## 2 Methodology
### Independent Component Analysis of HRTFs  
In the first step of the hybrid algorithm, ICA is applied to magnitudes of HRTFs to obtain independent components and weight coefficients. There are several algorithms to implement ICA. In order to achieve fast convergence, a fast fixed-point algorithm based on negative entropy maximization is adopted here (FastICA) [29]. The procedure is as follows:  
Step 1: HRTF data of \(n\) subjects are denoted as \(\mathbf{H}_{n,m}\), where \(m\) represents the sum of the frequency and directions of HRTF. It should be pointed out that, in order to fully extract subject-relevant individualized characteristics, all subjects constitute a vector direction of the matrix, that is:  
\[\mathbf{H}_{n,m}=[H_{1}\;H_{2}\;H_{3}\;H_{4}\;\ldots\;H_{n}]. \tag{2}\]  
Step 2: Centre (i.e., de-average) on \(\mathbf{H}_{n,m}\) to obtain a new matrix \(\overline{\mathbf{H}}_{n,m}\)  
\[\overline{\mathbf{H}}_{n,m}=\left[\overline{H}_{1}\;\overline{H}_{2}\;\overline{H }_{3}\;\ldots\;\overline{H}_{n}\right]. \tag{3}\]  
Step 3: Whiten (i.e., sphere) on \(\overline{\mathbf{H}}_{n,m}\); here, we use principal component analysis (PCA) to obtain the direct result as  
\[\mathbf{V}{=}\;\mathbf{\Lambda}^{-1/2}\mathbf{U} \tag{4}\]  
where \(\mathbf{\Lambda}\) and \(\mathbf{U}\) represent the eigenvector matrix and the eigenvalue matrix of the covariance matrix \(\mathbf{C}=\mathbf{E}\left(\overline{\mathbf{H}\mathbf{H}}^{T}\right)\), respectively. That is, a whitened linear transformation \(\mathbf{V}\) is  
Step 9: Next, the FastICA algorithm is performed 10 times, and the run with the independent components that have the highest kurtosis (non-gaussianity) is selected. Step 10: Finally, reconstruct HRTFs from the selected independent component matrix and the weight coefficient matrix  
\[\mathbf{H}_{n,m}^{\prime}=\mathbf{A}_{n,d}\mathbf{IC}_{d,m}. \tag{8}\]  
If \(d=n\), Equation (8) means complete reconstruction, that is, the reconstructed \(\mathbf{H}^{\prime}\) is as same as the original \(\mathbf{H}\); if \(d<n\), \(\mathbf{H}^{\prime}\) is an approximation of the original \(\mathbf{H}\). This work used the CIPIC HRTF database [7]. The database was established in 2001 by the UC Davis CIPIC Interface Laboratory. It contains HRTF data (44.1 kHz sampling frequency, 200 points sampling length) for 45 subjects, including the Knowles Electronics Manikin for Acoustic Research (KEMAR) with large and small ears, with 1250 directions per person (25 azimuths \(\times\) 50 elevations). Meanwhile, it also includes 27 anthropometric parameters of 35 subjects (17 head and torso anthropometric parameters and 10 pinna anthropometric parameters). As an illustrative example of the proposed algorithm, we selected data from 35 subjects with 27 complete anthropometric parameters, and divided them into a validation group (containing five subjects, numbers 003, 010, 018, 020, and 027) and the computational group (the remaining 30 subjects). The role of the computational group (also known as the baseline) is to implement the flow algorithm in Figure 1 to obtain an HRTF ICA decomposition and multiple linear regression formula between weight coefficients and anthropometric parameters. On the other hand, the role of the validation group is to input the anthropometric parameters of the subjects into the constructed HRTF ICA decomposition and multiple linear regression formula of weights and anthropometric parameters to predict their individualized HRTFs. Finally, we compare it with the measured individualized HRTFs to show the performance of the proposed algorithm. In the process of HRTF decomposition using ICA, we calculated HRTF data below 20 kHz (232 frequency points) from 30 subjects at 25 elevation directions (an interval of 5.625\({}^{\circ}\)) in the front of the median plane.