论文标题
半金融的放松，用于在Stiefel歧管上的异构二次形式的总和
A Semidefinite Relaxation for Sums of Heterogeneous Quadratic Forms on the Stiefel Manifold
论文作者
论文摘要
我们研究了在Stiefel歧管上的异质二次形式总和的最大化，这是一个非凸型问题，它是在几种现代信号处理和机器学习应用中引起的，例如异性概率概率的主成分分析（HPPCA）。在这项工作中，我们得出了一种新颖的半决赛计划（SDP）对原始问题的放松，并研究了其一些理论特性。我们通过双重证书证明了针对原始非凸问题的全球最佳证书，这导致了一个简单的可行性问题，可以证明在Stiefel歧管上候选解决方案的全球最佳性。此外，我们的放松还将减少到针对共同对角线问题的分配线性程序，因此在这种情况下已知很紧。我们概括了这个结果，以表明它对于几乎共同的对角线问题也很紧，我们表明HPPCA问题具有此特征。数值结果验证了我们的全球最佳证书和足够的条件，以便在各种问题设置中SDP紧张时。
We study the maximization of sums of heterogeneous quadratic forms over the Stiefel manifold, a nonconvex problem that arises in several modern signal processing and machine learning applications such as heteroscedastic probabilistic principal component analysis (HPPCA). In this work, we derive a novel semidefinite program (SDP) relaxation of the original problem and study a few of its theoretical properties. We prove a global optimality certificate for the original nonconvex problem via a dual certificate, which leads to a simple feasibility problem to certify global optimality of a candidate solution on the Stiefel manifold. In addition, our relaxation reduces to an assignment linear program for jointly diagonalizable problems and is therefore known to be tight in that case. We generalize this result to show that it is also tight for close-to jointly diagonalizable problems, and we show that the HPPCA problem has this characteristic. Numerical results validate our global optimality certificate and sufficient conditions for when the SDP is tight in various problem settings.
