非参数核方法的样本外扩展研究

Kernel methods are widely applied to artificial intelligence. The performance of kernel methods critically relies on the selection of kernel. Empirical results show that non-parametric kernels have flexible structure and good performance. However, most non-parametric kernel methods cannot deal with the new data directly, thus are restricted in the real applications. To tackle this problem, one needs to extend the non-parametric kernel matrix to the corresponding kernel function. Researchers have explored this problem and obtained some achievements. However, the existing methods have limitations in practice. They are either restricted on the specific algorithms, or based on some prior assumptions of the data. How to extend a general non-parametric kernel matrix to the corresponding kernel function is still unsolvable. This project will develop a framework for converting the non-parametric kernel matrix to the corresponding kernel function, allowing us to deal with the out-of-sample extension problem. This framework is suitable for any non-parametric kernel method. The achievement of this project will widen the applications of non-parametric kernel methods which would be widely used for solving the practical problems in artificial intelligence.

核方法在人工智能领域有着广泛的应用，其表现性能依赖于核的选择。经验表明，非参数核具有灵活的结构和较好的表现。然而，大多数非参数核方法不能直接处理新样本，使其在实际应用中受到制约。解决此问题的关键在于把非参数核矩阵扩展成相应的核函数。科研工作者对此进行了一些探索，取得了一定的成果。然而，已有方法在实际应用中存在着局限性。它们要么限制在特定的算法上，要么需要数据符合某些先验的假设。对于一般的非参数核矩阵，如何寻找相应的核函数依旧是未解决的问题。本项目拟建立非参数核矩阵转化为相应的核函数的框架，以解决样本外扩展问题。此框架适用于任意的非参数核方法。本项目的研究成果将极大地丰富非参数核方法的应用前景，使其能更广泛地被用来解决人工智能领域的实际问题。

核方法广泛应用于人工智能领域。非参数核因其灵活性，受到广泛关注。然而，大多数非参数核无法对未知样本进行计算，限制了其在实际中的应用。解决此问题的关键在于把非参数核矩阵扩展成相应的核函数。然而，已有的研究只能处理特殊的非参数核矩阵。本项目研究如何解决任意的非参数核矩阵的样本外扩展问题。我们考虑由核函数生成的超再生核Hilbert空间，采用回归的方法，将核矩阵扩展为相应的核函数。通过采样等策略，降低了算法的计算复杂度。理论分析指出，分类算法性能依赖于非参数核矩阵的近似程度以及正则化程度。数值实验表明，本项目的方法能很好地对新样本进行扩展。与常用的参数核方法相比，我们的非参数核方法在人脸识别方面表现地更好。本项目的研究成果极大地丰富了非参数核方法的应用前景，使其能更广泛地被用来解决人工智能领域的实际问题。