We consider Fair Principal Component Analysis (FPCA) and search for a low dimensional subspace that spans multiple target vectors in a fair manner. FPCA is defined as a non-concave maximization of the worst projected target norm within a given set. The problem arises in filter design in signal processing, and when incorporating fairness into dimensionality reduction schemes. The state of the art approach to FPCA is via semidefinite programming followed by rank reduction methods. Instead, we propose to address FPCA using simple sub-gradient descent. We analyze the landscape of the underlying optimization in the case of orthogonal targets. We prove that the landscape is benign and that all local minima are globally optimal. Interestingly, the SDR approach leads to sub-optimal solutions in this orthogonal case. Finally, we discuss the equivalence between orthogonal FPCA and the design of normalized tight frames.

中文翻译：

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公平的主成分分析和滤波器设计


我们考虑公平主成分分析（FPCA）并以公平的方式搜索跨越多个目标向量的低维子空间。 FPCA 被定义为给定集合内最差预测目标范数的非凹最大化。该问题出现在信号处理中的滤波器设计以及将公平性纳入降维方案中时。最先进的 FPCA 方法是通过半定规划，然后采用降阶方法。相反，我们建议使用简单的次梯度下降来解决 FPCA。我们分析了正交目标情况下底层优化的情况。我们证明环境是良性的，并且所有局部最小值都是全局最优的。有趣的是，在这种正交情况下，SDR 方法会导致次优解决方案。最后，我们讨论了正交 FPCA 与归一化紧框架设计之间的等价性。