## 3 Tensor Decomposition of Near-Field HRTF
The aforementioned three matrices satisfy the following orthogonality:  
\[(\mathbf{u}^{d})_{D^{\prime}\times D}^{T}\mathbf{u}_{D^{\prime}\times D^{\prime}}^{d}= \mathbf{I}_{D^{\prime}\times D^{\prime}}\quad(\mathbf{u}^{m})_{M^{\prime}\times M}^{T} \mathbf{u}_{M\times M^{\prime}}^{m}=\mathbf{I}_{M^{\prime}\times M^{\prime}}\quad(\mathbf{ u}^{n})_{N^{\prime}\times N}^{T}\mathbf{u}_{N\times N^{\prime}}^{n}=\mathbf{I}_{N^{ \prime}\times N^{\prime}}, \tag{6}\]  
where superscript \"\(T\)\" denotes the transpose of the matrix; \(\mathbf{I}\) is an identity matrix. The orthogonality in Equation (6) indicates that each eigen mode within a set of variable-related modes is independent. For example, the orthogonality of matrix \(\mathbf{u}_{N\times N^{\prime}}^{n}\) indicates that each eigen time-related mode is independent. The \(D^{\prime}\times M^{\prime}\times N^{\prime}\) core tensor \(\mathbf{w}_{D^{\prime}\times M^{\prime}\times N^{\prime}}\) represents the possibly complex interaction among distance-, direction-, and time-related modes. Its entries depend on distance-, direction-, and time-related modes but are independent from distance, direction, and time. Equation (5) is an exact representation of the full HRIR dataset. Given the tensor \(\mathbf{h}_{D\times M\times N}\) of HRIR data, the core tensor \(\mathbf{w}_{D^{\prime}\times M^{\prime}\times N^{\prime}}\) and three matrixes \(\mathbf{u}_{D^{\prime}\times D^{\prime}}^{d}\), \(\mathbf{u}_{M\times M^{\prime}}^{m}\), and \(\mathbf{u}_{N\times N^{\prime}}^{n}\) can be derived according to the method in the Appendix A, and the accurate tensor decomposition of near-field HRTFs in Equation (5) is obtained. For each variable-related mode, different eigen modes contribute differently to the related variation in HRIRs. A highly efficient representation of HRIRs can be obtained by retaining the eigen modes that contribute more to the variation in HRIRs and omitting the eigen modes that contribute less to the variation in HRIRs. To improve the efficiency of the algorithm of generating a virtual source, we are especially interested in the simplification of the time- and direction-related eigen modes in the tensor decomposition of HRIRs. Suppose that the columns of matrix \(\mathbf{u}_{N\times N^{\prime}}^{n}\) are arranged in the descending order according to the contribution of the time-related eigen modes to the variation in HRIRs. Then, the matrix \(\mathbf{u}_{N\times N^{\prime}}^{n}\) can be truncated up to the preceding \(N^{\prime\prime}<N^{\prime}\) column, resulting in a new \(N\times N^{\prime\prime}\) matrix of time-related modes. Similarly, the matrix \(\mathbf{u}_{M\times M^{\prime}}^{m}\) can be truncated up to the preceding \(M^{\prime\prime}<M^{\prime}\) column, resulting in a new \(M\times M^{\prime\prime}\) matrix of direction-related modes. At the same time, the dimensionality of direction- and time-related modes in the core tensor \(\mathbf{w}_{D^{\prime}\times M^{\prime}\times N^{\prime}}\) can also be truncated up to \(M^{\prime\prime}\) and \(N^{\prime\prime}\), respectively. The distance-related eigen modes can also be truncated in a similar way. However, the original (measured or calculated) HRIRs usually include the data of a few distances, which can be represented by a few distance-related eigen modes. Thus, it is not necessary to further truncate the distance-related eigen modes. Then, Equation (5) can be approximated as:  
\[\mathbf{\hat{h}}_{D\times M\times N}\approx\mathbf{w}_{D^{\prime}\times M^{\prime\prime }\times N^{\prime\prime}}\times_{D^{\prime}}\mathbf{u}_{D\times D^{\prime}}^{d} \times_{M^{\prime\prime}}\mathbf{u}_{M\times M^{\prime\prime}}^{m}\times_{N^{ \prime\prime}}\mathbf{u}_{N\times N^{\prime\prime\prime}}^{n}, \tag{7}\]  
where \(\mathbf{\hat{h}}\) represents the approximated tensor representation of HRIRs after truncation to distinguish the accurate tensor representation of HRIRs given by Equation (5).