Angular Gaussian Distributions Research Articles

Learning Representations on the Unit Sphere: Investigating Angular Gaussian and Von Mises-Fisher Distributions for Online Continual Learning

We use the maximum a posteriori estimation principle for learning representations distributed on the unit sphere. We propose to use the angular Gaussian distribution, which corresponds to a Gaussian projected on the unit-sphere and derive the associated loss function. We also consider the von Mises-Fisher distribution, which is the conditional of a Gaussian in the unit-sphere. The learned representations are pushed toward fixed directions, which are the prior means of the Gaussians; allowing for a learning strategy that is resilient to data drift. This makes it suitable for online continual learning, which is the problem of training neural networks on a continuous data stream, where multiple classification tasks are presented sequentially so that data from past tasks are no longer accessible, and data from the current task can be seen only once. To address this challenging scenario, we propose a memory-based representation learning technique equipped with our new loss functions. Our approach does not require negative data or knowledge of task boundaries and performs well with smaller batch sizes while being computationally efficient. We demonstrate with extensive experiments that the proposed method outperforms the current state-of-the-art methods on both standard evaluation scenarios and realistic scenarios with blurry task boundaries. For reproducibility, we use the same training pipeline for every compared method and share the code at https://github.com/Nicolas1203/ocl-fd.

A Robust and Flexible EM Algorithm for Mixtures of Elliptical Distributions with Missing Data

This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A classical imputation method, the Expectation Maximization (EM) algorithm for Gaussian mixture models, has shown interesting properties when compared to other popular approaches such as those based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -nearest neighbors or on multiple imputations by chained equations. However, Gaussian mixture models are known to be non-robust to heterogeneous data, which can lead to poor estimation performance when the data is contaminated by outliers or have non-Gaussian distributions. To overcome this issue, a new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. This paper shows that this problem reduces to the estimation of a mixture of angular Gaussian distributions under generic assumptions ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , each sample is drawn from a mixture of elliptical distributions, which is possibly different for one sample to another). In that case, the complete-data likelihood associated with mixtures of elliptical distributions is well adapted to the EM framework with missing data thanks to its conditional distribution, which is shown to be a multivariate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -distribution. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data. Furthermore, experiments conducted on real-world datasets show that this algorithm is very competitive when compared to other classical imputation methods.

Symmetric nonconfocal Fabry–Perot cavity with a stable long optical path length and improved tolerance for angular alignment

Engineering of optical alignment tolerance between a laser source and a Fabry–Perot (FP) cavity is of great importance in optimizing a laser analysis system and saving costs in the additional alignment task. We propose a modification of the conventional confocal FP (CFP) cavity–a symmetric nonconfocal FP (SNFP) cavity with two identical spherical mirrors separated by a distance equal to half of the common radius of curvature. Numerical analysis of the two cavities reveals that the SNFP cavity has better off-axis stability and much better angular misalignment tolerance than the CFP cavity, even with a longer optical path length (OPL). The SNFP cavity provides a six-transit path length for a basic cavity mode that supports a 1.5-fold long OPL within the same volume as the CFP cavity, which is favorable for applications requiring limits on the footprint and weight of the cavity. In addition, a ray-tracing simulation with a disk-type source having an angular Gaussian distribution (α = β = 5 deg) reveals that the emission beam profile from the SNFP cavity is narrower than that from the CFP cavity. These results are expected to help reduce cavity loss and to improve alignment efficiency between the laser source and the FP cavity.

Spherical regression models with general covariates and anisotropic errors

Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example that they are either scalar or are themselves on the unit sphere, and/or that the error distribution is isotropic. In many practical situations, such models are too inflexible. Here, we develop richer parametric spherical regression models in which the covariates can have quite general structure (for example, they may be on the unit sphere, in Euclidean space, categorical or some combination of these) and in which the errors are anisotropic. We consider two anisotropic error distributions—the Kent distribution and the elliptically symmetric angular Gaussian distribution—and two parametrisations of each which enable distinct ways to model how the response depends on the covariates. Various hypotheses of interest, such as the significance of particular covariates, or anisotropy of the errors, are easy to test, for example by classical likelihood ratio tests. We also introduce new model-based residuals for evaluating the fitted models. In the examples we consider, the hypothesis tests indicate strong evidence to favour the novel models over simpler existing ones.

An elliptically symmetric angular Gaussian distribution

We define a distribution on the unit sphere mathbb {S}^{d-1} called the elliptically symmetric angular Gaussian distribution. This distribution, which to our knowledge has not been studied before, is a subfamily of the angular Gaussian distribution closely analogous to the Kent subfamily of the general Fisher–Bingham distribution. Like the Kent distribution, it has ellipse-like contours, enabling modelling of rotational asymmetry about the mean direction, but it has the additional advantages of being simple and fast to simulate from, and having a density and hence likelihood that is easy and very quick to compute exactly. These advantages are especially beneficial for computationally intensive statistical methods, one example of which is a parametric bootstrap procedure for inference for the directional mean that we describe.

Refinement of the orientational distribution function in plasticn-butane guided by computer simulations

Neutron powder diffraction data for the plastic crystalline phase of n-butane are refined with a model suggested from molecular dynamics (MD) computer simulation work. A small decrease in the R factor is achieved, corresponding to a reduction of the discrepancy between fit and experiment of about 30%. This supports the MD result that the orientational distribution function (ODF) describing the disorder is an asymmetric sum of four angular Gaussian distributions, one of which is dominant. The orientational disorder is uniaxial about the axis between the end C atoms of a molecule. Refinement of the four-Gaussian function is stable, and results in a distribution closely related to that obtained from the MD work. It is suggested that the ODF is temperature dependent, and that this dependence could be found by a further series of high-quality experiments.

Distributions on the circle and sphere

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.

Distributions on the circle and sphere

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.