# Hyperspherical Dirac Mixture Reapproximation

@article{Li2021HypersphericalDM, title={Hyperspherical Dirac Mixture Reapproximation}, author={Kailai Li and Florian Pfaff and Uwe D. Hanebeck}, journal={ArXiv}, year={2021}, volume={abs/2110.10411} }

We propose a novel scheme for efficient Dirac mixture modeling of distributions on unit hyperspheres. A so-called hyperspherical localized cumulative distribution (HLCD) is introduced as a local and smooth characterization of the underlying continuous density in hyperspherical domains. Based on HLCD, a manifold-adapted modification of the Cramér–von Mises distance (HCvMD) is established to measure the statistical divergence between two Dirac mixtures of arbitrary dimensions. Given a (source…

## 3 Citations

### Dirac Mixture Reduction Using Wasserstein Distances on Projected Cumulative Distributions

- Computer Science2022 25th International Conference on Information Fusion (FUSION)
- 2022

The Wasserstein distance is established as a suitable measure to compare two Dirac mixtures and an iterative algorithm is proposed to minimize the sliced Wasserstone distance between the given distribution and approximation.

### Circular Discrete Reapproximation

- Computer Science2022 25th International Conference on Information Fusion (FUSION)
- 2022

The circular Cramer-von Mises distance (CCvMD) is proposed to measure the statistical divergence between two circular discrete models based on a smooth characterization of the localized cumulative distribution.

### Deterministic Sampling on the Circle Using Projected Cumulative Distributions

- Mathematics2022 25th International Conference on Information Fusion (FUSION)
- 2022

This work compares cumulatives of probability densities in the Radon space and proposes a method for deterministic sampling of arbitrary continuous angular density functions that can draw arbitrary numbers of deterministic samples and therefore improve the quality of state estimation.

## References

SHOWING 1-10 OF 57 REFERENCES

### Clustering on the Unit Hypersphere using von Mises-Fisher Distributions

- Computer ScienceJ. Mach. Learn. Res.
- 2005

A generative mixture-model approach to clustering directional data based on the von Mises-Fisher distribution, which arises naturally for data distributed on the unit hypersphere, and derives and analyzes two variants of the Expectation Maximization framework for estimating the mean and concentration parameters of this mixture.

### Optimal Reduction of Multivariate Dirac Mixture Densities

- MathematicsAutom.
- 2015

An alternative to the classical cumulative distribution, the Localized Cumulative Distribution, is defined, as a smooth characterization of discrete random quantities (on continuous domains), which provides the basis for various efficient nonlinear estimation and control methods.

### Hyperspherical Deterministic Sampling Based on Riemannian Geometry for Improved Nonlinear Bingham Filtering

- Mathematics, Computer Science2019 22th International Conference on Information Fusion (FUSION)
- 2019

A geometry-adaptive sampling scheme for generating equally weighted deterministic samples of Bingham distributions in arbitrary dimensions that gives better tracking accuracy and robustness for nonlinear orientation estimations.

### Progressive von Mises–Fisher Filtering Using Isotropic Sample Sets for Nonlinear Hyperspherical Estimation †

- MathematicsSensors
- 2021

A novel scheme for nonlinear hyperspherical estimation using the von Mises–Fisher distribution with deterministic sample sets with an isotropic layout is presented, considerably enhancing the filtering performance under strong nonlinearity.

### Synchronizing Probability Measures on Rotations via Optimal Transport

- Computer Science, Mathematics2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- 2020

A new paradigm, `measure synchronization', for synchronizing graphs with measure-valued edges is introduced, which aims at estimating marginal distributions of absolute orientations by synchronizing the `conditional' ones on the Riemannian manifold of quaternions.

### Grid-Based Quaternion Filter for SO(3) Estimation

- Computer Science, Engineering2020 European Control Conference (ECC)
- 2020

A novel discrete Bayesian filtering scheme is proposed on the manifold of unit quaternions for rotation estimation that allows non-parametric modeling of the underlying uncertainty using Dirac mixtures located on a hyperspherical grid.

### Nonlinear von Mises–Fisher Filtering Based on Isotropic Deterministic Sampling

- Mathematics2020 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI)
- 2020

A novel deterministic sampling approach for von Mises–Fisher distributions of arbitrary dimensions where samples of configurable size are drawn isotropically on the hypersphere while preserving the mean resultant vector of the underlying distribution.

### Directional Statistics with the Spherical Normal Distribution

- Mathematics2018 21st International Conference on Information Fusion (FUSION)
- 2018

This work develops efficient inference techniques for data distributed by the curvature-aware spherical normal distribution, and derives closed-form expressions for the normalization constant when the distribution is isotropic, and a fast and accurate approximation for the anisotropic case on the two-sphere.

### Unscented Orientation Estimation Based on the Bingham Distribution

- Computer ScienceIEEE Transactions on Automatic Control
- 2016

This work develops a recursive filter to estimate orientation in 3D, represented by quaternions, using directional distributions that use the Bingham distribution, the first deterministic sampling scheme that truly reflects the nonlinear manifold of orientations.