Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects

This paper studies fundamental aspects of modelling data using multivariate Watson distributions. Although these distributions are natural for modelling axially symmetric data (i.e., unit vectors where \pm \x are equivalent), for high-dimensions using them can be difficult. Why so? Largely because for Watson distributions even basic tasks such as maximum-likelihood are numerically challenging. To tackle the numerical difficulties some approximations have been derived---but these are either grossly inaccurate in high-dimensions (\emph{Directional Statistics}, Mardia & Jupp. 2000) or when reasonably accurate (\emph{J. Machine Learning Research, W. & C.P., v2}, Bijral \emph{et al.}, 2007, pp. 35--42), they lack theoretical justification. We derive new approximations to the maximum-likelihood estimates; our approximations are theoretically well-defined, numerically accurate, and easy to compute. We build on our parameter estimation and discuss mixture-modelling with Watson distributions; here we uncover a hitherto unknown connection to the "diametrical clustering" algorithm of Dhillon \emph{et al.} (\emph{Bioinformatics}, 19(13), 2003, pp. 1612--1619).Comment: 24 pages; extensively updated numerical result

Recurrent Pixel Embedding for Instance Grouping

We introduce a differentiable, end-to-end trainable framework for solving pixel-level grouping problems such as instance segmentation consisting of two novel components. First, we regress pixels into a hyper-spherical embedding space so that pixels from the same group have high cosine similarity while those from different groups have similarity below a specified margin. We analyze the choice of embedding dimension and margin, relating them to theoretical results on the problem of distributing points uniformly on the sphere. Second, to group instances, we utilize a variant of mean-shift clustering, implemented as a recurrent neural network parameterized by kernel bandwidth. This recurrent grouping module is differentiable, enjoys convergent dynamics and probabilistic interpretability. Backpropagating the group-weighted loss through this module allows learning to focus on only correcting embedding errors that won't be resolved during subsequent clustering. Our framework, while conceptually simple and theoretically abundant, is also practically effective and computationally efficient. We demonstrate substantial improvements over state-of-the-art instance segmentation for object proposal generation, as well as demonstrating the benefits of grouping loss for classification tasks such as boundary detection and semantic segmentation