From 53477ad823eba36be59e96156fee0cd65ddce94e Mon Sep 17 00:00:00 2001 From: Bang-Shien Date: Wed, 25 Sep 2024 12:53:06 +0800 Subject: [PATCH 1/2] :tada: Add supplement materials --- appendix/README.md | 7 ++ appendix/appx_A/.gitignore | 2 + appendix/appx_A/README.md | 92 +++++++++++++++++++++ appendix/appx_A/fracgm.py | 66 +++++++++++++++ appendix/appx_A/main.py | 9 ++ appendix/appx_B/README.md | 30 +++++++ appendix/appx_B/docs/registration_loss.png | Bin 0 -> 66620 bytes appendix/appx_B/docs/rotation_loss.png | Bin 0 -> 68844 bytes appendix/appx_C/README.md | 11 +++ appendix/appx_C/docs/perturbation.png | Bin 0 -> 73571 bytes 10 files changed, 217 insertions(+) create mode 100644 appendix/README.md create mode 100644 appendix/appx_A/.gitignore create mode 100644 appendix/appx_A/README.md create mode 100644 appendix/appx_A/fracgm.py create mode 100644 appendix/appx_A/main.py create mode 100644 appendix/appx_B/README.md create mode 100644 appendix/appx_B/docs/registration_loss.png create mode 100644 appendix/appx_B/docs/rotation_loss.png create mode 100644 appendix/appx_C/README.md create mode 100644 appendix/appx_C/docs/perturbation.png diff --git a/appendix/README.md b/appendix/README.md new file mode 100644 index 0000000..154212c --- /dev/null +++ b/appendix/README.md @@ -0,0 +1,7 @@ +# Supplement materials + +[:page_facing_up: Appendix A.](appx_A) A simple example of FracGM with global optimal guarantees + +[:page_facing_up: Appendix B.](appx_B) Tightness of linear relaxation + +[:page_facing_up: Appendix C.](appx_C) Sensitivity of initial guess \ No newline at end of file diff --git a/appendix/appx_A/.gitignore b/appendix/appx_A/.gitignore new file mode 100644 index 0000000..fc79902 --- /dev/null +++ b/appendix/appx_A/.gitignore @@ -0,0 +1,2 @@ +# Byte-compiled / optimized / DLL files +__pycache__/ \ No newline at end of file diff --git a/appendix/appx_A/README.md b/appendix/appx_A/README.md new file mode 100644 index 0000000..dec48e7 --- /dev/null +++ b/appendix/appx_A/README.md @@ -0,0 +1,92 @@ +### 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: + +$$ + \min_x\ \frac{f(x)}{h(x)}=\frac{x^2}{x^2+1}, +$$ + +where the residual $r(x)=x$ and $c=1$ in sense of Geman-McClure. The dual problem $(6)$ is as follows: + +$$ + \min_x\ \mu\big(f(x)-\beta h(x)\big)=\mu\big(x^2-\beta(x^2+1)\big)=\mu x^2-\mu\beta x^2-\mu\beta, +$$ + +where $\beta\in\mathbb{R}$ and $\mu\in\mathbb{R}$. Let $\boldsymbol{\alpha}=(\beta,\mu)^\top\in\mathbb{R}^2$ and $x_{\boldsymbol{\alpha}}$ be the optimal solution of Problem $(6)$ given $\boldsymbol{\alpha}$, we obtain $x_{\boldsymbol{\alpha}}=0$ by setting the derivative to zero as follows: + +$$ + \frac{\partial}{\partial x}(\mu x^2-\mu\beta x^2-\mu\beta)=2\mu(1-\beta)x=0\implies x^*=x_{\boldsymbol{\alpha}}=0. +$$ + +Thereby we can write + +$$ + \psi(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})= + \begin{pmatrix} + -f(x_{\boldsymbol{\alpha}})+\beta h(x_{\boldsymbol{\alpha}})\\ + -1+\mu h(x_{\boldsymbol{\alpha}}) + \end{pmatrix}= + \begin{pmatrix} + -x_{\boldsymbol{\alpha}}^2+\beta x_{\boldsymbol{\alpha}}^2+\beta\\ + -1+\mu x_{\boldsymbol{\alpha}}^2+\mu + \end{pmatrix}= + \begin{pmatrix} + \beta\\ + \mu-1 + \end{pmatrix}. +$$ + +It is trivial that both component functions $\psi_1(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})=\beta$ and $\psi_2(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})=\mu-1$ are differentiable, thus $\psi(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})$ is differentiable. Given $\boldsymbol{\alpha}_1=(\beta_1,\mu_1)^\top$ and $\boldsymbol{\alpha}_2=(\beta_2,\mu_2)^\top$, + +$$ + \\|\psi(\boldsymbol{\alpha}\_1,x_{\boldsymbol{\alpha}\_1})-\psi(\boldsymbol{\alpha}\_2,x_{\boldsymbol{\alpha}_2})\\|= + \begin{Vmatrix} + \beta_1-\beta_2\\\\\mu_1-1-(\mu_2-1) + \end{Vmatrix}\leq\begin{Vmatrix} + \beta_1-\beta_2\\\\\mu_1-\mu_2 + \end{Vmatrix}, +$$ + +then $\psi(\boldsymbol{\alpha},x_{\boldsymbol{\alpha}})$ is Lipschitz continuous. Solving such (simple) case by FracGM guarantees that the solution is global optimal. + + +### Usage +``` +cd appendix/appx_A +python ./main.py +``` + +We test various random initial guess between -100000 and 100000, all solutions converge to 0. + +``` +========================= +initial guess: -90760 +solution: 0.0 +========================= +initial guess: 90371 +solution: 0.0 +========================= +initial guess: 71362 +solution: 0.0 +========================= +initial guess: 25425 +solution: 0.0 +========================= +initial guess: -82805 +solution: 0.0 +========================= +initial guess: 42735 +solution: 0.0 +========================= +initial guess: -174 +solution: 0.0 +========================= +initial guess: -12185 +solution: 0.0 +========================= +initial guess: 33929 +solution: 0.0 +========================= +initial guess: -23746 +solution: 0.0 +``` \ No newline at end of file diff --git a/appendix/appx_A/fracgm.py b/appendix/appx_A/fracgm.py new file mode 100644 index 0000000..95ce0b6 --- /dev/null +++ b/appendix/appx_A/fracgm.py @@ -0,0 +1,66 @@ +import numpy as np +from scipy.optimize import minimize_scalar + +class FracGM: + """ + FracGM solver for the optimization problem: + min x^2 / (x^2 + 1), + where the optimal solution is x* = 0. + """ + def __init__(self, max_iteration=1000, tolerance=1e-7, verbose=True): + + self.max_iter = max_iteration + self.tol = tolerance + self.verbose = verbose + + @staticmethod + def f(x): + return x**2 + + @staticmethod + def h(x): + return x**2 + 1 + + def solve_beta_mu(self, x): + + beta = self.f(x)/self.h(x) + mu = 1/self.h(x) + + return beta, mu + + def solve_x(self, beta, mu): + + def dual_problem(x): + return mu*(self.f(x) - beta*self.h(x)) + + # Scipy minimizer for scalar function of one variable + res = minimize_scalar(dual_problem) + + return res.x + + def compute_psi_norm(self, beta, mu, x): + + psi_1 = -self.f(x) + beta*self.h(x) + psi_2 = -1 + mu*self.h(x) + + return np.linalg.norm([psi_1, psi_2]) + + def solve(self, initial): + + if self.verbose: + print("initial guess:", initial) + + x = initial + + for i in range(self.max_iter): + beta, mu = self.solve_beta_mu(x) + x = self.solve_x(beta, mu) + + norm = self.compute_psi_norm(beta, mu, x) + if norm < self.tol: + break + + if self.verbose: + print("solution:", x) + + return x \ No newline at end of file diff --git a/appendix/appx_A/main.py b/appendix/appx_A/main.py new file mode 100644 index 0000000..6573680 --- /dev/null +++ b/appendix/appx_A/main.py @@ -0,0 +1,9 @@ +import random +from fracgm import FracGM + +# random initial guess +initial_list = random.sample(range(-100000, 100000), 10) + +for initial_guess in initial_list: + print("="*25) + FracGM().solve(initial=initial_guess) \ No newline at end of file diff --git a/appendix/appx_B/README.md b/appendix/appx_B/README.md new file mode 100644 index 0000000..af693fa --- /dev/null +++ b/appendix/appx_B/README.md @@ -0,0 +1,30 @@ +### 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. +Taking our rotation estimation tasks on synthetic dataset for example, we quantify the tightness by an orthogonal loss $\mathcal{L}_o$ and a determinant loss $\mathcal{L}_d$ defined as follows: + +$$ +\begin{aligned} +\mathcal{L}_o&=\|(\mathbf{R}^\prime)^\top\mathbf{R}^\prime - \mathbf{I}_3\|_F\\ +\mathcal{L}_d&=|\text{det}(\mathbf{R}^\prime)-1| +\end{aligned} +$$ + +where $\|\cdot\|_F$ is the Frobenius norm, and we verify the orthogonality and determinant of the rotation part of solutions $\mathbf{R}^\prime$ before SVD projection: + +![](./docs/rotation_loss.png) + +Similarly, we verify the rotation part of solutions for registration tasks on synthetic dataset as follows: + +![](./docs/registration_loss.png) + +Above evaluations reveal that FracGM-based rotation and registration solvers are empirically tight mostly in our experiments. +We conjecture that the solution tends to approach the original solution space with sufficient number of measurements, in which a similar discussion can be found in [[34](#ref1), Section 2.1]. +Though, unfortunately we are unable to clarify the conditions to theoretically guarantee tightness and global optimality for both FracGM-based rotation and registration solvers. +Besides the issue of tightness, the global optimality of the relaxed problem is also subject to the conditions specified in Proposition 3, which is difficult to verify as it depends on the input data and requires discussions on various situations. +We appreciate the reviewer for raising this important issue, which we also consider to be one of the difficulties but crucial research problems in our future work. + + +[34] +C. Olsson and A. P. Eriksson, “Solving quadratically constrained geometrical problems using lagrangian duality,” in ICPR. 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changed, 19 insertions(+), 22 deletions(-) diff --git a/appendix/appx_A/fracgm.py b/appendix/appx_A/fracgm.py index 95ce0b6..cc3bae4 100644 --- a/appendix/appx_A/fracgm.py +++ b/appendix/appx_A/fracgm.py @@ -1,58 +1,55 @@ import numpy as np from scipy.optimize import minimize_scalar + class FracGM: - """ + """ FracGM solver for the optimization problem: min x^2 / (x^2 + 1), where the optimal solution is x* = 0. """ + def __init__(self, max_iteration=1000, tolerance=1e-7, verbose=True): - self.max_iter = max_iteration - self.tol = tolerance + self.tol = tolerance self.verbose = verbose @staticmethod def f(x): return x**2 - + @staticmethod def h(x): return x**2 + 1 def solve_beta_mu(self, x): - - beta = self.f(x)/self.h(x) - mu = 1/self.h(x) - + beta = self.f(x) / self.h(x) + mu = 1 / self.h(x) + return beta, mu def solve_x(self, beta, mu): - def dual_problem(x): - return mu*(self.f(x) - beta*self.h(x)) - + return mu * (self.f(x) - beta * self.h(x)) + # Scipy minimizer for scalar function of one variable res = minimize_scalar(dual_problem) - + return res.x def compute_psi_norm(self, beta, mu, x): - - psi_1 = -self.f(x) + beta*self.h(x) - psi_2 = -1 + mu*self.h(x) - + psi_1 = -self.f(x) + beta * self.h(x) + psi_2 = -1 + mu * self.h(x) + return np.linalg.norm([psi_1, psi_2]) def solve(self, initial): - if self.verbose: print("initial guess:", initial) x = initial - for i in range(self.max_iter): + for i in range(self.max_iter): beta, mu = self.solve_beta_mu(x) x = self.solve_x(beta, mu) @@ -62,5 +59,5 @@ def solve(self, initial): if self.verbose: print("solution:", x) - - return x \ No newline at end of file + + return x diff --git a/appendix/appx_A/main.py b/appendix/appx_A/main.py index 6573680..c670db5 100644 --- a/appendix/appx_A/main.py +++ b/appendix/appx_A/main.py @@ -5,5 +5,5 @@ initial_list = random.sample(range(-100000, 100000), 10) for initial_guess in initial_list: - print("="*25) - FracGM().solve(initial=initial_guess) \ No newline at end of file + print("=" * 25) + FracGM().solve(initial=initial_guess)