再生核希尔伯特空间中自适应滤波新方法及应用

Recently, a family of nonlinear adaptive filtering algorithms, known as the kernel adaptive filtering (KAF) algorithms, becomes an emerging area of research. The KAF algorithms are developed in reproducing kernel Hilbert spaces (RKHS), by using the linear structure of this space to implement well-established linear adaptive algorithms and to obtain nonlinear filters in the original input space. These algorithms include the kernel least mean square (KLMS), kernel recursive least squares (KRLS), and kernel affine projection algorithms (KAPA), etc. When the kernel is radial, they naturally build a growing RBF network, where the weights are directly related to the errors at each sample. Compared with other nonlinear adaptive filtering algorithms, the KAF algorithms have some advantages; but till now, the related theory and algorithms about KAF are far from mature, and there are still many problems that need to be solved. This research project aims to address several key problems in kernel adaptive filtering, including the active sparsification, reduction of the network size, kernel selection and online optimization of the kernel, new topological structures, and the applications in cognitive modeling and brain signal analysis and processing. This research will improve the efficiency and applicability of the kernel adaptive filters, and promote the development, popularization, and application of this new technique.

近年来，核自适应滤波算法作为一类新型非线性自适应滤波算法越来越受到研究者们关注。核自适应滤波算法是通过在再生核希尔伯特空间(RKHS)中实现传统线性自适应滤波算法，进而利用核技巧（Kernel trick）得到原输入空间中非线性滤波算法。目前这类算法有核最小均方算法（KLMS），核递归最小二乘算法（KRLS），核仿射投影算法（KAPA），等等。与其他非线性自适应滤波方法比较，核自适应滤波具有很多优点，但其理论和算法还远未成熟，有许多关键问题需要解决。本项目主要研究核自适应滤波器主动稀疏化、网络规模的缩减、核函数选取与在线优化算法、新型拓扑结构以及核自适应滤波在认知建模和脑信号分析及处理中的应用。项目研究将提高核自适应滤波算法的高效性和实用性，进一步完善核自适应滤波理论和算法，促进该类新型滤波算法的发展、普及和应用。

时域信号的非线性、非高斯等特性是物理世界大多数信号存在的基本性质，而时域信号参数估计与滤波在雷达、声纳、通信、遥感数据处理和生物医学工程等众多领域中有着极其重要的应用意义。本项目研究核自适应滤波新方法来解决非线性、非高斯时域信号估计与滤波问题。项目针对核自适应滤波器的稀疏化、网络缩减、核函数参数优化、新型拓扑结构，及其在脑信号分析与处理方面的应用进行了深入研究，提出了自组织的核自适应滤波算法、核宽度自适应调整的核最小均方算法、密度依赖的量化核自适应滤波算法、基于最大互相关熵准则的核自适应滤波算法等新型核自适应滤波方法，有效解决了传统非线性滤波算法计算复杂度高、参数难以确定、受噪声影响大等难题。此外，项目通过在核空间中定义新型相似性度量很好解决了非高斯噪声环境下滤波算法的鲁棒性问题，突破了传统方法的局限性，实现对非线性、非高斯时域信号的高效、鲁棒的分析与处理。上述研究结果具有突破性、创新性、科学性和实用性，丰富和发展了自适应信号处理的参数估计与滤波的理论与方法。项目研究成果丰硕，在国际著名刊物及会议发表学术论文55篇，其中SCI期刊论文34篇。研究成果受到国际学术界相关领域著名学者在国际权威期刊或学术会议上的积极评价，所发表学术论文中有4篇论文为ESI高被引论文（前1%），充分体现了本项目研究的重要科学意义和价值。