Least squares regression (LSR) has attracted widespread attention in the fields of statistics, machine learning, and pattern recognition. However, it utilizes strict zero-one regression targets, which leads to inferior performance on classification tasks. Furthermore, LSR ignores the local manifold structures of data and lacks robustness. To address these issues, this paper proposes a general regression framework called RLRR, where a low-rank constraint is imposed on regression matrices to explore the underlying correlation structures of classes. Strict zero-one regression targets are redirected to more feasible variable matrices for the purpose of margin amplification of different classes. Additionally, rather than using a pre-constructed weighted graph, the proposed framework dynamically updates the neighborhood structures of data to preserve original manifold structures. By utilizing this framework as a general platform, we developed two dynamic neighborhood-structure-based regression models called RLRRM and RLRRP. RLRRM integrates a reconstruction error minimization term into the proposed RLRR framework, whereas RLRRP aims to preserve the local geometric structures of data in a low-dimensional subspace. Both RLRRM and RLRRP use the ${\ell }_{2,1}$-norm penalty to replace the traditional F-norm penalty for the projection matrix for the sake of self-adaptive feature selection. Instead of directly solving the resultant optimization problems with non-convex constraints, we adopt the variable-splitting and penalty techniques to derive an equivalent solution. Analysis of the corresponding convergence and computational complexity characteristics is also presented. Extensive experiments on several well-known datasets demonstrate the promising performance of the proposed models.

最小二乘回归（LSR）在统计，机器学习和模式识别领域引起了广泛的关注。但是，它使用严格的零一回归目标，这导致分类任务的性能较差。此外，LSR忽略了数据的局部流形结构，并且缺乏鲁棒性。为了解决这些问题，本文提出了一个称为RLRR的通用回归框架，其中对回归矩阵施加了低秩约束，以探索类的底层相关结构。严格的零一回归目标将重定向到更可行的变量矩阵，以扩大不同类别的边距。此外，与其使用预先构造的加权图，提出的框架动态更新数据的邻域结构以保留原始的流形结构。通过将该框架用作通用平台，我们开发了两个基于动态邻域结构的回归模型，称为RLRRM和RLRRP。RLRRM将重构误差最小化术语整合到了所提出的RLRR框架中，而RLRRP旨在保留低维子空间中数据的局部几何结构。RLRRM和RLRRP都使用${\ell }_{2，\mathrm{1个}}$-范数罚分以取代传统的F-范数罚分用于投影矩阵，以便进行自适应特征选择。代替直接解决具有非凸约束的结果优化问题，我们采用变量分解和惩罚技术来得出等效解。还介绍了相应的收敛性和计算复杂度特征的分析。在几个著名的数据集上的大量实验证明了所提出模型的有希望的性能。