# Kernel Geometric Mean Metric Learning
###### Abstract  
Geometric mean metric learning (GMML) algorithm is a novel metric learning approach proposed recently. It has many advantages such as unconstrained convex objective function, closed form solution, faster computational speed, and interpretability over other existing metric learning technologies. However, addressing the nonlinear problem is not effective enough. The kernel method is an effective method to solve nonlinear problems. Therefore, a kernel geometric mean metric learning (KGMML) algorithm is proposed. The basic idea is to transform the input space into a high-dimensional feature space through nonlinear transformation, and use the integral  
pairs of samples more compact in the new metric space, while the distribution of dissimilar pairs is more discrete. The proposal of this algorithm marked the real development of metric learning, and many subsequent research works were inspired by this algorithm. [PERSON] [14] proposed an information theoretic metric learning algorithm. The basic idea of the algorithm is to assume the existence of a priority metric matrix \(M_{0}\), while ensuring that the distance between similar pairs of samples is less than a threshold and the distance between dissimilar pairs of samples is greater than a threshold. By minimizing the relative entropy between the multivariate Gaussian distributions corresponding to \(M\) and \(M_{0}\), [PERSON] [15] proposed an information geometry metric learning algorithm. The basic idea of the algorithm is to use the category labeling information of the training samples to construct a kernel matrix that can reflect the desired distance between samples. Construct an actual kernel matrix describing the realistic distance relationship between the samples using the eigenvectors of the samples as well as the metric matrix \(M\). The metric matrix \(M\) is then solved by minimizing the distance between these two kernel matrices. [PERSON] [16] proposed a metric learning algorithm based on the maximum margin. The basic idea is to define an interval function between similar samples of each sample and other classes of samples. The metric matrix is then solved by minimizing the distance between similar pairs of samples and maximizing the defined interval. Geometric mean metric learning algorithm was proposed by [PERSON] [17] in 2016. The essence of most metric learning algorithms is to minimize the distance between similar pairs of samples rather than maximize the distance between similar pairs of samples. Therefore, the sign of the term corresponding to the dissimilar pair of samples in the objective function is usually negative, while the strategy of the geometric mean metric learning model is to use the inverse matrix of the metric matrix \(M\) to represent the distance between dissimilar pairs of samples. The advantage of this approach is that the sign of the item corresponding to the dissimilar pair of samples in the objective function becomes positive, which reduces the difficulty of solving the model. Its objective function is as follows:  
\[\min_{M>0}\sum_{(x_{1i},x_{2i})\in D^{+}}d_{M}(x_{1i},x_{2i})+\sum_{(\overline{ x}_{1i},\overline{x}_{2i})\in D^{-}}d_{M^{-1}}\big{(}\overline{x}_{1j}, \overline{x}_{2j}\big{)}, \tag{2}\]  
where the similar pair sample set \(D^{+}\) and the dissimilar pair sample set \(D^{-}\) can be expressed as  
\[\left\{ \begin{array}{l}D^{+}=\Big{\{}(x_{1i},x_{2i})\mid x_{1i},x_{2i}\in R ^{d}\text{ are in the same class },i=1,2,\ldots,n\Big{\}},\\ D^{-}=\Big{\{}\big{(}\overline{x}_{1i},\overline{x}_{2j})\mid\overline{x}_{1 j},\overline{x}_{2j}\in R^{d}\text{ are in different class },j=1,2,\ldots,m\Big{\}}.\end{array} \right. \tag{3}\]  
Although the GMML algorithm has some advantages, such as unconstrained convex objective function, closed-form solution and interpretability, and faster calculation speed [18], it is actually a linear learning method and does not work well to address nonlinear problems.