Regularized nonnegative low-rank approximations such as sparse Nonnegative Matrix Factorization or sparse Nonnegative Tucker Decomposition are an important branch of dimensionality reduction models with enhanced interpretability. However, from a practical perspective, the choice of regularizers and regularization coefficients, as well as the design of efficient algorithms, is challenging because of the multifactor nature of these models and the lack of theory to back these choices. This paper aims at improving upon these issues. By studying a more general model called the Homogeneous Regularized Scale-Invariant, we prove that the scale-invariance inherent to low-rank approximation models causes an implicit regularization with both unexpected beneficial and detrimental effects. This observation allows to better understand the effect of regularization functions in low-rank approximation models, to guide the choice of the regularization hyperparameters, and to design balancing strategies to enhance the convergence speed of dedicated optimization algorithms. Some of these results were already known but restricted to specific instances of regularized low-rank approximations. We also derive a generic Majorization Minimization algorithm that handles many regularized nonnegative low-rank approximations, with convergence guarantees. We showcase our contributions on sparse Nonnegative Matrix Factorization, ridge-regularized Canonical Polyadic decomposition and sparse Nonnegative Tucker Decomposition.

通过研究称为均匀正则化比例不变的更一般模型，本文证明了低秩逼近模型中的比例不变性带来了隐式正则化，具有意想不到的有益和有害效果，并根据这一观察加强了正则化函数在低秩逼近模型中的作用理解，以引导正则化超参数的选择，并设计平衡策略以提高优化算法的收敛速度。

正则化非负标度不变低秩逼近模型的高效算法