This paper proposes a framework for optimizing cost functions of orthonormal basis learning problems, such as principal component analysis (PCA), subspace recovery, orthogonal dictionary learning, etc. The optimization algorithm is derived using the majorization-minimization framework in conjunction with orthogonal projection reformulations to deal with the orthonormality constraint in a systematic manner. In this scope, we derive surrogate functions for various standard objectives that can then be used as building blocks, with examples for robust learning costs and sparsity enforcing penalties. To illustrate this point, we propose a new set of algorithms for sparse PCA driven by this methodology, whose objective function is composed of an $M$ -estimation type subspace fitting term plus a regularizer that promotes sparsity. Simulations and experiments on real data illustrate the interest of the proposed approach, both in terms of performance and computational complexity.

中文翻译：

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Stiefel流形上的主化-最小化及其在鲁棒的稀疏PCA中的应用

本文提出了一个优化正交函数基础学习问题的成本函数的框架，例如主成分分析（PCA），子空间恢复，正交字典学习等。该优化算法是使用主化最小化框架与正交投影重构相结合而得出的。以系统的方式处理正交性约束。在此范围内，我们导出了各种标准目标的替代功能，然后可以将其用作构建块，并提供了示例性的强大学习成本和稀疏性实施惩罚措施。为了说明这一点，我们提出了一套新的稀疏PCA算法，该算法由该方法驱动，其目标函数由一个$ M $ -估计类型的子空间拟合项，加上一个促进稀疏性的正则化器。对真实数据的仿真和实验从性能和计算复杂性两方面说明了该方法的重要性。