一种基于高等代数的线性最小二乘问题的解法

A Solution to the Linear Least Squares Problem Based on Advanced Algebra

自从19世纪初勒让德提出线性最小二乘问题以来,诸多数学家已经就它的意义、误差分析及解法进行了广泛的研究,但基于高等代数理论的解法在许多资料上只有零散的讨论,为此提出了一种基于高等代数理论的线性最小二乘问题的解法。基于严格的证明,得到了适用于数据集为列满秩和非列满秩时的解法,并进一步推算出解集中的最优最小二乘解。实验证明该算法确实可以得到最优最小二乘解,并且在数据集属性数较少的情况下优于负梯度下降法。

Since Legendre proposed the linear least squares problem in the early 19 th century,many mathematicians have conducted extensive research on its significance,error analysis,and solutions,but solutions based on higher algebra theory have only been scattered discussion on many materials.To this end,a solution to the linear least squares problem based on higher algebra theory is proposed.Based on strict proofs,a solution suitable for the data set with column full rank and non-column full rank is obtained,and the optimal least squares solution in the solution set is further deduced.Experiments show that the algorithm can indeed obtain the optimal least squares solution,and it is better than the negative gradient descent method when the number of attributes in the data set is small.