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Recursive Projection-Free Identification with Binary-Valued Observations

Tianning Han, Ying Wang, Yanlong Zhao

TL;DR

This work addresses parameter identification for FIR systems with binary-valued observations while minimizing computational cost. It introduces a recursive projection-free algorithm (RPFI) that uses a cut-off coefficient and an adaptive acceleration to achieve mean-square and almost-sure convergence with a rate of $O\left(\frac{1}{k}\right)$ under mild conditions. Building on CR-lower-bound ideas, it then formulates an Information-Matrix Projection-Free (IMPF) algorithm that attains asymptotic efficiency for first-order FIR systems, with simulations suggesting similar behavior for higher orders and substantially lower complexity than projection-based methods. The results offer practical, scalable strategies for set-valued measurements and point to future work on high-order efficiency proofs and extensions to more complex systems such as ARMAX.

Abstract

This paper is concerned with parameter identification problem for finite impulse response (FIR) systems with binary-valued observations under low computational complexity. Most of the existing algorithms under binary-valued observations rely on projection operators, which leads to a high computational complexity of much higher than O(n^2). In response, this paper introduces a recursive projection-free identification algorithm that incorporates a specialized cut-off coefficient to fully utilize prior information, thereby eliminating the need for projection operators. The algorithm is proved to be mean square and almost surely convergent. Furthermore, to better leverage prior information, an adaptive accelerated coefficient is introduced, resulting in a mean square convergence rate of O(1/k) , which matches the convergence rate with accurate observations. Inspired by the structure of the Cramer-Rao lower bound, the algorithm can be extended to an information-matrix projection-free algorithm by designing adaptive weight coefficients. This extension is proved to be asymptotically efficient for first-order FIR systems, with simulations indicating similar results for high order FIR systems. Finally, numerical examples are provided to demonstrate the main results.

Recursive Projection-Free Identification with Binary-Valued Observations

TL;DR

This work addresses parameter identification for FIR systems with binary-valued observations while minimizing computational cost. It introduces a recursive projection-free algorithm (RPFI) that uses a cut-off coefficient and an adaptive acceleration to achieve mean-square and almost-sure convergence with a rate of under mild conditions. Building on CR-lower-bound ideas, it then formulates an Information-Matrix Projection-Free (IMPF) algorithm that attains asymptotic efficiency for first-order FIR systems, with simulations suggesting similar behavior for higher orders and substantially lower complexity than projection-based methods. The results offer practical, scalable strategies for set-valued measurements and point to future work on high-order efficiency proofs and extensions to more complex systems such as ARMAX.

Abstract

This paper is concerned with parameter identification problem for finite impulse response (FIR) systems with binary-valued observations under low computational complexity. Most of the existing algorithms under binary-valued observations rely on projection operators, which leads to a high computational complexity of much higher than O(n^2). In response, this paper introduces a recursive projection-free identification algorithm that incorporates a specialized cut-off coefficient to fully utilize prior information, thereby eliminating the need for projection operators. The algorithm is proved to be mean square and almost surely convergent. Furthermore, to better leverage prior information, an adaptive accelerated coefficient is introduced, resulting in a mean square convergence rate of O(1/k) , which matches the convergence rate with accurate observations. Inspired by the structure of the Cramer-Rao lower bound, the algorithm can be extended to an information-matrix projection-free algorithm by designing adaptive weight coefficients. This extension is proved to be asymptotically efficient for first-order FIR systems, with simulations indicating similar results for high order FIR systems. Finally, numerical examples are provided to demonstrate the main results.

Paper Structure

This paper contains 9 sections, 8 theorems, 94 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. System Description
  4. Assumption
  5. The RPFI algorithm and its convergence properties
  6. Asymptotically efficient algorithm
  7. Simulation
  8. Concluding remarks
  9. Appendix

Key Result

Theorem 1

If Assumptions A1-A4 hold, the RPFI algorithm is both mean-square and almost surely convergent, i.e., where $\widetilde{\theta}_k=\hat{\theta}_k-\theta$ is the estimation error.

Figures (4)

  • Figure 1: The convergence of the RPFI algorithm
  • Figure 2: Convergence rate of the RPFI algorithm
  • Figure 3: $\Delta_k^{-1}(E\widetilde{\theta}_k^2-\Delta_k)$ of the IMPF algorithm, RPFI algorithm and the algorithm in Ke2022RecursiveIO
  • Figure 4: Comparison between the empirical variance of the IMPF algorithm and CR lower bound .

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • ...and 3 more