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Robust Estimation of Location in Matrix Manifolds Using the Projected Frobenius Median

Houren Hong, Kassel Liam Hingee, Janice L. Scealy, Andrew T. A. Wood

Abstract

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and Grassmann manifolds, Kendall shape spaces as well as to projective Stiefel manifolds, a type of quotient space of a Stiefel manifold. Our approach involves computation of the Frobenius median in an ambient Euclidean space followed by projection onto the relevant matrix manifold. Our estimation method is computationally attractive, has a unique solution provided the sample data are not colinear in the ambient Euclidean space, has desirable robustness features and has appropriate equivariance properties under natural groups of transformations. We establish asymptotic normality under mild conditions and derive the influence function for matrix manifolds of interest. Simulation studies on the rank-1 complex Grassmann manifold and the projective Stiefel manifold further show the applicability and robustness of our method. We also apply our method to a real-world earthquake moment tensor dataset.

Robust Estimation of Location in Matrix Manifolds Using the Projected Frobenius Median

Abstract

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and Grassmann manifolds, Kendall shape spaces as well as to projective Stiefel manifolds, a type of quotient space of a Stiefel manifold. Our approach involves computation of the Frobenius median in an ambient Euclidean space followed by projection onto the relevant matrix manifold. Our estimation method is computationally attractive, has a unique solution provided the sample data are not colinear in the ambient Euclidean space, has desirable robustness features and has appropriate equivariance properties under natural groups of transformations. We establish asymptotic normality under mild conditions and derive the influence function for matrix manifolds of interest. Simulation studies on the rank-1 complex Grassmann manifold and the projective Stiefel manifold further show the applicability and robustness of our method. We also apply our method to a real-world earthquake moment tensor dataset.
Paper Structure (25 sections, 8 theorems, 46 equations, 6 figures, 3 tables)

This paper contains 25 sections, 8 theorems, 46 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The projected Frobenius median
  4. Outline of the paper
  5. Matrix manifolds of interest
  6. Stiefel manifolds
  7. Grassmann manifolds
  8. Complex projective space
  9. Projective Stiefel manifolds
  10. Other matrix manifolds of potential interest
  11. The Frobenius median on matrix manifolds
  12. The main idea
  13. Calculation of Frobenius median
  14. The projection step
  15. Equivariance properties of the projected Frobenius median
  16. ...and 10 more sections

Key Result

Proposition 1

(i) Consider $\mathcal{M}=\mathcal{V}_{k,r}$, the Stiefel manifold. Suppose the sample Frobenius median $\hat{A}_s$ has the singular value decomposition (SVD) $\hat{A}_s=\sum_{j=1}^r \hat{\rho}_j \hat{s}_j \hat{t}_j^\top$, where $\hat{\rho}_1 > \cdots > \hat{\rho}_r \geq 0$, the $k \times 1$ vectors

Figures (6)

  • Figure 1: Left: Illustrations of the original configurations (top row) and five samples from the complex Bingham distribution (bottom row). Right: Estimated configurations in the presence of $90$ outliers, obtained by EMedian (first column), IMean (second column), IMedian (third column) and MoM (last column).
  • Figure 2: Boxplots of the logarithm of estimation errors for planar shape data (Shape 1, 20 outliers, 500 replicates).
  • Figure 3: The estimates of $\hat{M}_{\text{mean}}$ and $\hat{M}_{\text{median}}$ for Case 1 with $\kappa=(5,5,5), n=50$ and 5 outliers, where the red circles are the 95% pivotal bootstrap confidence ellipses of $\hat{M}_{\text{median}}$. The population axial frame lies at the origin.
  • Figure 4: The orthogonal axial frames of moment tensors in region 2 (21 events). Left: axes of the moment tensors (up to sign, suspected outliers are circled in red and labelled in numbers). Middle: the largest eigenvalues $\lambda_1$ of the moment tensors. Right: the second largest eigenvalues $\lambda_2$ of the moment tensors.
  • Figure 5: Scatter plots (top row) and magnified aerial views (bottom row) of the T, B and P axes of moment tensors in region 2. Outliers in region 2 are highlighted with red circles. The shaded areas (top row) and red circles (bottom row) represent 95% confidence regions of the spatial median.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 1
  • Theorem 2
  • Theorem 3