The kernel method of machine learning is to transform data from data space to reproducing kernel Hilbert space (RKHS) and then perform machine learning in RKHS, while kernel learning is to select the best RKHS for specific applications and given learning samples. Since RKHS can be generated from kernel functions, kernel learning is to learn kernel functions. At present, the dilemma of kernel learning is that there are few kinds of kernel functions available for learning. The first contribution of this paper is to propose a new framework of kernel functions, in which the given learning samples can be embedded. Moreover, the framework contains a learnable part which can be optimized for specific applications. Symmetric positive definite (SPD) matrix data are more and more common in machine learning. However, SPD data space does not constitute a linear space and dictionary learning involves a lot of linear operations. Therefore, dictionary learning cannot be performed directly on SPD data space. The second contribution of this paper is to apply the proposed framework of kernel functions to dictionary learning of SPD data, in which SPD data are first transformed to the RKHS produced by the proposed framework, and then, both dictionary and the learnable part of the framework are learned simultaneously in RKHS. The experimental results on 4 landmark datasets show that the proposed algorithm performs better than 6 other algorithms published recently in top academic journals.

机器学习的内核方法是将数据从数据空间转换为可再生的内核希尔伯特空间（RKHS），然后在RKHS中执行机器学习，而内核学习则是为特定应用和给定的学习样本选择最佳的RKHS。由于RKHS可以从内核函数生成，因此内核学习就是学习内核函数。当前，内核学习的困境是几乎没有可用的内核函数。本文的第一个贡献是提出了一个新的内核函数框架，其中可以嵌入给定的学习样本。此外，该框架包含一个可学习的部分，可以针对特定应用进行优化。对称正定（SPD）矩阵数据在机器学习中越来越普遍。然而，SPD数据空间不构成线性空间，而字典学习涉及许多线性运算。因此，字典学习不能直接在SPD数据空间上执行。本文的第二个贡献是将提出的内核函数框架应用于SPD数据的字典学习，其中SPD数据首先转换为所提出的框架生成的RKHS，然后将字典和框架的可学习部分都转换为是在RKHS中同时学习的。在4个地标数据集上的实验结果表明，该算法的性能优于最近在顶级学术期刊上发布的其他6个算法。本文的第二个贡献是将提出的内核函数框架应用于SPD数据的字典学习，其中SPD数据首先转换为所提出的框架生成的RKHS，然后将字典和框架的可学习部分都转换为是在RKHS中同时学习的。在4个地标数据集上的实验结果表明，该算法的性能优于最近在顶级学术期刊上发布的其他6个算法。本文的第二个贡献是将提出的内核函数框架应用于SPD数据的字典学习，其中SPD数据首先转换为所提出的框架生成的RKHS，然后将字典和框架的可学习部分都转换为是在RKHS中同时学习的。在4个地标数据集上的实验结果表明，该算法的性能优于最近在顶级学术期刊上发表的其他6个算法。