## 2 Pattern-Coupled Bayesian Model for Sound Field Reconstruction
### Sound Field Model
In matrix form, Equation (2) can be reformulated as:  
\[\mathbf{p}_{m}=\mathbf{D}_{m}^{\prime}\star\mathbf{W}, \tag{4}\]  
where the \(\star\) operator in Equation (4) denotes a two-dimensional cross-correlation between matrices(\(\mathbb{R}^{l_{1}\times l_{1}}\star\mathbb{R}^{l_{2}\times l_{2}}\rightarrow\mathbb{R}^{(l_{1} -l_{2}+1)\times(l_{1}-l_{2}+1)}\)), and \(\mathbf{D}_{m}^{\prime}=\left[\mathbf{0}^{L\times(N-1)},\mathbf{D}_{m}\right] \in\mathbb{R}^{L\times(2N-1)}\) is obtained by adding zero padding to the left side of \(\mathbf{D}_{m}\in\mathbb{R}^{L\times N}\), the dictionary of basis functions evaluated at location \(\mathbf{x}_{m}\), with each element \(\left(\mathbf{D}\right)_{m,l,n}=\phi_{m,l,n}\). The matrix \(\mathbf{W}\in\mathbb{R}^{L\times N}\) contains the unknown coefficients, with each element \(\left(\mathbf{W}\right)_{l,n}=w_{l,n}\). In three-dimensional tensor form, the full sound pressure field at all measurement locations, denoted as \(\mathbf{P}\), is expressed as:  
\[\mathbf{P}=\mathbf{D}^{\prime}\star\mathbf{W}= \begin{bmatrix}\mathbf{D}_{1}^ {\prime}\star\mathbf{W}\\ \vdots\\ \mathbf{D}_{M}^{\prime}\star\mathbf{W}\end{bmatrix} , \tag{5}\]where \(\mathbf{D}^{\prime}\in\mathbb{R}^{M\times L\times(2N-1)}\) is a tensor obtained by adding zero padding to the third dimension of the dictionary \(\mathbf{D}\in\mathbb{R}^{M\times L\times N}\). Here, \(\mathbf{D}\) represents a three-dimensional tensor dictionary constructed by stacking the matrices \(\mathbf{D}_{m}\). The first dimension of \(\mathbf{D}\) corresponds to different measurement channels. The \(\star\) operator in Equation (5) connects a three-dimensional tensor and a matrix, acting as a multichannel two-dimensional cross-correlation operator (\(\mathbb{R}^{K\times I_{1}\times I_{1}}\star\mathbb{R}^{I_{2}\times I_{2}} \rightarrow\mathbb{R}^{K\times(I_{1}-I_{2}+1)\times(I_{1}-I_{2}+1)}\)). Therefore, the shape of \(\mathbf{D}^{\prime}\star\mathbf{W}\) is \(M\times 1\times N\), with the second dimension unsqueezed to a size of 1, corresponding to the third term in Equation (5). Adopting a three-dimensional tensor dictionary is motivated by two primary reasons. First, compared to conventional convolution, implementing the multichannel two-dimensional cross-correlation in Equation (5) is more straightforward in programming environments, particularly in machine learning frameworks. Second, since the dictionary typically requires the most memory, using a three-dimensional tensor representation helps conserve memory. This representation requires storing \(ML(2N-1)\) floating-point numbers, of which \(MLN\) are nonzero. In contrast, the matrix form dictionary proposed in Ref. [34] requires storing \(MLN^{2}\) floating-point numbers, with \(MLN(N-1)/2\) being nonzero. Consequently, the three-dimensional tensor dictionary reduces memory requirements by approximately a factor of \(2/N\). To reconstruct the sound field at \(k\) locations \(\mathbf{x}_{\star k}\), the unknown coefficients \(\mathbf{W}\) are estimated by solving the inverse problem represented by:  
\[\widetilde{\mathbf{P}}=\mathbf{D}^{\prime}\star\mathbf{W}+\mathbf{E}. \tag{6}\]  
Once the estimated coefficients \(\hat{\mathbf{W}}\) are obtained, the reconstructed sound field is expressed as:  
\[\mathbf{P}_{\bullet}=\mathbf{D}^{\prime}_{\bullet}\star\hat{\mathbf{W}}, \tag{7}\]  
where \(\mathbf{P}_{\bullet}=\left[\mathbf{p}^{T}_{\bullet 1},\cdots,\mathbf{p}^{T}_{ \bullet K}\right]^{T}\in\mathbb{R}^{K\times N}\) represents the estimated sound field at the target locations. The tensor \(\mathbf{D}^{\prime}_{\bullet}\in\mathbb{R}^{K\times L\times(2N-1)}\) is obtained by zero padding the third dimension of the dictionary \(\mathbf{D}_{\bullet}\in\mathbb{R}^{K\times L\times N}\), which itself is constructed using elements \(\phi_{k,l,n}\). The calculation of \(\phi_{k,l,n}\) follows the same process as in Equation (3), with \(\mathbf{x}_{m}\) replaced by \(\mathbf{x}_{\star k}\) to adapt to the reconstruction locations.