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EB-RANSAC: Random Sample Consensus based on Energy-Based Model

Muneki Yasuda, Nao Watanabe, Kaiji Sekimoto

Abstract

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.

EB-RANSAC: Random Sample Consensus based on Energy-Based Model

Abstract

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.
Paper Structure (16 sections, 3 theorems, 37 equations, 6 figures)

This paper contains 16 sections, 3 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Robust Estimation based on RANSAC
  3. Proposed Method: EB-RANSAC
  4. Energy-based model
  5. Conditional distributions
  6. EB-RANSAC estimator
  7. Theoretical Analysis of EB-RANSAC
  8. Numerical Experiments
  9. Application to linear regression problem
  10. Application to maximum likelihood estimation
  11. Gaussian distribution
  12. Exponential distribution
  13. Conclusion and Future Study
  14. Proofs of Theorems
  15. Proof of Theorem \ref{['thm:minimum_solution']}
  16. ...and 1 more sections

Key Result

Theorem 1

Given $q(\bm{x})$, the distribution $p_{\theta}(\bm{x})$ that minimizes equation (eqn:EBM-RANSAC_MLE) is as follows: where $\mathop{\mathrm{relu}}\nolimits(x):= \max(x, 0)$ is the rectified linear unit and $T_{ \mathrm{cut} }(\beta)$ is the solution to

Figures (6)

  • Figure 1: Illustration of the EB-RANSAC estimator in equation (\ref{['eqn:solution_EBM-RANSAC_MLE']}). Top panel: the empirical distribution of $D$ that has outliers' probability in its tail. Bottom panel: the EB-RANSAC estimator obtained by removing low probability region (lower than the cut-off threshold) from the empirical distribution.
  • Figure 2: (a) Estimators obtained from the EB-RANSAC with $\beta = 5$ and standard LMS methods. (b) MSE plot for various $\beta$s. The blue broken line represents the MSE of the LMS estimator.
  • Figure 3: (a) Gaussians obtained from the EB-RANSAC with $\beta = 5$ and standard MLE methods. The bars represent the histogram of the dataset that is normalized such that the maximum value is equal to two for easier comparison with the Gaussians. (b) KLD between the EB-RANSAC estimator and the inlier Gaussian. The blue broken line represents the KLD of the ML estimator.
  • Figure 4: EB-RANSAC estimators for $\beta = -2,-1,0,1$. The bars represent the data histogram that is the same as that shown in figure \ref{['fig:MLE_Gauss']}(a).
  • Figure 5: (a) Normalized histogram of the dataset. (b) Estimators obtained from the EB-RANSAC (with $\beta = 4$) and the standard MLE methods. The red broken line represents the inlier distribution $\mathcal{E} (x \mid 2)$. (c) Values of $\lambda$ obtained from the EB-RANSAC method for $\beta \in [4,8]$. For comparison, the value of $\lambda$ obtained from the standard MLE method ($\lambda \approx 0.653$) also plotted (the blue broken line). A jump of $\lambda$ is observed at $\beta_c \approx 6.65$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof 1