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Sequential Interval Passing for Compressed Sensing

Salman Habib, Remi Chou, Taejoon Kim

TL;DR

This work addresses reconstructing sparse signals from underdetermined measurements $\mathbf{y}=\mathbf{A}\mathbf{x}$ by enhancing the interval passing algorithm (IPA) with sequential CN scheduling, yielding SIPA. By updating CNs serially within each iteration, SIPA leverages fresher information to reduce total message updates and accelerate convergence while preserving reconstruction accuracy; theoretical results show no worse FER than flooding IPA under certain conditions and empirical results confirm substantial complexity reductions (up to ~36%) with faster convergence. The paper demonstrates, both analytically and via simulations on LDPC-based measurement matrices, that SIPA maintains performance and can outperform FIPA in efficiency, with potential upper bounds on FER for specific matrix families. Practical impact lies in enabling faster, lower-cost sparse reconstruction suitable for real-time or resource-constrained compressed sensing applications.

Abstract

The reconstruction of sparse signals from a limited set of measurements poses a significant challenge as it necessitates a solution to an underdetermined system of linear equations. Compressed sensing (CS) deals with sparse signal reconstruction using techniques such as linear programming (LP) and iterative message passing schemes. The interval passing algorithm (IPA) is an attractive CS approach due to its low complexity when compared to LP. In this paper, we propose a sequential IPA that is inspired by sequential belief propagation decoding of low-density-parity-check (LDPC) codes used for forward error correction in channel coding. In the sequential setting, each check node (CN) in the Tanner graph of an LDPC measurement matrix is scheduled one at a time in every iteration, as opposed to the standard ``flooding'' interval passing approach in which all CNs are scheduled at once per iteration. The sequential scheme offers a significantly lower message passing complexity compared to flooding IPA on average, and for some measurement matrix and signal sparsity, a complexity reduction of 36% is achieved. We show both analytically and numerically that the reconstruction accuracy of the IPA is not compromised by adopting our sequential scheduling approach.

Sequential Interval Passing for Compressed Sensing

TL;DR

This work addresses reconstructing sparse signals from underdetermined measurements by enhancing the interval passing algorithm (IPA) with sequential CN scheduling, yielding SIPA. By updating CNs serially within each iteration, SIPA leverages fresher information to reduce total message updates and accelerate convergence while preserving reconstruction accuracy; theoretical results show no worse FER than flooding IPA under certain conditions and empirical results confirm substantial complexity reductions (up to ~36%) with faster convergence. The paper demonstrates, both analytically and via simulations on LDPC-based measurement matrices, that SIPA maintains performance and can outperform FIPA in efficiency, with potential upper bounds on FER for specific matrix families. Practical impact lies in enabling faster, lower-cost sparse reconstruction suitable for real-time or resource-constrained compressed sensing applications.

Abstract

The reconstruction of sparse signals from a limited set of measurements poses a significant challenge as it necessitates a solution to an underdetermined system of linear equations. Compressed sensing (CS) deals with sparse signal reconstruction using techniques such as linear programming (LP) and iterative message passing schemes. The interval passing algorithm (IPA) is an attractive CS approach due to its low complexity when compared to LP. In this paper, we propose a sequential IPA that is inspired by sequential belief propagation decoding of low-density-parity-check (LDPC) codes used for forward error correction in channel coding. In the sequential setting, each check node (CN) in the Tanner graph of an LDPC measurement matrix is scheduled one at a time in every iteration, as opposed to the standard ``flooding'' interval passing approach in which all CNs are scheduled at once per iteration. The sequential scheme offers a significantly lower message passing complexity compared to flooding IPA on average, and for some measurement matrix and signal sparsity, a complexity reduction of 36% is achieved. We show both analytically and numerically that the reconstruction accuracy of the IPA is not compromised by adopting our sequential scheduling approach.

Paper Structure

This paper contains 7 sections, 8 theorems, 15 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Sequential Interval Passing Algorithm
  4. Complexity Reduction Via Sequential CN Scheduling
  5. FER Comparison Between FIPA and SIPA
  6. Simulation Results
  7. Conclusion

Key Result

Lemma 1

There exists a $x(v_i)\in \mathbb{R}_{\geq 0}$ and $\mathbf{A} \in \mathbb{R}_{\geq 0}^{m\times n}$ for which SIPA propagates $\mu_{v_i \rightarrow c_j}^{(1,t_j)}>0$ and $M_{c_j \rightarrow v_i}^{(1,t_j)}\leq y(c_j)/A_{c_j,v_i}$.

Figures (2)

  • Figure 1: Example of $\mathcal{G}_{\mathbf{A}}$ corresponding to a $(2,3)$ LDPC measurement matrix with $m=2$ and $n=3$. Circles represent VNs and squares represent CNs. The intervals passed by VNs $v_i$ and $c_j$ are also shown.
  • Figure 2: Reconstruction performance of $\mathbf{A}_1$ using different interval passing schemes.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 3
  • Theorem 2