The nonlinearity of power flow (PF) equations challenges the analysis and optimization of power systems. Both model-based and data-driven approach was recently applied to linearize the PF equations. The data-driven approach relies heavily on the quality of the measurement data, where measurement noise may cause large modeling errors. This paper tackles the challenges of the hidden measurement noise in the data-driven PF linearization problem. We transform the problem into a regression model where the structure of the AC power flow equations is exploited. Jacobian matrix guided constraints are added to shrink the search space greatly. This regression model is formulated as three linearly constrained quadratic programming problems and is solved in an iterative manner. The effectiveness of the proposed approach is demonstrated through case studies on several IEEE standard test systems and a practical provincial system.