## 2 Related Work
### MTR Algorithm  
The existing MTR algorithms can be broadly classified into two categories: problem transformation methods and algorithm adaptation methods [24]. The former is to split the MTR problem into several single-target regression subproblems and solve them independently. The main drawback is that when predicting a single objective independently, the dependency relationship between different objectives is neglected, which affects the overall prediction performance. Therefore, in order to overcome these drawbacks, existing problem transformation methods such as stacked single-target (SST) borrow the idea of stacked generalization in multi-label classification to stack the prediction results of single-target regression [25]; random linear target combination (RLC) constructs new target variables by random linear combination of targets to improve the regression performance of the model when there are dependencies between targets [26]; ensemble of regressor chains (ERC) is based on the idea of classifier chains in multi-label learning, by constructing random target chains, linking single target models to make predictions in turn, and solving the chain order sensitivity problem by integration [25]. Although the above problem transformation methods perform well in some cases, their performance often depends on the chosen regression algorithm, and the complexity of the training process also causes a large computational cost when the data output is of high dimensionality or large size. Unlike the problem transformation approach, the basic idea of algorithm adaptation is to construct a structured model that predicts all outputs simultaneously, and the correlation between inputs and outputs as well as the correlation between different targets can be handled in a single framework so that the model can consider the effects of both factors during the learning process. Most of the existing algorithmic adaptation methods explore the inter-target correlations by imposing corresponding regularization constraints on the regression coefficient matrix, which are often limited to the linear regression case. By assuming a priori knowledge of the output structure, constraints such as low rank, sparsity or streamlining are imposed on the corresponding coefficient matrices [27]. However, these linear models cannot handle complex nonlinear input-output relationships, and the above prior knowledge assumptions are often only effective in domain-specific problems, which cannot automatically and adaptively extract correlation knowledge from datasets in different domains, resulting in a lack of model adaptability. In recent years, various nonlinear based algorithms have been proposed to better handle the nonlinear correlation between input and output in data. In the literature [28; 29; 30; 31], kernel methods are introduced to the MTR problem to decouple the highly complex nonlinear input-output relationships. In [31], a learning output kernels (OKL) algorithm is proposed to reveal the complex correlation structure in the output space through kernel methods to improve the learning performance of the algorithm. In [32], the coefficient matrix in multitask regression is reshaped into a vector to explore the correlation between task goals, which not only fails to distinguish the correlation between task goals and intra-task correlation, but also ignores the negative correlation between goals. However, the covariance matrix needs to satisfy the a priori assumption that the matrix variables obey normal distribution. In [33], the covariance structure of the potential model parameters and the conditional covariance structure between the outputs are learned from the data so that the algorithm does not require the corresponding a priori knowledge. In the literature [27], a multi-layer multi-target regression (MMR) algorithm is proposed by combining robust low-rank learning methods and kernel methods.