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.

翻译：我们提出了一种带外推的块主化最小化方法（BMMe），用于求解一类多凸优化问题。BMMe的外推参数通过一种新颖的自适应更新规则进行更新。通过将块主化最小化重新表述为一种块镜像下降方法（其中每次迭代自适应更新布雷格曼散度），我们建立了BMMe的序列收敛性。我们利用该方法设计了高效算法，以解决$\beta$-散度下的非负矩阵分解问题（$\beta$-NMF，其中$\beta\in [1,2]$）。这些算法（即带外推的乘法更新）得益于我们提供收敛保证的新颖结果。我们还通过大量实验实证说明了BMMe在$\beta$-NMF中的显著加速效果。