## 1 Introduction  
Sound field reconstruction (SFR) is an essential area in audio signal processing that aims to determine the room impulse response (RIR) at positions where direct measurements are impractical or impossible. By inferring the RIR at unobserved points, SFR enables numerous applications, such as sound field synthesis [1, 2], sound field control [3, 4, 5, 6, 7], and sound source localization [8, 9, 10, 11] in a room environment. The core challenge of SFR arises because it is infeasible to measure RIRs at every possible location, given spatial constraints and limited microphone availability [12]. Consequently, effective algorithms are required to bridge the gap between limited measurement data and the desired dense reconstruction of the acoustic field. Despite significant progress, accurately reconstructing sound fields in complex spaces remains challenging. Real-world rooms often exhibit complex reverberation patterns, frequency-dependent boundary conditions, and numerous interfering reflections. On top of that, the inverse problem underlying SFR is commonly underdetermined [13], which means there are far more unknown coefficients than available observations; additionally, it is ill-conditioned [14], causing small perturbations in the measurements to result in significant errors in the reconstructed response. Therefore, developing robust and high-accuracy methods that effectively utilize prior information is crucial for practical sound field reconstruction. Many solutions have been proposed [15; 16; 17; 18; 19; 20; 21; 22; 23; 24], among which sparse Bayesian learning (SBL) is a widely used and highly effective approach. To deal with the underdetermined issue, SBL introduces the assumption of sparsity [25] in the coefficients used to represent the RIRs. In many acoustic contexts, certain physical processes tend to induce sparse patterns in the underlying representation of sound fields. Enforcing sparsity can help improve reconstruction accuracy. SBL has emerged as a powerful framework for achieving this objective. To deal with ill-conditioned issues, SBL [26] adopts a Bayesian perspective, incorporating hyperpriors that automatically adjust the regularization strength to better capture the underlying signal structure. Unlike purely deterministic methods that rely on II regularization (e.g., lasso) [19; 27; 28], this hierarchical approach enhances both the accuracy and robustness of the reconstructed RIRs. Although SBL has demonstrated effectiveness in many domains [26], further refinements can substantially improve its performance when structural dependencies exist among neighboring coefficients [25; 29; 30; 31]. In sound field reconstruction, these dependencies often manifest as clusters of active parameters corresponding to related physical phenomena, such as coherent reflections. For this reason, recent studies have explored approaches to incorporate structured priors into the SBL framework. One particularly noteworthy approach is pattern-coupled SBL, which has been shown in Refs. [25; 32; 33] and outperformed standard SBL in scenarios where coefficients tend to form block-sparse or spatially correlated patterns. Pattern-coupled SBL enhances performance by incorporating coupling terms in the hyperprior, thereby encouraging correlated coefficients to share information about their relevance. As a result, the algorithm can more effectively detect and preserve important clusters within the RIR, avoiding the risk of overfitting and error amplification. Nevertheless, the mathematical formulation of pattern-coupled hyperpriors admits a variety of possible coupling structures, each of which may exhibit distinct advantages or drawbacks for sound field reconstruction. Depending on the specific topology used to define these couplings, the outcome can vary. While Ref. [32] has already reported promising results using certain predefined coupling patterns, there is no guarantee that these patterns are universally optimal.