Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two-Sample Tests for Optimal Lifts, Manifold Stability and Reverse Labeling Reflection Shap

Do Tran Van, Susovan Pal, Benjamin Eltzner, Stephan F. Huckemann

TL;DR

The paper addresses inference on shape data living on quotient manifolds $Q = M/G$ by introducing optimal lifts of quotient data back to the ambient manifold $M$ to mitigate curvature-induced bias in Fréchet-mean analysis. It establishes measurability, almost-everywhere uniqueness, and smoothness of optimal lifts, proves a manifold-stability theorem ensuring Fréchet means lie on the manifold part $Q^*$ with positive probability, and proves a strong law for optimal lifts that underpins new two-sample tests. Building four testing procedures based on lifting each sample optimally (with pooled and individual variants, intrinsic versions, and bootstrap calibrations), the work demonstrates superior performance over classical Procrustes-type tests on simulated data and in a reverse-labeling planar shape space modeling filament-like biology. It further introduces reverse labeling reflection shape spaces to model end-to-end label reversals in filament data and applies the methods to microtubule–intermediate filament interactions, offering a practical approach for detecting group differences in curved shape spaces. Overall, the framework provides theoretically grounded, computationally feasible tools for hypothesis testing on non-Euclidean shape data with tangible applications in cell biology.

Abstract

We consider a quotient of a complete Riemannian manifold modulo an isometrically and properly acting Lie group and lifts of the quotient to the manifolds in optimal position to a reference point on the manifold. With respect to the pushed forward Riemannian volume onto the quotient we derive continuity and uniqueness a.e. and smoothness to large extents also with respect to the reference point. In consequence we derive a general manifold stability theorem: the Fréchet mean lies in the highest dimensional stratum assumed with positive probability, and a strong law for optimal lifts. This allows to define new two-sample tests utilizing individual optimal lifts which outperform existing two-sample tests on simulated data. They also outperform existing tests on a newly derived reverse labeling reflection shape space, that is used to model filament data of microtubules within cells in a biological application.

Two-Sample Tests for Optimal Lifts, Manifold Stability and Reverse Labeling Reflection Shap

TL;DR

The paper addresses inference on shape data living on quotient manifolds by introducing optimal lifts of quotient data back to the ambient manifold to mitigate curvature-induced bias in Fréchet-mean analysis. It establishes measurability, almost-everywhere uniqueness, and smoothness of optimal lifts, proves a manifold-stability theorem ensuring Fréchet means lie on the manifold part with positive probability, and proves a strong law for optimal lifts that underpins new two-sample tests. Building four testing procedures based on lifting each sample optimally (with pooled and individual variants, intrinsic versions, and bootstrap calibrations), the work demonstrates superior performance over classical Procrustes-type tests on simulated data and in a reverse-labeling planar shape space modeling filament-like biology. It further introduces reverse labeling reflection shape spaces to model end-to-end label reversals in filament data and applies the methods to microtubule–intermediate filament interactions, offering a practical approach for detecting group differences in curved shape spaces. Overall, the framework provides theoretically grounded, computationally feasible tools for hypothesis testing on non-Euclidean shape data with tangible applications in cell biology.

Abstract

We consider a quotient of a complete Riemannian manifold modulo an isometrically and properly acting Lie group and lifts of the quotient to the manifolds in optimal position to a reference point on the manifold. With respect to the pushed forward Riemannian volume onto the quotient we derive continuity and uniqueness a.e. and smoothness to large extents also with respect to the reference point. In consequence we derive a general manifold stability theorem: the Fréchet mean lies in the highest dimensional stratum assumed with positive probability, and a strong law for optimal lifts. This allows to define new two-sample tests utilizing individual optimal lifts which outperform existing two-sample tests on simulated data. They also outperform existing tests on a newly derived reverse labeling reflection shape space, that is used to model filament data of microtubules within cells in a biological application.

Paper Structure

This paper contains 15 sections, 16 theorems, 92 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Manifold Stability and Optimal Lifts
  4. Smoothness of Optimal Lifts
  5. The Strong Law for Optimal Lifts
  6. Almost Everywhere Continuity and Uniqueness of Optimal Lifts
  7. Two-Sample Tests Using Optimal Lifts
  8. Reverse Labeling Reflection Shape Spaces
  9. Classical shape spaces and reflection shape spaces
  10. Planar shape and planar reflection shape
  11. The new reverse labeling reflection shape spaces
  12. For planar reverse labeling reflection shape
  13. Simulations and Application
  14. Simulations on $\Sigma_2^5$.
  15. Application to the interaction of microtubules and intermediate filaments in biological cells.

Key Result

Theorem 3.1

Let $\nu \in \mathop{\mathrm{argmin}}\limits_{q\in Q} F^{\pi \circ X}(q)$ and $\mathbb{P}\{\pi \circ X \in Q^*\} > 0$ then

Figures (4)

  • Figure 1: Depicting the counterexample from Remark \ref{['rm:foci-closed-set']}: All points on the circular segment $S$ are closest to any point in the lower right quadrant which is $C_S$ in the incomplete $M$ which is the plane minus a slit along the positive first axis.
  • Figure 2: Power curves for the data and five tests (labeling explained in Table \ref{['tab:perfomance_shape_space']}. The values at $0.06$ correspond to the ones recorded under power in Table \ref{['tab:perfomance_RR_shape_space']}. Top row for equal sample sizes (rows 4 -- 7 in Table \ref{['tab:perfomance_RR_shape_space']}) and bottom row for different sample sizes (rows 8 -- 10 in Table \ref{['tab:perfomance_RR_shape_space']}). In contrast to Table \ref{['tab:perfomance_RR_shape_space']} each test has been repeated $500$ times. The horizontal axis records the reverse relabeling reflection shape distance between the corresponding two population Fréchet means and the vertical axis depicts the percentage of rejections.
  • Figure 3: Determining $5$ landmarks on a typical microtubules filament at mathematically characteristic locations for buckle structure. Landmarks 1 and 5 maximize the ratio between Euclidean distance and curve length between them times the fourth root of the maximal normal distance along the buckle, which gives the 3rd landmark. Landmark 2 is furthest from this normal line (on the side of Landmark 1). Landmark 4 maximizes normal distance from the line connecting Landmarks 5 and 3. Afterwards, all landmarks (2,3,4) have been shifted minimally to make them more equidistant .
  • Figure 4: Typical microtubules buckle structures discretized by 5 landmarks. Upper row: without intermediate (vimentin) filaments. Lower row: in the presence of intermediate (vimentin) filaments (generating stiffness).

Theorems & Definitions (40)

  • Theorem 3.1: Manifold Stability
  • proof
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4: Existence of optimal lifts
  • proof
  • Definition 3.5
  • Theorem 3.6: LeBarden2014, Corollaries 2 and 3
  • Lemma 3.7
  • proof
  • ...and 30 more