Online Nonlinear Estimation via Iterative L2-Space Projections: Reproducing Kernel of Subspace

TL;DR

This paper introduces a novel online nonlinear function estimation method using iterative L2-space projections and reproducing kernels, offering improved efficiency and convergence properties over traditional kernel-based approaches.

Contribution

It presents a new online learning paradigm that operates in L2 space with a dictionary subspace, differing from conventional kernel methods by not requiring the entire space to be a RKHS.

Findings

The proposed algorithm achieves monotone approximation and asymptotic optimality.

Numerical experiments demonstrate superior performance over Kalman filter and batch machine learning methods.

Efficient updates of the Gram matrix enable scalable online learning.

Abstract

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.