Optimal Modeling in Curved Reproducing Kernel Hilbert Spaces

摘要本研究的目的是在信息理论的基础上进一步理解非线性系统的成本函数优化问题。最近将熵解释为特征空间中投影样本的均方，提供了与再现核希尔伯特空间（RKHS）理论的联系，并提出了一个假设，即与目前使用搜索程序的估计不同，可以解析计算非线性系统的最优解。本文将讨论两种最广泛使用的滤波模型，即维纳滤波和卡尔曼滤波的解决方案。该方法也是新颖的，因为它将利用RKHS的内积结构和使用微分几何方法的估计过程的几何形状。知识的优点。该建议的智力优点是提出了一种基于信息几何的新方法，以适应具有直接操纵数据中的信息的成本函数的系统，其预期结果是在从数据创建模型和提供对数据的理解方面优于传统方法。更广泛的好处。我们这个技术驱动的世界正在以惊人的速度创造数据。然而，人类对信息感兴趣，而不是数据，这在医学、商业甚至工程领域造成了巨大的瓶颈。自适应系统是从数据中创建模型并提供理解的最有前途的方法之一。PI还将培养一批应用于信号处理的微分几何新领域的研究生，他们需要帮助解决这一信息瓶颈。

Principe AbstractThe objective of this research is to understand further the optimization of nonlinear systems with cost functions based on information theory. The recent interpretation of entropy as the mean square of the projected samples in feature space provides a link to the theory of reproducing kernels Hilbert spaces (RKHS), and raises the hypothesis that it may be possible to analytically compute the optimal solution of nonlinear systems, unlike the current estimates that use search procedures. The solution of the two most widely used models in filtering, the Wiener and the Kalman filters will be addressed. The approach is also novel because it will exploit both the inner product structure of the RKHS and the geometry of the estimation process using a differential geometry approach. Intellectual Merits. The intellectual merit of the proposal is to propose a new methodology based on information geometry to adapt systems with cost functions that directly manipulate information in the data, with the expected outcome of improving performance over the conventional methods in creating models from data and providing understanding of data. Broader Benefits. Our technology driven world is creating data at alarming rates. However, humans are interested in information, not data, and this is creating a tremendous bottleneck in medicine, business and even in engineering. Adaptive systems are one of the most promising methods to create models from data and provide understanding. The PI will also educate a breed of graduate students in the new area of differential geometry applied to signal processing who are needed to help solve this information bottleneck.