We propose a Block Majorization Minimization method with Extrapolation (BMMe) for solving a class of multi-convex optimization problems. The extrapolation parameters of BMMe are updated using a novel adaptive update rule. By showing that block majorization minimization can be reformulated as a block mirror descent method, with the Bregman divergence adaptively updated at each iteration, we establish subsequential convergence for BMMe. We use this method to design efficient algorithms to tackle nonnegative matrix factorization problems with the $\beta$-divergences ($\beta$-NMF) for $\beta\in [1,2]$. These algorithms, which are multiplicative updates with extrapolation, benefit from our novel results that offer convergence guarantees. We also empirically illustrate the significant acceleration of BMMe for $\beta$-NMF through extensive experiments.

我们提出了一种用于解决一类多凸优化问题的块主导最小化方法及其外推算法（BMMe）。该方法通过使用一种新颖的自适应更新规则来更新BMMe的外推参数。通过将块主导最小化重新表述为一种块镜像下降方法，并在每次迭代中自适应地更新Bregman散度，我们为BMMe建立了逐次收敛性。我们使用该方法设计了高效算法来解决带有$\beta$-散度（$\beta$-NMF）的非负矩阵分解问题。这些算法使用外推的乘法更新，并从我们的新颖结果中获得了收敛保证。我们还通过大量实验证明了BMMe在$\beta$-NMF中的显著加速。

块主导性最小化方法的外推与应用: β-NMF