Fair Principal Component Analysis and Filter Design

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.