The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of sketch and project methods. Recently, a block Gaussian version was proposed that uses a block Gaussian sketch, enjoying the regularization properties of Gaussian sketching, combined with the acceleration of the block variants. Theoretical analysis was only provided for the non-block version of the Gaussian sketch method. Here, we provide theoretical guarantees for the block Gaussian Kaczmarz method, proving a number of convergence results showing convergence to the solution exponentially fast in expectation. On the flip side, with this theory and extensive experimental support, we observe that the numerical complexity of each iteration typically makes this method inferior to other iterative projection methods. We highlight only one setting in which it may be advantageous, namely when the regularizing effect is used to reduce variance in the iterates under certain noise models and convergence for some particular matrix constructions.

由于其简单性和计算效率，Kaczmarz算法是解决大规模超定线性系统最流行的方法之一。可以将这种方法视为草图和项目方法的更一般类的特殊实例。最近，有人提出了一种使用块高斯草图的块高斯版本，该块高斯草图具有高斯草图的正则化特性，并具有块变体的加速度。仅对非块版本的高斯素描方法提供了理论分析。在这里，我们为块高斯Kaczmarz方法提供了理论保证，证明了许多收敛结果，表明以期望的指数速度收敛到解。另一方面，有了这一理论和广泛的实验支持，我们观察到，每次迭代的数值复杂度通常会使此方法劣于其他迭代投影方法。我们仅强调一种可能有利的设置，即在某些噪声模型下使用正则化效果减少迭代的方差时，以及对于某些特定矩阵构造的收敛时。