved by representing the set as a histogram over bins found by k-means. We show how this approach can be improved and generalised using Gaussian Mixture Models (GMMs). Alternatively, the set of keypoints can be represented directly as a probability density function, over which a kernel can be de ned. This approach is shown to give state of the art categorisation performance

Spectral Thresholds in the Bipartite Stochastic Block Model

We consider a bipartite stochastic block model on vertex sets $V_1$ and $V_2$, with planted partitions in each, and ask at what densities efficient algorithms can recover the partition of the smaller vertex set. When $|V_2| \gg |V_1|$, multiple thresholds emerge. We first locate a sharp threshold for detection of the partition, in the sense of the results of \cite{mossel2012stochastic,mossel2013proof} and \cite{massoulie2014community} for the stochastic block model. We then show that at a higher edge density, the singular vectors of the rectangular biadjacency matrix exhibit a localization / delocalization phase transition, giving recovery above the threshold and no recovery below. Nevertheless, we propose a simple spectral algorithm, Diagonal Deletion SVD, which recovers the partition at a nearly optimal edge density. The bipartite stochastic block model studied here was used by \cite{feldman2014algorithm} to give a unified algorithm for recovering planted partitions and assignments in random hypergraphs and random $k$-SAT formulae respectively. Our results give the best known bounds for the clause density at which solutions can be found efficiently in these models as well as showing a barrier to further improvement via this reduction to the bipartite block model.Comment: updated version, will appear in COLT 201

A new method for automatic Multiple Partial Discharge Classification

A new wavelet based feature parameter have been developed to represent the characteristics of PD activities, i.e. the wavelet decomposition energy of PD pulses measured from non-conventional ultra wide bandwidth PD sensors such as capacitive couplers (CC) or high frequency current transformers (HFCT). The generated feature vectors can contain different dimensions depending on the length of recorded pulses. These high dimensional feature vectors can then be processed using Principal Component Analysis (PCA) to map the data into a three dimensional space whilst the first three most significant components representing the feature vector are preserved. In the three dimensional mapped space, an automatic Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is then applied to classify the data cluster(s) produced by the PCA. As the procedure is undertaken in a three dimensional space, the obtained clustering results can be easily assessed. The classified PD sub-data sets are then reconstructed in the time domain as phase-resolved patterns to facilitate PD source type identification. The proposed approach has been successfully applied to PD data measured from electrical machines and power cables where measurements were undertaken in different laboratories

Equivalence of weak and strong modes of measures on topological vector spaces

A strong mode of a probability measure on a normed space $X$ can be defined as a point $u$ such that the mass of the ball centred at $u$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $E$ of $X$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with measures that are finite on bounded sets, the density of $E$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.Comment: 22 pages, 3 figure