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Multi-Source Approximate Message Passing with Random Semi-Unitary Dictionaries

Burak Çakmak, Giuseppe Caire

TL;DR

This work develops a multi-source AMP framework for matrix-valued observations where dictionaries are drawn from a random semi-unitary ensemble ${\bf S}_u=\sqrt{\alpha_u}\,{\bf P}_u{\bf O}_u$ and studies finite-sample high-dimensional behavior via state evolution. The proposed AMP algorithm uses divergence-free nonlinearities built from Bayesian posterior-mean denoisers, leading to decoupled Gaussian dynamics with recursion relations for covariances ${\bf C}_{\psi_u}^{(t,s)}$ and ${\bf C}_{\phi_u}^{(t,s)}$, and a fixed point compatible with replica-symmetric predictions in the RS setting. The paper also conjectures universality for dictionaries formed from randomly signed Fourier matrices, enabling efficient dictionary construction with ${\cal O}(L)$ cost and ${\cal O}(L\log L)$ per-iteration complexity. As a proof of concept, the framework is applied to unsourced random access in cell-free wireless networks, demonstrating reliable message detection and channel estimation and showing performance gains over i.i.d. dictionaries under modest system loads, with simulations corroborating the asymptotic predictions.

Abstract

Motivated by the recent interest in approximate message passing (AMP) for matrix-valued linear observations with superposition of \emph{multiple statistically asymmetric signal sources}, we introduce a multi-source AMP framework in which the dictionary matrices associated with each signal source are drawn from a \emph{random semi-unitary ensemble} (rather than the standard Gaussian matrix ensemble.) While a similar model has been explored by Vehkaper{ä}, Kabashima, and Chatterjee (2016) using the replica method, here we present an AMP algorithm and provide a high-dimensional yet \emph{finite-sample} analysis. As a proof of concept, we show the effectiveness of the proposed approach on the problem of \emph{message detection and channel estimation} in an unsourced random access scenario in wireless communication.

Multi-Source Approximate Message Passing with Random Semi-Unitary Dictionaries

TL;DR

This work develops a multi-source AMP framework for matrix-valued observations where dictionaries are drawn from a random semi-unitary ensemble and studies finite-sample high-dimensional behavior via state evolution. The proposed AMP algorithm uses divergence-free nonlinearities built from Bayesian posterior-mean denoisers, leading to decoupled Gaussian dynamics with recursion relations for covariances and , and a fixed point compatible with replica-symmetric predictions in the RS setting. The paper also conjectures universality for dictionaries formed from randomly signed Fourier matrices, enabling efficient dictionary construction with cost and per-iteration complexity. As a proof of concept, the framework is applied to unsourced random access in cell-free wireless networks, demonstrating reliable message detection and channel estimation and showing performance gains over i.i.d. dictionaries under modest system loads, with simulations corroborating the asymptotic predictions.

Abstract

Motivated by the recent interest in approximate message passing (AMP) for matrix-valued linear observations with superposition of \emph{multiple statistically asymmetric signal sources}, we introduce a multi-source AMP framework in which the dictionary matrices associated with each signal source are drawn from a \emph{random semi-unitary ensemble} (rather than the standard Gaussian matrix ensemble.) While a similar model has been explored by Vehkaper{ä}, Kabashima, and Chatterjee (2016) using the replica method, here we present an AMP algorithm and provide a high-dimensional yet \emph{finite-sample} analysis. As a proof of concept, we show the effectiveness of the proposed approach on the problem of \emph{message detection and channel estimation} in an unsourced random access scenario in wireless communication.

Paper Structure

This paper contains 21 sections, 10 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Proposed AMP Algorithm and Its High-Dimensional Analysis
  3. The High Dimensional Representation
  4. On Constructing the non-linear mappings Lg
  5. Compatibility with the RS Ansatz
  6. Conjecture on Universality of Dictionaries with Randomly Signed Fourier Matrices
  7. Application to Message Detection and Channel Estimation For Unsources Random Access in Wireless Networks
  8. Message Detection and its High-Dimensional Analysis
  9. Channel Estimation
  10. Genie-Aided MMSE Channel Estimation
  11. Simulation Results
  12. Conclusion and Outlook
  13. The Proof of Theorem \ref{['the1']}
  14. The Random-Matrix-Free Equivalent
  15. The High-Dimensional Representation
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose Assumption as1 hold. For all $(u,t)\in[U]\times [T]$, let the functions $f_{u,t}$ be differentiable and Lipschitz-continuous and satisfying the divergence-free property onsager. Let ${\hbox{\boldmath$\Theta$}}_u\doteq {\bf S}_u{\bf X}_u$. Then, we have for any $t\in[T]$ where ${\hbox{\boldmath$\Psi$}}_u^{(t)}\sim_{\text{i.i.d.}}{\hbox{\boldmath$\psi$}}_u^{(t)}$ and ${\hbox{\boldmath$\Phi$}

Figures (3)

  • Figure 1: A toy (Wyner) model: $B = 2$ and $U = 2$cakmak2024joint. $g_{ub}$ denotes the large-scale fading coefficient between a user in the $u$th location and the $M$-antenna array of RU $b$.
  • Figure 2: $2$-Locations: Comparison of asymptotic missed detection and false alarm rates with $\alpha_u=1$ and $\sigma^2=0.1$.
  • Figure 4: Randomly-signed Fourier dictionaries: 2-Locations, $L=2^{13}$, $\alpha_u = \frac{3}{2}$, and $\sigma^2 = 0.1$. Simulations for each pair $(\lambda_1, \lambda_2)$ are based on a single, independent instance of the AMP dynamics.

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 3 more