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A Slices Perspective for Incremental Nonparametric Inference in High Dimensional State Spaces

Moshe Shienman, Ohad Levy-Or, Michael Kaess, Vadim Indelman

TL;DR

This work introduces a slice-based nonparametric inference framework for high-dimensional state spaces, enabling incremental probabilistic reasoning without intermediate reconstructions or KDEs. By treating the joint distribution as a high-dimensional surface and directly extracting conditional slices, the method efficiently approximates both marginals and conditionals during forward and backward passes, even when unary factors are absent. An incremental extension with a maximum mean discrepancy (MMD) based early stopping heuristic reduces computation, enabling real-time operation. Empirical results on synthetic and real SLAM datasets show superior accuracy and runtime reductions—often by an order of magnitude—compared to existing online nonparametric methods, while matching offline state-of-the-art performance. The approach is general and applicable to tracking, sensor fusion, BA/SfM, and SLAM, offering a practical path toward real-time nonparametric inference in complex systems.

Abstract

We introduce an innovative method for incremental nonparametric probabilistic inference in high-dimensional state spaces. Our approach leverages \slices from high-dimensional surfaces to efficiently approximate posterior distributions of any shape. Unlike many existing graph-based methods, our \slices perspective eliminates the need for additional intermediate reconstructions, maintaining a more accurate representation of posterior distributions. Additionally, we propose a novel heuristic to balance between accuracy and efficiency, enabling real-time operation in nonparametric scenarios. In empirical evaluations on synthetic and real-world datasets, our \slices approach consistently outperforms other state-of-the-art methods. It demonstrates superior accuracy and achieves a significant reduction in computational complexity, often by an order of magnitude.

A Slices Perspective for Incremental Nonparametric Inference in High Dimensional State Spaces

TL;DR

This work introduces a slice-based nonparametric inference framework for high-dimensional state spaces, enabling incremental probabilistic reasoning without intermediate reconstructions or KDEs. By treating the joint distribution as a high-dimensional surface and directly extracting conditional slices, the method efficiently approximates both marginals and conditionals during forward and backward passes, even when unary factors are absent. An incremental extension with a maximum mean discrepancy (MMD) based early stopping heuristic reduces computation, enabling real-time operation. Empirical results on synthetic and real SLAM datasets show superior accuracy and runtime reductions—often by an order of magnitude—compared to existing online nonparametric methods, while matching offline state-of-the-art performance. The approach is general and applicable to tracking, sensor fusion, BA/SfM, and SLAM, offering a practical path toward real-time nonparametric inference in complex systems.

Abstract

We introduce an innovative method for incremental nonparametric probabilistic inference in high-dimensional state spaces. Our approach leverages \slices from high-dimensional surfaces to efficiently approximate posterior distributions of any shape. Unlike many existing graph-based methods, our \slices perspective eliminates the need for additional intermediate reconstructions, maintaining a more accurate representation of posterior distributions. Additionally, we propose a novel heuristic to balance between accuracy and efficiency, enabling real-time operation in nonparametric scenarios. In empirical evaluations on synthetic and real-world datasets, our \slices approach consistently outperforms other state-of-the-art methods. It demonstrates superior accuracy and achieves a significant reduction in computational complexity, often by an order of magnitude.
Paper Structure (13 sections, 1 theorem, 23 equations, 7 figures)

This paper contains 13 sections, 1 theorem, 23 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. BACKGROUND AND NOTATIONS
  4. METHODOLOGY
  5. Inference Using Slices
  6. Forward Pass
  7. Backward Pass
  8. Incremental Inference with Early Stopping Heuristic
  9. Experiments
  10. Synthetic Dataset - Multi Modal Four Doors
  11. Real World Dataset - Plaza
  12. CONCLUSIONS
  13. APPENDIX: Proof of Lemma \ref{['lemma: non unary factors']}

Key Result

Lemma 1

Given a factor graph $G = \left( \mathcal{F}, \Theta, \mathcal{E} \right)$ and an elimination order $\mathcal{O}$, if each eliminated variable $\theta_j \in \Theta$ either has a unary factor connected to it, i.e. $\exists f(\theta_j) \in \mathcal{F}_{j-1} (\theta_j)$ or, $\exists \theta_i \in \Theta

Figures (7)

  • Figure 1: (a) The factor graph of a small probabilistic inference problem containing two variable nodes $X,Y$ and two factor nodes $f(X), f(X,Y)$ indicated as small solid black circles; (b) A high-dimensional surface representing the joint distribution $\mathbb{P}({X,Y})$. Each red$\textit{slice}$, at a specific realization $Y=y$, represents a conditional distribution $\mathbb{P}({X | Y=y})$. The marginal distribution $\mathbb{P}({X})$, shown in blue, is calculated by integrating over all conditional $\textit{slices}$.
  • Figure 2: (a) A factor graph formulation of a SLAM problem with five variable nodes and seven factor nodes. Factor nodes represent probabilistic information over random variables. In this example, factor nodes include: a prior $f(x_1)$, odometry measurements $f(x_1,x_2), f(x_2,x_3)$ and landmark measurements $f(x_1,l_1), f(x_2,l_1), f(x_2,l_2), f(x_3,l_2)$. The factor graph represents a factorization of the joint distribution as a product of all factors; (b) The corresponding Bayes net after performing variable elimination on the factor graph using the elimination order $\mathcal{O} = \{\theta_1=x_1, \theta_2=l_1, \theta_3=x_2, \theta_4=l_2, \theta_5=x_3\}$. The joint distribution is expressed as the product of conditionals produced in each step as $\mathbb{P}({x_1,x_2,x_3,l_1,l_2}) = \mathbb{P}({x_1|x_2,l_1}) \cdot \mathbb{P}({l_1|x_2}) \cdot \mathbb{P}({x_2|x_3,l_2}) \cdot \mathbb{P}({l_2|x_3}) \cdot \mathbb{P}({x_3})$.
  • Figure 3: Generating samples during variable elimination; (a) A subset of three variables $x_1,x_2,l_1$ from a larger factor graph. The given elimination order is $\mathcal{O} = \{\theta_1=x_1, \theta_2=l_1, ...\}$. When eliminating $x_1$, samples of $x_1$ are directly obtained from $f(x_1)$ to approximate the new factor $\hat{f}_{new} (l_1, x_2|D_1)$ with $\textit{slices}$ via (13); (b) When eliminating $l_1$, samples of $l_1$ are obtained according to Lemma \ref{['lemma: non unary factors']} from $\hat{f}_{new} (l_1, x_2|D_1)$. Even without a unary factor on $l_1$, our $\textit{slices}$ approach provides a method to directly generate samples of $l_1$.
  • Figure 4: Four doors synthetic SLAM dataset Fourie16iros marginal posterior distributions for the first and last robot poses. A black solid line indicates the analytic solution when considering only the mode which corresponds to the ground truth. Our $\textit{slices}$ approach directly approximates these marginal via (17). In mm-ISAM, KDEs are used to approximate these marginal distributions, whereas in NF-iSAM and NSFG, they are derived solely from samples and approximated using KDEs based on these samples, as demonstrated in Huang22ral. In each method, 200 samples were utilized to approximate the distributions.
  • Figure 5: Four doors synthetic SLAM dataset Fourie16iros results. We report RMSE (m) and run time (sec) as functions of the number of samples used by all methods.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 1