## 2 Methodology
### Independent Component Analysis of HRTFs
The directions and the frequency points (25 directions \(\times\) 232 frequency points, \(m\) = 5800) were combined, and the HRTF data from the left and right ears (30 subjects \(\times\) 2, \(n\) = 60) were combined, so the final variable is a two-dimensional matrix \(\mathbf{H}_{60,5800}\). Our previous study has demonstrated the effectiveness of ICA decomposition and reconstruction of the vertical KEMAR HRTF [30]. The results indicated that the combination of the preceding six independent components can effectively characterize the main spectral characteristics of the median plane without introducing perceptible auditory defects. Considering that more individual-dependent variations are contained in this work, the preceding 10 independent components \(d\) = 10 were selected in the HRTF ICA decomposition for 30 subjects. Thus, the ICA decomposition of the median-plane HRTFs is obtained as  
\[\hat{\mathbf{H}}_{60,5800}=\mathbf{A}_{60,10}\mathbf{IC}_{10,5800}. \tag{9}\]  
In Equation (9), \(\mathbf{A}_{60,10}\) is the individual-dependent weight coefficients, which are used in the subsequent linear regression analysis; \(\mathbf{IC}_{10,5800}\) is the direction and frequency-dependent base vector, which is used for the subsequent reconstruction of the individualized HRTF. During implementation, PCA is implemented before applying FastICA to reduce the HRTFs to 10 dimensions. The decorrelation approach is a symmetric method that estimates all the independent components in parallel. The nonlinearity used in the fixed-point algorithm is \(g(u)=tanh(u)\). We plot one example of the input HRTF and the output independent components in Figure 2. It can be observed that the original data and the independent components share the same spectral characteristics, such as pinna-related spectral notches and torso-related spectral cues. Spectral distortion (SD) is used to evaluate the deviation between the original and reconstructed HRTFs. For a specific subject, SD is defined as [31]:  
\[SD(\phi)=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left(20\times lg\frac{\left|H\left(f_{k }\phi\right)\right|}{\left|H\left(f_{k}\phi\right)\right|}\right)^{2}}\left( \mathrm{dB}\right) \tag{10}\]  
where \(N=232\) represents the number of calculated frequency points. For different spatial directions, the average SD is defined as:  
\[\overline{SD}=\frac{\sum_{j=1}^{M}SD\left(\phi_{j}\right)}{M}(\mathrm{dB}) \tag{11}\]  
where \(M=25\) is the number of involved spatial directions. The closer the values of SD and the \(\overline{\mathrm{SD}}\) value are to zero, the lower the deviation between the original and reconstructed HRTFs. Figure 3 shows the \(\overline{\mathrm{SD}}\) value of the original and reconstructed HRTFs from 30 subjects in the baseline. It indicates that the \(\overline{\mathrm{SD}}\) value is substantially around 4 dB in the 25 directions, implying that the HRTF decomposition using the preceding 10 independent components is stable and effective for reconstruction at all directions in the median plane. Figure 3: Mean spectral distortion across 25 elevations in the median plane when using 10-order independent component analysis (ICA) reconstructive HRTFs. Values are averaged among 30 subjects in the illustrative set. Figure 2: Independent components of head-related transfer functions (HRTFs).