A novel and concise Shu’s two–point method for fast least squares and regression analysis and its straightedge–and–compass construction

The method of least squares is a core technique widely used in statistics and machine learning to determine the best-fit line through a set of observed data points. Traditional implementations rely on matrix inversion and decomposition; these methods are computationally intensive and inefficient for large-scale data environments. This paper presents a novel, computationally efficient, and easy-to-understand two-point method for fast least squares and regression analysis. This approach determines the best-fit line by joining the geometric center and the mass center of the data, eliminating the need to solve a large system of equations; the paper demonstrates that this method is an algebraically equivalent reconstruction of the method of least squares. Crucially, this approach requires only basic arithmetic operations (addition, subtraction, multiplication, and division), making it highly suitable for teaching basic statistics and machine learning in primary and secondary schools. This stands in stark contrast to the current schoolbook way of least squares and regression analysis based on classical Gaussian approach in statistics, as well as gradient descent in machine learning, all requiring university-level advanced calculus knowledge. Furthermore, this two-point method can be implemented using straightedge-and-compass construction and extended to manage measurement errors involving multiple independent and dependent variables. Its geometric intuitiveness and computational simplicity make it well-suited for big data applications in machine learning, such as knowledge-intensive engineering informatics applications.