论文标题
公平的主体组件分析和过滤器设计
Fair Principal Component Analysis and Filter Design
论文作者
论文摘要
我们考虑公平的主成分分析（FPCA），并搜索以公平方式跨越多个目标向量的低维子空间。 FPCA定义为给定集中最差的投影目标标准的非符号最大化。该问题在信号处理中以及将公平性纳入维度降低方案中时出现。 FPCA的最先进方法是通过半决赛松弛，涉及多项式但计算昂贵的优化。为了允许可伸缩性，我们建议使用幼稚的次级下降来解决FPCA。在正交目标的情况下，我们分析了基础优化的景观。我们证明了景观是良性的，并且所有局部最小值在全球范围内都是最佳的。有趣的是，在这种简单的情况下，SDR方法导致了亚最佳解决方案。最后，我们讨论了正交FPCA与归一化紧密帧的设计之间的等效性。
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 relaxation and involves a polynomial yet computationally expensive optimization. To allow scalability, we propose to address FPCA using naive 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 simple case. Finally, we discuss the equivalence between orthogonal FPCA and the design of normalized tight frames.
