📝 Update appendix

README.md

@@ -22,7 +22,7 @@ of the proposed solver can be guaranteed under given conditions.
 
 For more, please refer to our paper:
 
-- [Bang-Shien Chen](https://github.com/doggydoggy0101), [Yu-Kai Lin](https://github.com/StephLin), [Jian-Yu Chen](https://github.com/Jian-yu-chen), [Chih-Wei Huang](https://sites.google.com/ce.ncu.edu.tw/cwhuang/), [Jann-Long Chern](https://math.ntnu.edu.tw/~chern/), [Ching-Cherng Sun](https://www.dop.ncu.edu.tw/en/Faculty/faculty_more/9), **FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust Estimator**. _submitted to IEEE Robotics and Automation Letters (RA-L)_, 2024. (paper) ([preprint](https://arxiv.org/abs/2409.13978)) ([code](https://github.com/StephLin/FracGM))
+- [Bang-Shien Chen](https://dgbshien.com/), [Yu-Kai Lin](https://github.com/StephLin), [Jian-Yu Chen](https://github.com/Jian-yu-chen), [Chih-Wei Huang](https://sites.google.com/ce.ncu.edu.tw/cwhuang/), [Jann-Long Chern](https://math.ntnu.edu.tw/~chern/), [Ching-Cherng Sun](https://www.dop.ncu.edu.tw/en/Faculty/faculty_more/9), **FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust Estimator**. _submitted to IEEE Robotics and Automation Letters (RA-L)_, 2024. (paper) ([preprint](https://arxiv.org/abs/2409.13978)) ([code](https://github.com/StephLin/FracGM))
 
 **Table of Contents**
 

appendix/README.md

@@ -1,4 +1,4 @@
-# Supplement materials
+# :coffee: Supplementary Materials
 
 [:page_facing_up: Appendix A.](appx_A) A simple example of FracGM with global optimal guarantees
 

appendix/appx_A/README.md

@@ -1,6 +1,7 @@
-### A simple example of FracGM with global optimal guarantees
+# :page_facing_up: A simple example of FracGM with global optimal guarantees
 
-Here we give a simple example that one can verify the differentiability and Lipschitz continuity of $\psi(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})$. Suppose we have an optimization problem:
+We give a simple example that satisfies Proposition 3 as follows. 
+Suppose we have an optimization problem:
 
 $$
     \min_x\ \frac{f(x)}{h(x)}=\frac{x^2}{x^2+1},
@@ -50,7 +51,21 @@ $$
 then $\psi(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})$ is Lipschitz continuous. Solving such (simple) case by FracGM guarantees that the solution is global optimal.
 
 
-### Usage
+To verify the above statement empirically, we feed various initial guesses to FracGM to examine the global optimality as follows:
+
+| Initial Guess | FracGM's 1$^\text{st}$ Iteration | FracGM's 2$^\text{nd}$ Iteration |
+|---------------|----------------------------------|----------------------------------|
+| $-10^{5}$     | $-2.20\times 10^{-4}$            | $0.00\times 10^{-13}$            |
+| $-10^{3}$     | $-1.92\times 10^{-10}$           | $0.00\times 10^{-13}$            |
+| $-10^{1}$     | $-5.10\times 10^{-8}$            | $0.00\times 10^{-13}$            |
+| $-10^{0}$     | $-2.73\times 10^{-9}$            | $0.00\times 10^{-13}$            |
+| $10^{0}$      | $-2.73\times 10^{-9}$            | $0.00\times 10^{-13}$            |
+| $10^{1}$      | $-5.10\times 10^{-8}$            | $0.00\times 10^{-13}$            |
+| $10^{3}$      | $-1.92\times 10^{-10}$           | $0.00\times 10^{-13}$            |
+| $10^{5}$      | $-2.20\times 10^{-4}$            | $0.00\times 10^{-13}$            |
+
+
+## :running: Run
 ```
 cd appendix/appx_A
 python ./main.py

appendix/appx_B/README.md

@@ -1,4 +1,4 @@
-### Tightness of linear relaxation
+# :page_facing_up: Tightness of linear relaxation
 
 For FracGM-based rotation and registration solvers, the relaxation is tight if solutions of the relaxed program in $\mathbb{R}^{3\times 3}$ and $\mathbb{R}^{4\times 4}$ are also within $\text{SO}(3)$ and $\text{SE}(3)$ respectively.
 In practice, we can evaluate the tightness of the relaxation by checking the orthogonality and determinant of a solution given by the relaxed program.

appendix/appx_C/README.md

@@ -1,11 +1,11 @@
-### Sensitivity of initial guess
+# :page_facing_up: Sensitivity of initial guess
 
 If a FracGM method satisfies Proposition 3, then it is a global solver, and it should not be sensitive to initial guesses.
 We do not think the FracGM-based rotation and registration solvers are insensitive to initial guesses for any kinds of input data, due to the fact that we are unable to verify Proposition 3 at the moment. Nevertheless, empirical studies on experiments show that our FracGM-based solvers are mostly insensitive to initial guesses.
 
 ![](./docs/perturbation.png)
 
-We compare our FracGM-based rotation solver with the other Geman-McClure solver, GNC-GM [[17]](#ref1), on the synthetic dataset. We construct various initial guesses by adding some degree of perturbation to the ground truth rotation matrix. With an outlier rate of 50\%, both FracGM and GNC-GM seems insensitive to initial guesses. However, in extreme cases where the outlier rate is 90\%, GNC-GM produces different solutions due to different initial guesses, while FracGM is still insensitive to initial guesses.
+We compare our FracGM-based rotation solver with the other Geman-McClure solver, GNC-GM [[17]](#ref1), on the synthetic dataset. We construct various initial guesses by adding some degree of perturbation to the ground truth rotation matrix. With an outlier rate of 50\%, both FracGM and GNC-GM seems insensitive to initial guesses. However, in extreme cases where the outlier rate is 90\%, GNC-GM produces different solutions due to different initial guesses, while FracGM remains insensitive to initial guesses.
 
 <a id="ref1">[17]</a> 
 H. Yang, P. Antonante, V. Tzoumas, and L. Carlone, “Graduated nonconvexity for robust spatial perception: From non-minimal solvers to global outlier rejection,” IEEE Robotics Autom. Lett., vol. 5, no. 2, pp. 1127–1134, 2020.