基于迭代L2-Space投影的在线非线性估计：子空间的再生核
Online Nonlinear Estimation via Iterative L2-Space Projections: Reproducing Kernel of Subspace
摘要：
我们提出了一种新的在线学习范式的非线性函数估计任务的基础上的迭代投影在L2空间的概率测度反映输入信号的随机性。所提出的学习算法利用了所谓的字典子空间的再生核，基于这样一个事实，即任何有限维的函数空间具有由格拉姆矩阵表征的再生核。L2空间几何结构原则上提供了最佳的去相关特性。所提出的学习范式是显着不同于传统的基于核的学习范式在两个意义上：（i）整个空间是不是一个再生核希尔伯特空间和（ii）的最小均方误差估计给出了所需的非线性函数在字典子空间的最佳逼近。它保留了计算内积的效率，以及在更新的Gram矩阵时，字典的增长。基于变尺度自适应投影次梯度法，分析了算法的单调逼近性、渐近最优性和收敛性。数值例子表明，所提出的算法的有效性，实际数据的各种方法，包括扩展的卡尔曼滤波器和许多批处理机器学习方法，如多层感知器。
英文摘要：
We propose a novel online learning paradigm for nonlinear-function estimation tasks based on the iterative projections in the L2 space with probability measure reflecting the stochastic property of input signals. The proposed learning algorithm exploits the reproducing kernel of the so-called dictionary subspace, based on the fact that any finite-dimensional space of functions has a reproducing kernel characterized by the Gram matrix. The L2-space geometry provides the best decorrelation property in principle. The proposed learning paradigm is significantly different from the conventional kernel-based learning paradigm in two senses: (i) the whole space is not a reproducing kernel Hilbert space and (ii) the minimum mean squared error estimator gives the best approximation of the desired nonlinear function in the dictionary subspace. It preserves efficiency in computing the inner product as well as in updating the Gram matrix when the dictionary grows. Monotone approximation, asymptotic optimality, and convergence of the proposed algorithm are analyzed based on the variable-metric version of adaptive projected subgradient method. Numerical examples show the efficacy of the proposed algorithm for real data over a variety of methods including the extended Kalman filter and many batch machine-learning methods such as the multilayer perceptron.
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