## 3 Problem Statement and the Dimension Reduction with Iovp
### Problem Statement and the Proposed Iovp  
In previous works, the JIO approach reported in [16] outperforms the Krylov-based method with efficient adaptive implementations; however, there was a problem in this approach. Specifically, the theoretical JIO dimensionality reduction transform matrix, Equation (63) in [16], is in fact rank-one. Consequently, when the reduced dimension is selected as greater than one, so that the JIO projection matrix has more than one column, pre-processing with the JIO projection matrix will yield a singular,non-invertible reduced-rank covariance matrix, which means that the reduced dimensional weights will not exist; the rank-one column space of the JIO matrix is precisely the MVDR line, and the rank selection may fail to work. In order to address this issue, in the following, we detail a set of novel reduced-rank algorithms based on the proposed IOVP design of beamformers. The proposed IOVP design strategy jointly optimizes a projection matrix \(\mathbf{S}_{D}[i]\) and a reduced-rank beamformer \(\bar{\mathbf{\omega}}_{k}[i]\) by introducing several independently-generated small perturbations of the assumed steering vector and recursively updates a different column of the projection matrix to ensure a desired rank. The bank of adaptive beamformers in the front-end is responsible for performing dimensionality reduction, which is followed by a reduced-rank beamformer, which effectively forms the beam in the direction of the SoI. This two-stage scheme allows the adaptation with different update rates, which could lead to a significant reduction in the computational complexity per update. Specifically, this complexity reduction can be obtained as the dimensionality reduction performed by the rank reduction matrix could be updated less frequently than the reduced-rank beamformer. The principle of the proposed IOVP reduced rank scheme is depicted in Figure 1, which employs a projection matrix \(\mathbf{S}_{D}[i]\in\mathbb{C}^{M\times D}\) to perform dimensionality reduction on data vector \(\mathbf{x}[i]\in\mathbb{C}^{M\times 1}\). The rank-reduced filter \(\bar{\mathbf{\omega}}_{k}[i]\in\mathbb{C}^{D\times 1}\) processes the reduced-rank data vector \(\bar{\mathbf{x}}[i]\in\mathbb{C}^{D\times 1}\) to obtain a scalar estimate \(\bar{y}_{k}[i]\) of the \(k\)-th desired signal. The design criterion of MVDR-IOVP beamformer is given by the following optimization problem:  
\[ \begin{split}\min_{\mathbf{\omega},\mathbf{s}_{d}}&\bar{ \mathbf{\omega}}^{H}\bar{\mathbf{R}}\bar{\mathbf{\omega}}=\mathbf{\omega}^{H}\mathbf{S}_{D}^{H}\bm {R}\mathbf{S}_{D}\mathbf{\omega},\\ \text{subject to}&\bar{\mathbf{\omega}}^{H}\sum_{d=1}^{D }\mathbf{q}_{d}\mathbf{s}_{d}^{H}\mathbf{a}_{d}=1\end{split}  \tag{18}\]  
where \(\mathbf{R}\) is the covariance matrix obtained from sensors and vector \(\mathbf{q}_{d}\) with dimension \(D\times 1\) is a zero vector except its \(d\)-th element being one. The vector \(\mathbf{s}_{d}\in\mathbb{C}^{M\times 1}\) is the \(d\)-th column of the projection matrix \(\mathbf{S}_{D}\in\mathbb{C}^{M\times D}\). The vectors \(\mathbf{a}_{d},d=1\ldots D\) represent the assumed steering vector and \(D-1\) independently-generated small perturbations of the assumed steering vector. Different from the JIO approach in [16], where the columns of \(\mathbf{S}_{D}\) are jointly designed under the same criterion, the proposed IOVP approach (18) uses the vector \(\mathbf{q}_{d}\) to orthogonalize the columns of the projection matrix, and the columns can be independently updated with the perturbation vector in each recursion. The scheme  
Figure 1: Schematic of the proposed reduced-rank scheme. updates a different column of the projection matrix in each recursion and concatenates these columns to form the projection matrix \(\mathbf{S}_{D}\); the concatenation procedure ensures that the projection matrix has a desired rank. According to Equation (28) in Section 3.4, an increased rank of \(\mathbf{S}_{D}\) is obtained for higher \(d\), and the rank-one problem in [16] can be avoided.