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Dual NUP Representations and Min-Maximization in Factor Graphs

Yun-Peng Li, Hans-Andrea Loeliger

TL;DR

This paper proposes a new iterative forward-backward algorithm that is dual to a recently proposed backward-forward algorithm and augmenting factor graphs with convex-dual variables and pertinent NUP representations.

Abstract

Normals with unknown parameters (NUP) can be used to convert nontrivial model-based estimation problems into iterations of linear least-squares or Gaussian estimation problems. In this paper, we extend this approach by augmenting factor graphs with convex-dual variables and pertinent NUP representations. In particular, in a state space setting, we propose a new iterative forward-backward algorithm that is dual to a recently proposed backward-forward algorithm.

Dual NUP Representations and Min-Maximization in Factor Graphs

TL;DR

This paper proposes a new iterative forward-backward algorithm that is dual to a recently proposed backward-forward algorithm and augmenting factor graphs with convex-dual variables and pertinent NUP representations.

Abstract

Normals with unknown parameters (NUP) can be used to convert nontrivial model-based estimation problems into iterations of linear least-squares or Gaussian estimation problems. In this paper, we extend this approach by augmenting factor graphs with convex-dual variables and pertinent NUP representations. In particular, in a state space setting, we propose a new iterative forward-backward algorithm that is dual to a recently proposed backward-forward algorithm.
Paper Structure (22 sections, 31 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 22 sections, 31 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background on NUP Representations
  3. Basic Idea and Iteratively Reweighted Linear-Gaussian Estimation (IRLGE)
  4. Example and Issues
  5. Iterated Backward Filtering Forward Deciding (IBFFD)
  6. Harnessing Convex Duality and Min-maximization
  7. Background on the Legendre Transform
  8. System Model with Convex-Dual Variables
  9. Working with Dual NUP Representations
  10. Iterated Forward Filtering Backward Dual Deciding
  11. Example of Dual NUP: Half-Space Constraint
  12. Min-Maximization by Gaussian Message Passing
  13. Gaussian Message Passing with Primal and Dual Variables
  14. IFFBDD by Gaussian Message Passing
  15. Conclusion
  16. ...and 7 more sections

Figures (7)

  • Figure 1: Factor graph of (\ref{['eqn:GenSystemModel']}) with (\ref{['eqn:NUPforSys']}). The arrows serve as a (arbitrary) reference for the message passing notation LDHKLK:fgsp2007.
  • Figure 2: Factor graph of (\ref{['eqn:fSSM']}).
  • Figure 3: Fig. \ref{['fig:SysNUP']} augmented with convex-dual variables $\tilde{Z}_1,\ldots,\tilde{Z}_n$.
  • Figure 4: Fig. \ref{['fig:SysDuals']} with dashed boxes replaced by dual NUP representations (\ref{['eqn:dualMinNUP']}).
  • Figure 5: Factor graph of the state space model of Section \ref{['sec:IFFBDD:GaussianAlg']}.
  • ...and 2 more figures