Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Support of Continuous Smeary Measures on Spheres

Susovan Pal

Abstract

We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures $μ$ with density $ρ$ on spheres $\mathbb{S}^m$. First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius $R_m>π/2$, where $R_m=π/2+O(1/m)$. In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius $π/2$, however, past the support radius $π/2,$ full smeariness may break down, but directional smeariness breaks down only past the support radius $R_m.$ Second, we prove sharpness of this threshold. For every $\varepsilon>0$, we show there exists $m_0(\varepsilon)$ such that for all $m\ge m_0(\varepsilon)$ there exists a rotationally symmetric continuous smeary probability measure on $\mathbb{S}^m$ whose support lies in a ball of radius $π/2+\varepsilon$ around the Fréchet mean. Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius $π/2+\varepsilon$ whose Fréchet function has Hessian of rank one. Finally, we study finite sample smeariness. We show that any continuous non-smeary distribution supported in a geodesic ball of radius $π/2$ is necessarily Type~I finite sample smeary, i.e. its variance modulation $m_n$ satisfies $\lim_{n\to\infty} m_n>1$. In the rotationally symmetric case, we further prove a curse-of-dimensionality phenomenon: the variance modulation increases with the dimension and can become arbitrarily large depending on the support.

Support of Continuous Smeary Measures on Spheres

Abstract

We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures with density on spheres . First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius , where . In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius , however, past the support radius full smeariness may break down, but directional smeariness breaks down only past the support radius Second, we prove sharpness of this threshold. For every , we show there exists such that for all there exists a rotationally symmetric continuous smeary probability measure on whose support lies in a ball of radius around the Fréchet mean. Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius whose Fréchet function has Hessian of rank one. Finally, we study finite sample smeariness. We show that any continuous non-smeary distribution supported in a geodesic ball of radius is necessarily Type~I finite sample smeary, i.e. its variance modulation satisfies . In the rotationally symmetric case, we further prove a curse-of-dimensionality phenomenon: the variance modulation increases with the dimension and can become arbitrarily large depending on the support.
Paper Structure (24 sections, 22 theorems, 360 equations, 1 figure)

This paper contains 24 sections, 22 theorems, 360 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fréchet function, smeariness and directional smeariness.
  3. Contribution of the paper to smeariness and directional smearineess:
  4. Finite sample smeariness.
  5. Contribution of the paper to finite sample smeariness:
  6. Acknowledgements and funding.
  7. Taylor expansion of the Riemannian squared distance of $\mathbb{S}^m$ up to order $4$
  8. Taylor expansion of the Riemannian squared distance function in the tangent space using the exponential map
  9. Second and fourth partial derivative tensors of the Riemannian squared distance in normal coordinates
  10. Fréchet function for probability measures on $\mathbb{S}^m$ with rotationally symmetric densities
  11. Setup
  12. Hessian and fourth derivative tensors of the Fréchet function
  13. Location and asymptotics of zeros of $b_m, h_m$ in Equations \ref{['eqn:b']}, \ref{['eqn:h']}
  14. Super-hemispherical obstruction for smeariness for rotationally symmetric densities
  15. Constructing rotationally symmetric densities on high-dimensional spheres support in arbitrarily thin open annuli around the equator
  16. ...and 9 more sections

Key Result

Lemma 2.2

Let $B:V^4\to\mathbb{R}$ be symmetric $4$--linear and set $p(w):=B(w,w,w,w)$. Then for all $w_1,w_2,w_3,w_4\in V$, In particular, $B$ is uniquely determined by the diagonal map $w\mapsto p(w)$.

Figures (1)

  • Figure 1: Finite-sample modulation versus dimension.

Theorems & Definitions (59)

  • Definition 1.1: Smeariness and directional smeariness
  • Definition 1.2: Variance modulation and finite sample smeariness
  • Remark 1.3: Smeariness, directional smeariness and divergence of the variance modulation
  • Definition 2.1: Symmetric $(4,0)$--tensor
  • Lemma 2.2: Polarization identity for symmetric $4$--linear forms
  • Proposition 2.3: Second and fourth derivatives of $g(\,\cdot\,;R\Theta)$ at $u=0$
  • proof
  • Proposition 3.2: Hessian and fourth derivative tensor of the Fréchet function in case of rotationally symmetric densities, and their vanishing
  • Remark 3.3
  • proof
  • ...and 49 more