## 4 Reduction of Computational Costs with Data-Driven Sparse Sampling  
The sound source localization using beamforming requires the exhaustive calculation of the propagation time and attenuation rate at each point of the sound source localization  
Figure 4: The distributions of propagation time and attenuation rate in the \(X_{n}-Y_{n}\) plane (\(Z_{n}=0\)). area, as described in Section 3. However, this requires high calculation costs if the number of the grid points in the source area is large, such as 200 \(\times\) 200 points, as in the present study. Because the Amiet method [5] involves iterative calculations, the calculation takes approximately eight minutes for 40,000 source points of one microphone. Hundreds of microphones are sometimes used in the experiments and, therefore, the exhaustive calculation of the propagation time and the attenuation rate seems to be difficult in the online quick check of experiments. Therefore, this study proposes a reduction in the calculation cost using data-driven sparse sampling. The data-driven sparse sampling [13] is a method for reconstructing the entire field of the high-dimensional data using a few dominant modes that are based on the modal decomposition, such as SVD. \[\hat{\mathbf{P}}=\tilde{\mathbf{U}}\mathbf{\hat{X}}, \tag{27}\]  
where \(\hat{\mathbf{P}}\in\mathbb{R}^{40,000}\), \(\tilde{\mathbf{U}}\in\mathbb{R}^{40,000\times r}\), and, and \(\mathbf{X}\in\mathbb{R}^{r}\) are the estimated propagation time or attenuation rate distributions of 40,000 grid point, the spatial-mode matrix, and the latent-state vector (the mode coefficient), respectively. Here, \(r\) is the number of mode and \(p\) is the number of sampling points. If the spatial mode shape in \(\tilde{\mathbf{U}}\) is appropriately selected, then the distributions of 40,000 grid points can be expressed by a small number of latent variable. If this approximation works well, the latent variable can be determined by the observations at suboptimized sparse sampling points:  
\[\mathbf{Y}=\mathbf{H}\mathbf{\tilde{U}}\mathbf{X}, \tag{28}\]  
where \(\mathbf{Y}\in\mathbb{X}^{p}\) and \(\mathbf{H}\in\mathbb{R}^{p\times 40,000}\) are the observation vector and sparse-sampling point matrix indicating the positions of \(p\) sampling points that were highly sensitive to the latent state variables. Here, \(\mathbf{H}\) is determined by the problem choosing \(p\) sampling points out of 40,000 grid points for the better estimation of the latent state variables that express the mode coefficients. The estimated mode coefficients \(\mathbf{\hat{X}}\) can be obtained by multiplying the pseudo inverse matrix of \(\mathbf{H}\mathbf{\tilde{U}}\), as follows. \[\mathbf{\hat{X}}=\mathbf{(H}\mathbf{\tilde{U}})^{+}\mathbf{Y} \tag{29}\]  
Finally, \(\hat{\mathbf{P}}\in\mathbb{R}^{40,000}\), corresponding to the overall propagation time or attenuation rate distributions of 40,000 grid points, can be reconstructed by substituting \(\mathbf{\hat{X}}\) into Equation (27). SVD constructs the spatial mode matrix \(\mathbf{U}\) in the present study. The training data matrix \(\mathbf{P}\) that consists of the propagation time or the attenuation rate of 40,000 grid points at randomly selected 100 conditions in the range of Table 1 is constructed. Subsequently, SVD is applied to the training data matrix \(\mathbf{P}\), of which the size is 40,000 \(\times\) 100, and Equation (30) is obtained,  
\[\mathbf{P}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{*}\approx\mathbf{\tilde{U}}\mathbf{\tilde{\Sigma}}\bm {\tilde{V}}^{*}, \tag{30}\]  
where the columns of \(\mathbf{U}\) and \(\mathbf{V}\) are orthogonal and called left and right singular vectors, respectively. The matrix \(\mathbf{\Sigma}\) contains decreasing nonnegative diagonal entries, called singular values. Here, \(\mathbf{\tilde{U}}\mathbf{\Sigma}\mathbf{\tilde{V}}^{*}\) is a low-dimensional description of the original data approximated with a low-rank matrix.