Optimization of Independent Vector Analysis Algorithm: Comparison
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With the advent of the era of big data information, artificial intelligence (随着大数据信息时代的到来，人工智能（AI) methods have become extremely promising and attractive. It has become extremely important to extract useful signals by decomposing various mixed signals through blind source separation (BSS). BSS has been proven to have prominent applications in multichannel audio processing. For multichannel speech signals, independent component analysis (ICA) requires a certain statistical independence of source signals and other conditions to allow blind separation. independent vector analysis (IVA) is an extension of ICA for the simultaneous separation of multiple parallel mixed signals. IVA solves the problem of arrangement ambiguity caused by independent component analysis by exploiting the dependencies between source signal components and plays a crucial role in dealing with the problem of convolutional blind signal separation.）方法变得极具前景和吸引力。通过盲源分离（BSS）分解各种混合信号来提取有用的信号变得极其重要。BSS已被证明在多声道音频处理中具有突出的应用。对于多通道语音信号，独立分量分析（ICA）要求源信号和其他条件具有一定的统计独立性，以允许盲目分离。独立矢量分析（IVA）是ICA的扩展，用于同时分离多个并行混合信号。IVA利用源信号分量之间的依赖关系，解决了独立分量分析引起的排列模糊问题，在处理卷积盲信号分离问题中起着至关重要的作用。

  • blind source separation (BSS)
  • independent vector analysis (IVA)
  • optimization update rule

1. Introduction简介

With the advent of the era of big data information, people’s access to information has become more and more abundant. However, researchers usually only obtain the mixed information collected from the receiver, and the whole mixed information needs to be separated or extracted from the latent signals. The subsequent problem is how to effectively obtain useful signals from the received signals, which leads to the technology related to blind source separation (随着大数据信息时代的到来，人们获取信息的方式越来越丰富。然而，研究人员通常只获得从接收器收集的混合信息，并且需要从潜在信号中分离或提取整个混合信息。随之而来的问题是如何有效地从接收信号中获取有用的信号，这就引出了盲源分离（BSS) ）的相关技术。[1].
The theory of BSS can be traced back to the cocktail party problem, which has attracted much attention for decades. The cocktail party problem is when you are at a cocktail party and there are all kinds of people chatting around, but you can only concentrate on one of the discussions, or focus on the conversation of one of the people. BSS theory refers to observing the mixed signals of different sources and using these mixed signals to restore the original signal, and the prior information of the source signal and its mixed signal is minimal. A large number of applications of BSS in communication, speech, and medical signal processing has received extensive attention in recent years 的理论可以追溯到鸡尾酒会问题，这个问题几十年来一直备受关注。鸡尾酒会的问题是，当你在鸡尾酒会上，有各种各样的人在周围聊天，但你只能专注于其中一个讨论，或者专注于其中一个人的谈话。BSS理论是指观察不同声源的混合信号，并利用这些混合信号还原原始信号，源信号及其混合信号的先验信息极少。近年来，BSS在通信、语音和医疗信号处理方面的大量应用受到广泛关注[2]. It is of great significance to realize blind estimation, blind equalization, and adaptive signal processing through blind characteristics.通过盲特性实现盲估计、盲均衡和自适应信号处理具有重要意义。
Independent component analysis 独立成分分析[3][4][5] (（ICA) is one of the most important methods first proposed to deal with BSS. This is a classic BSS technology based on statistical independence of source signals and is the mainstream technology of BSS. ICA requires that source signals be statistically independent of each other. It is an unsupervised, data-driven signal processing technique based on non-Gaussian maximization to separate time-invariant mixture signals in the time domain.）是首次提出的处理BSS的最重要方法之一。这是一种基于源信号统计独立性的经典BSS技术，是BSS的主流技术。ICA 要求源信号在统计上彼此独立。它是一种基于非高斯最大化的无监督、数据驱动的信号处理技术，用于在时域中分离时不变的混合信号。
However, consider that in a real scenario, the signal is often mixed with reverberation in the form of convolution. However, 但是，考虑到在真实场景中，信号通常以卷积的形式与混响混合。但是，ICA cannot separate the common form of convolution mixing. Moreover, the convolution mixed signal is processed in the time domain with high computational complexity and a huge amount of computation, and the convergence speed is slow, which greatly reduces the separation performance. Taking advantage of the properties of convolution mixing: the convolution in the time domain is equal to the product in the frequency domain, a frequency domain 无法分离卷积混合的常见形式。而且卷积混合信号在时域处理，计算复杂度高，计算量大，收敛速度慢，大大降低了分离性能。利用卷积混合的特性：时域中的卷积等于频域中的乘积，频域ICA [6][7] (提出（FD-ICA) algorithm is proposed. The entire convolutional mixed signal is converted from the time domain to the frequency domain for separation by the short-time Fourier transform (STFT). Compared with the time-domain convolution operation, the frequency-domain product operation has the advantages of convenient calculation, small computational complexity, and fast convergence speed.）算法。整个卷积混合信号从时域转换为频域，通过短时傅里叶变换（STFT）进行分离。与时域卷积运算相比，频域积运算具有计算方便、计算复杂度小、收敛速度快等优点。
To solve the above-mentioned problems of 为解决ICA, the independent vector analysis (IVA) 的上述问题，独立矢量分析（IVA）[8][9] algorithm is proposed. It generalizes 提出了算法。它通过利用数据集之间的统计依赖关系将ICA to multiple datasets by exploiting statistical dependencies across datasets, addressing some of the uncertainty in the output of signal separation.推广到多个数据集，解决了信号分离输出中的一些不确定性。

2. Optimizing优化 IVA Algorithm—Optimizing Update Rules算法 - 优化更新规则

GD 广东[10]是最原始的优化算法之一。梯度下降是一种通过在与目标函数 is one of the mostI 的梯度相反的方向上更新模型参数来最小化 primitive optimization algorithms. Gradient descent is a method that minimizes I by updating the model parameter in the opposite direction of the gradient of the objective function I. The learning rate 的方法。学习率η determines the size of the step size chosen to reach the local minimum, in other words, the descending hill along the slope of the surface produced by the objective function until a valley is reached. This is a separation method obtained by minimizing (决定了选择达到局部最小值的步长的大小，换句话说，沿着目标函数产生的表面斜率的下坡，直到到达谷。这是通过最小化（5), a simple ）得到的分离方法，一个简单的GD method is extrapolated as follows:方法外推如下：
Δ W ( k ) = − ∂ I ∂ W ( k )
Its main variants are batch gradient (它的主要变体是批量梯度（BG), stochastic gradient (SG), and natural gradient (NG). Among them, the NG algorithm ），随机梯度（SG）和自然梯度（NG）。其中，NG算法[11][12] is an effective and one of the most commonly used algorithms to solve the problem of 是解决BSS. The main idea is to take the 问题的一种有效且最常用的算法之一。主要思想是将目标函数I的NG direction of the objective function I as the iterative direction so that the algorithm can quickly converge, so as to realize the separation of source signals. Additionally, it is proved that the best descent direction is not the 方向作为迭代方向，使算法能够快速收敛，从而实现源信号的分离。此外，证明最佳下降方向不是“负”规则梯度方向，而是“negative负” regular gradient direction but the "negative" Riemann gradient. It was first proposed in 黎曼梯度。它最初是在[13][14], and its main idea is to multiply the scaling matrix ，其主要思想是将缩放矩阵Q(（k) to modify the gradient in the original ）相乘，修改原GD method to obtain faster convergence speed. As Equation (方法中的梯度，以获得更快的收敛速度。如公式（7):）：
Δ W ( k ) = − ∂ I ∂ W ( k ) Q ( k )
The update for the separation matrix is:分离矩阵的更新为：

W ( k ) ← W ( k ) + η Δ W ( k )
 when solving the objective function. The choice of step size will directly affect the convergence speed and accuracy. In order to speed up the convergence speed of the algorithm, many scholars have also optimized and improved the classical 求解目标函数时。步长的选择将直接影响收敛速度和精度。为了加快算法的收敛速度，许多学者还对经典的NG algorithm. In 2011, Liang et al算法进行了优化和改进。2011年，梁等. [15] proposed a control mechanism that considers the step size to obtain fast and stable convergence. In 提出了一种考虑步长以获得快速稳定收敛的控制机制。2011, Zhang et al年，张等. [16] proposed an 提出了一种通过函数近似直接估计评分函数的NG blind separation algorithm that directly estimates the score function through function approximation, which uses a linear combination of a set of orthogonal polynomials to approximate the score function, and its performance is measured by the mean squared error. An improved momentum term method was proposed in 盲分离算法，该算法使用一组正交多项式的线性组合来近似评分函数，其性能由均方误差来衡量。在[17] which can speed up the algorithm’s convergence.这可以加快算法的收敛速度。
In 2018, Fu et al年，傅等. [18] proposed a blind separation algorithm for 提出了一种基于步长自适应的IVA based on step-size adaptation. The algorithm initializes the separation matrix using the feature matrix joint approximate diagonalization algorithm and adaptively optimizes the step-size parameter. That is, to avoid local convergence, it can also significantly improve the convergence speed of the algorithm and further improve the separation performance. According to the relationship between the iteration step size and the estimated cost function change. In 2012, Wang et al盲分离算法。该算法利用特征矩阵联合近似对角化算法初始化分离矩阵，自适应优化步长参数。即避免局部收敛，还可以显著提高算法的收敛速度，进一步提高分离性能。根据迭代步长与估计成本函数之间的关系变化。2012年，王等. [19] proposed a variable提出了一种基于最大块速度-step-size IVA gradient algorithm based on the most block speed step-size descent. Additionally, according to the relationship between the iterative step size and the change in the separation matrix to be obtained, a variable-step-size IVA gradient algorithm based on the estimation function is proposed. In 2010, Kim 步长下降的可变步长IVA梯度算法。此外，根据迭代步长与待得到的分离矩阵变化之间的关系，提出一种基于估计函数的变步长IVA梯度算法。2010年，金·[12] proposed a modified gradient and normalized 提出了一种具有非完全闭合约束的修正梯度和归一化IVA method with nonfully closed constraints. Gradient normalization improves the convergence speed, and nonholographically constrained gradients with lower computational complexity show better performance, while possessing simpler structures compared with other methods. In 2018, Koldovský et al方法。梯度归一化提高了收敛速度，计算复杂度较低的非全息约束梯度表现出更好的性能，同时与其他方法相比具有更简单的结构。2018年，科尔多夫斯基等人. [20], based on the independent vector extraction (基于IVE) of the IVA algorithm, proposed an IVE algorithm with an adaptive step-size method in complex non-Gaussian scenarios to speed up convergence.A算法的独立向量提取（IVE），提出了一种在复杂非高斯场景中采用自适应步长方法的IVE算法，以加快收敛速度。

3. Fast Fixed Point Method快速定点法

The fast fixed point method was derived by introducing Newton’s method. The iterative update rule based on fast fixed point 快速不动点法是通过引入牛顿法推导而来的。基于快速定点的迭代更新规则[21] was first proposed to optimize the objective function of 最初提出来优化ICA. It provides a very simple algorithm, one that does not depend on any defined parameters and that quickly converges to the most accurate update rule the data allow.的目标功能。它提供了一种非常简单的算法，该算法不依赖于任何定义的参数，并且可以快速收敛到数据允许的最准确的更新规则。
When优化基于负熵的目标函数时，最简单的方法是使用 optimizing a negative entropy-based objective function, the easiest way is to use GD. Although the GD-based method has a good separation effect, it is relatively simple to use. The overall convergence speed of this method is slow and depends on a good choice of the learning rate sequence, i.e., the step size per iteration. Although various optimizations for the step-size factor were summarized in the previous section, GD methods rely on a suitable step size for separation.GD。虽然基于GD的方法具有良好的分离效果，但使用起来相对简单。该方法的整体收敛速度很慢，并且取决于学习速率序列的良好选择，即每次迭代的步长。尽管上一节总结了步长因子的各种优化，但GD方法依赖于合适的步长进行分离。
Therefore, in practical applications, it is very important to make the entire convergence process faster and more reliable. Therefore, a fast fixed point iterative algorithm 因此，在实际应用中，使整个收敛过程更快、更可靠非常重要。因此，一种快速定点迭代算法[22] is proposed to achieve this. In fixed point algorithms, the entire computation is performed in batch or block mode, i.e., a large number of data points are used in one step of the algorithm. The fast fixed point algorithm has very attractive convergence properties, and in experiments, it converges much faster than the commonly used 建议实现这一目标。在定点算法中，整个计算以批处理或块模式执行，即在算法的一个步骤中使用大量数据点。快速定点算法具有非常吸引人的收敛特性，在实验中，它的收敛速度比常用的GD method. At the same time, in environments where fast real-time adaptation is not required, this method is a good alternative to adaptive learning rules. In 1997, Hyvarinen 方法快得多。同时，在不需要快速实时适应的环境中，这种方法是自适应学习规则的良好替代方案。1997年，海瓦里宁·[23] described a more heuristic derivation of it.描述了一种更启发式的推导。
In 2000, Bingham et al年，宾汉姆等. [24] proposed a 提出了一种能够分离复值线性混合源信号的FastICA algorithm capable of separating complex-valued linear mixed-source signals. The method shows good performance in the ICA algorithm. The same 算法。该方法在ICA算法中表现出良好的性能。一样[25] generalized fast fixed point method to the 广义快速定点法为IVA algorithm, which was developed based on the idea of 算法，基于FastICA and used to optimize the traditional IVA algorithm. Under this method, the update is expressed as:的思想开发，用于优化传统的IVA算法。在此方法下，更新表示为：


w n ( k ) ← E [ G ′ ( ∑ k | y n ( k ) | 2 ) + | y n ( k ) | 2 G ″ ( ∑ k | y n ( k ) | 2 ) ] w n ( k ) − E [ ( y n ( k ) ) * G ′ ( ∑ k | y n ( k ) | 2 ) x ( k ) ]
where其中 E denotes the expectation, 表示期望，G(（⋅)） denotes a nonlinear function, and 表示非线性函数，以及
G ( ∑ k | y n ( k ) | 2 ) = − log g ∧ s n ( y n )
After the updated matrix 通过更新规则得到更新后的矩阵W is obtained through the update rule, decorrelation needs to be performed to ensure orthogonality as follows:后，需要进行去相关，保证正交性，如下所示：
W [ k ] ← W [ k ] W [ k ] H − 1 / 2 W [ k ]
where其中 (⋅)H（⋅）H denotes表示 the（⋅） conjugate transpose of (⋅). To be able to directly apply Newton’s method to derive a fast algorithm for complex variables, a quadratic Taylor polynomial is introduced into the complex notation. Using this form of Taylor series expansion makes the derivation simpler and is useful for directly applying Newton’s method to objective functions of complex-valued variables. In 的共轭转置。为了能够直接应用牛顿方法推导出复变量的快速算法，在复数符号中引入了二次泰勒多项式。使用这种形式的泰勒级数展开使推导更简单，并且对于直接将牛顿方法应用于复值变量的客观函数很有用。2000, Yan et al年，闫等. [26] provided an independent equivalent.提供了独立的等效物。 Recently, in最近，在 2021, Koldovský et al. 年，科尔多夫斯基等人。[27] proposed an extended fast dynamic independent vector analysis (提出了一种基于FastDIVA) algorithm based on the FastICA and FastIVA static hybrid algorithms, used to blindly extract or separate one or more signal sources from a time-varying mixed signal. In a source-by-source separation mixture model that allows the desired source to move, the mixture is either in series or in parallel. The algorithm inherits the advantages of ICA和FastIVA静态混合算法的扩展快速动态独立矢量分析（FastDIVA）算法，用于从时变混合信号中盲目提取或分离一个或多个信号源。在允许所需源移动的逐源分离混合物模型中，混合物要么串联，要么并联。该算法继承了FastIVA, exhibits good performance in motion source separation, and exhibits superior convergence speed and ability to separate super-Gaussian and sub-Gaussian signals.的优点，在运动源分离方面表现出良好的性能，表现出优越的收敛速度和分离超高斯和亚高斯信号的能力。 In 2021, Amor et al年，阿莫尔等人. [28]使用 used FastDIVA for blind source extraction for mixture models with constant separation vector CSV. Additionally, it shows new potential and good separation performance in three environments: motion loudspeaker in a noisy environment, extraction of motion brain activity, and motion source. In 2021, Koldovský et al对具有恒定分离载体 CSV 的混合物模型进行盲源提取。此外，它在嘈杂环境中的运动扬声器、运动大脑活动的提取和运动源三种环境中显示出新的潜力和良好的分离性能。2021年，科尔多夫斯基等人. [29] proposed a new dynamic 提出了一种新的动态IVA algorithm. It is based on a mixed model in which the source-of-interest (SOI)-related mixing parameters are time-varying, and the separation parameters are time-invariant. The Newton–Raphson method is used to optimize the objective function based on the quasi-likelihood method, then the iterative update is performed without imposing orthogonality constraints, and then orthogonality is performed. This algorithm is an optimization of the fast fixed point algorithm, which is better than the gradient algorithm and the auxiliary function method in performance.算法。它基于一个混合模型，其中与兴趣源（SOI）相关的混合参数是时变的，分离参数是时不变的。采用牛顿-拉夫森方法在准似然法的基础上对目标函数进行优化，然后在不施加正交约束的情况下进行迭代更新，然后进行正交性。该算法是对快速定点算法的优化，在性能上优于梯度算法和辅助函数法。

4. Auxiliary Function辅助功能

The update method based on the auxiliary function technology is also a method that does not include tuning parameters such as step size, which is an iterative algorithm with a convergence guarantee. This is a stable and fast update rule derived from the majorize-minimization principle [30][31]. Find its minimum by exploiting the convexity of the function. When the objective function f(θ) is difficult to optimize, and the optimization algorithm used cannot directly find the optimal solution to the objective function, an easy-to-optimize objective function g(θ) can be found instead. Then, the substitution function is solved, and the optimal solution of g(θ) is close to the optimal solution of f(θ). In each iteration, a new surrogate function for the next iteration is reconstructed from the solution. Then, the new substitute function is optimized and solved to obtain the objective function of the next iteration. After several iterations, the optimal solution that is closer and closer to the original objective function that can be obtained. It was first proposed in the literature [32] to accelerate the convergence speed of the ICA algorithm. This rule consists of two optional updates:
  • The update of the weighted covariance matrix (that is, the auxiliary function variable).
  • The update of the separation matrix ensures that the objective function decreases monotonically at each update and finally achieves convergence.

Equation (12) is the auxiliary function variable update:

V n = E n [ U ′ ( ‖ y n ‖ 2 ) ‖ y n ‖ 2 x n ( x n ) H ]

Among them, Vn denotes a covariance matrix of the observed signals, U(⋅) denotes a continuous and differentiable function of a real variable · satisfying, and U′(⋅) usually takes the constant 1. ∥⋅∥2 denotes the 2-norm of ·. Equation (13) is the update of the unmixing matrix:

w n ( k ) = [ W V n ] − 1 e n e n T ( W n − H V n − 1 W n − 1 ) e n

In 2011, Nobutaka Ono [33] used the auxiliary function technique in the objective function of the IVA algorithm and similarly derived an efficient update rule suitable for the IVA algorithm, called AuxIVA. In 2012, Nobutaka Ono [34] proposed an AuxIVA algorithm based on a generalized Gaussian source model or a Gaussian source model with time-varying variance. In 2012 and 2013, Nobutaka Ono [35][36] proposed a faster algorithm that can update two separation vectors simultaneously by solving the generalized eigenvalue problem for the AuxIVA algorithm with two sources and two microphones. Compared with the one-by-one update method, this method has faster convergence speed and better performance. This pairwise update method is also applicable to the pairwise separation of vectors in the case of three or more sources [37]. In 2014, Taniguchi et al. [38] used the AuxIVA algorithm based on the auxiliary function method for online real-time blind speech separation. In experimental comparisons with commonly used real-time IVA algorithms, the proposed online algorithm achieves a higher signal-to-noise ratio without environment-sensitive tuning parameters such as step factor. In 2021, Brendel et al. [39] further optimized the IVA algorithm based on auxiliary functions under the same computational cost. The convergence speed of the AuxIVA algorithm is enhanced by three methods:
  • Turn the differential term into a tuning parameter via the differential term in the NG approximation algorithm.
  • Approximate the differential term as a matrix using the quasi-Newton method.
  • Use the square iteration method to speed it up.

5. EM Method

In signal processing, a common problem is estimating the parameters of a probability distribution function. The situation is more complicated in many parameter estimation problems because the data needed to estimate the parameters are not directly accessible, or some data are missing. EM-based optimization algorithms are well-suited for solving this class of problems because the EM algorithm produces maximum likelihood (ML) estimates of the parameters when there is a many-to-one mapping from the underlying distribution to the distribution of the control observations, while taking additive noise into account. The EM algorithm overcomes the problem of unanalyzable solutions and has been widely used in statistics, signal processing, and machine learning [40]. The EM algorithm is an iterative optimization method [41] that is used to estimate some unknown parameters given measurement data. The solution is divided into two steps. E-step: First assign an initial distribution to each hidden variable empirically, that is, assume distribution parameters. Then, according to the parameters of the distribution, the expectation of the hidden variables in each data tuple can be obtained, that is, the classification operation is performed. The posteriors of the source signal can be obtained by
log q ( x 1 ( k ) , … , x N ( k ) | s 1 ( k ) , … , s N ( k ) ) ∝ log g ( y 1 ( k ) , … , y N ( k ) | x 1 ( k ) , … , x N ( k ) ) + ( log g ( x 1 ( k ) | s 1 ( k ) ) + … + log g ( x N ( k ) | s N ( k ) ) ) + c o n s t .
where ∝ denotes it is proportional to the previous term, and q denotes posterior probability. M-step: Calculate the maximum likelihood value of the distribution parameter (vector) based on the classification result, and then in turn recalculate the expectation of the hidden variable for each data tuple based on this maximum likelihood value. The update rules for mixing matrices A are 
A ( k ) = ( ∑ k < y ( k ) ( x ( k ) ) T > q ) ( ∑ k < x ( k ) ( x ( k ) ) T > q ) − 1
where <⋅>q denotes expectation over q. Through the repetition of the above two steps, when the expectation of the hidden variable and the maximum likelihood value of the parameter tends to be stable, the entire iteration is completed. In 2004 and 2008, Varadhan et al. [42][43] used the square iteration method in the EM algorithm to accelerate its convergence speed. In 2008, Lee et al. [44] deduced the expectation-maximization algorithm, and the algorithm was used in the updated iteration of the IVA algorithm. The EM algorithm could estimate the parameters of the separation matrix and the unknown source at the same time, showing a good separation performance. In 2010, Hao et al. [45] proposed a unified probabilistic framework for the IVA algorithm with the Gaussian mixture model as the source prior model; this flexible prior source enables the IVA algorithm to separate different types of signals, deduce different EM algorithms, and test three models: noiseless IVA, online IVA, and noise IVA. The EM algorithm can effectively estimate the unmixing matrix without sensor noise. In online IVA, an online EM algorithm is derived to track the motion of the source under nonstationary conditions. Noise IVA includes sensor noise and denoising combined with separation. An EM algorithm suitable for this model is proposed which can effectively estimate the model parameters and separate the source signal at the same time. In 2019, Gu et al. [46] proposed a Gaussian mixture model IVA algorithm with time-varying parameters to accommodate temporal power fluctuations embedded in nonstationary speech signals, thus avoiding the pretraining process of the original Gaussian mixture model IVA (GMM-IVA) algorithm and using the corresponding improved EM algorithm to estimate the separation matrix and signal model. The experimental results confirm the effectiveness of the method in random initialization and the advantages in separation accuracy and convergence speed. In 2019, Rafique et al. [47] proposed a new IVA algorithm based on Student’s t-mixture model as a source before adapting to the statistical properties of different speech sources. At the same time, an efficient EM algorithm is derived which estimates the location parameters of the source prior matrix and the decomposition matrix together, thereby improving the separation performance of the IVA algorithm. In 2020, Tang et al. [48] proposed a complex generalized Gaussian mixture distribution with weighted variance to capture the non-Gaussian and nonstationary properties of speech signals to flexibly characterize real speech signals. At the same time, the optimization rules based on the EM method are used to estimate and update the mixing parameters.

6. BCD Method

Coordinate descent (CD) is a nongradient optimization algorithm. The algorithm does not need to calculate the gradient of the objective function and performs a linear search along a single dimension at a time. When a minimum value of the current dimension is obtained, different dimension directions are used repeatedly, and the optimal solution is finally converged. However, this algorithm is only suitable for smooth functions. When nonsmooth functions are used, they may fall into a nonstagnant point and fail to converge. In 2015, Wright [49] proposed block coordinate descent (BCD), a generalization of the coordinate descent algorithm. It decomposes the original problem into multiple subproblems by simultaneously optimizing a subset of variables. The order of updates during the descent can be deterministic or random. This algorithm is mainly used to solve the nonconvex function, of which the objective function’s global optimal value is difficult to obtain. Among them, the BCD algorithm has developed two methods with closed update formula for the BSS IVA algorithm’s [50] IP and ISS methods.

6.1. Iterative Projection

The IVA algorithm based on iterative projection was first introduced in the AuxIVA [33] algorithm. This update rule is derived by solving a quadratic system of equations obtained by differentiating the cost function concerning the separation vector. In 2004, Dégerine et al. [51] also proposed a similar scheme in the context of semiblind Gaussian source components. In 2016, Kitamura et al. [52] used the IP algorithm in a BSS algorithm combining IVA and NMF, which provided good convergence speed and separation effect. In 2018, Yatabe et al. [53] proposed an alternative to the AuxIVA-IP algorithm based on proximal splitting. In 2021, Nakashima et al. [54] optimized it based on IP and extended each row vector of the separation matrix to update one by one to two rows of the separation matrix per update, resulting in a faster IP-2. In 2020, Ikeshita et al. [55] deduced IP-1 and IP-2 and used these two update rules to accelerate the OverIVA algorithm, forming the OverIVA-IP and OverIVA-IP2 update rules. In 2021, Scheibler [56] proposed an iterative projection with adjustment (IPA) and a Newton conjugate gradient (NCG) to solve the hybrid exact-approximate diagonalization (HEAD) problem. IPA adopts a multiplicative update form, that is, the current separation matrix is multiplied by the rank 2 perturbation of the identity matrix. This method performs joint updates to the unmixing filters and additional rank-one updates to the remainder of the unmixing matrix. Simply put, the IPA optimization rule is a combination of IP and ISS methods. Updating one row and one column of the matrix in each update, performing IP- and ISS-style updates jointly, outperforms the IP and ISS methods.

6.2. Iterative Source Steering

ISS [57] is an alternative to IP. Although IP has the advantages of good performance and fast convergence speed, in the iterative update process, it needs to recalculate a covariance matrix and invert for each source and each iteration. This greatly increases the overall complexity of the algorithm. The complexity of the algorithm is three times the number of microphones used. In addition to that, inverting a matrix is an inherently dangerous operation that can lead to unstable convergence when iterating. On this basis, the proposed ISS algorithm can effectively reduce the computational cost and complexity brought by the IP algorithm. ISS can also minimize the same cost function as the AuxIVA algorithm. 
W ( k ) ← W ( k ) − v n ( k ) ( w n ( k ) ) H
This update rule, which does not require matrix inversion, is used in a new method for joint deredundancy and BSS [58]. This is a method based on an ILRMA framework, which combines the advantages of no inversion and low complexity of the ISS algorithm to achieve efficient BSS. In 2021, Du et al. [59] proposed a computationally efficient optimization algorithm for BSS of overdetermined mixtures, an improved ISS algorithm for OverIVA algorithm, namely OverIVA-ISS. The algorithm combines the technology in OverIVA-IP with the technology in AuxIVA-ISS, which is more computationally efficient than the OverIVA-IP algorithm and can guarantee convergence. Additionally, the computational complexity is reduced from O(M2) to O(MN). The overall performance of the ISS algorithm is better than the IP algorithm but inferior to the IP-2 algorithm. Therefore, an ISS-2 algorithm is proposed. In 2022, Ikeshita et al. [60] extended the ISS algorithm to ISS-2. At the same time, the advantage of the smaller time complexity of the ISS algorithm is maintained, and the separation performance is comparable to IP-2.

7. EVD Method

The EVD method is to find the most similar matrix to the original matrix. The optimization update rule based on EVD can be expressed as:
w [ k ] ← w [ k ] ‖ w [ k ] ‖ 2
and
w ( k ) = 1 λ M ( k ) u M ( k )
where λM and uM denote the smallest eigenvalue and eigenvector, respectively.
The IVA algorithm based on the EVD update rule was proposed in [61] for a fast independent vector extraction (FIVE) algorithm. By comparing with the OverIVA and AuxIVA algorithms experimentally, the proposed algorithm can obtain the optimal solution with only a few iterations and is far superior to other algorithms in terms of convergence performance. In 2021, Brendel et al. [62] extended the update rule of eigenvalue decomposition to an IVA source extraction algorithm with SOI mechanism. The proposed update rule achieves fast convergence at lower computational cost and outperforms the IP update rule in performance.

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