A study on orthogonal matrix optimization and application to blind source separation problems

在本研究项目中，我们研究了盲信号处理及其相关问题，包括：(i)盲源分离的正交矩阵优化，（ii）降阶矩阵上的二次矩阵优化，（iii）拟非扩张映射不动点集上的凸优化，（iv）自适应滤波问题的凸函数序列渐近最小化及其统一视图。(v) DS/CDMA系统的盲多址干扰减少问题；（vi）立体声回声消除问题。在数学上，上述问题是具有挑战性的优化问题，可分为以下几类。(i)和（ii）的问题是，在一定的非凸约束集上的优化问题。（iii）中开发的算法回答了V.Vasin在1995年提出的问题。（iv）中提出的算法（自适应投影子梯度法）可以解决一个广义优化问题，该问题在整个过程中代价函数本身是不断变化的。应用自适应投影子梯度法解决了(v)和（vi）问题

In this research project, we studied the blind signal processing and its related problems including (i)orthogonal matrix optimization for blind source separations, (ii)quadratic matrix optimization over the reduced rank matrices, (iii)convex optimization over the fixed point set of quasi-nonexpansive mapping, (iv)asymptotic minimization of sequence of convex functions and its unified view for adaptive filtering problems, (v)blind multiple access interfererence reduction problem for DS/CDMA systems, (vi)stereophonic acoustic echo canceling problem.Mathematically, the above problems are callenging optimization problems which can be classified as follows. The problems on (i)and (ii)are, optimization problems over certain nonconvex constraint sets. The algorithm developed in (iii)answers the question posed by V.Vasin in 1995. The algorithm (adaptive projected subgradient method) developed in (iv)can solve a generalized optimization problem, where the cost function itself keeps changing during the whole process. The problems (v)and (vi)have been tackled by applying the adaptive projected subgradient method