We investigate the theoretical foundations of a recently introduced entropy-based formulation of weighted least squares for the approximation of overdetermined linear systems, motivated by robust data fitting in the presence of sparse gross errors. The weight vector is interpreted as a discrete probability distribution and is determined by maximizing Shannon entropy under normalization and a prescribed mean squared error (MSE) constraint. Unlike classical ordinary least squares, where the error level is an output of the minimization process, here the MSE value plays the role of a control parameter, and entropy selects the least biased weight distribution achieving the prescribed accuracy. The resulting optimization problem is nonconvex due to the nonlinear coupling between the weights and the solution induced by the residual constraint. We analyze the associated optimality system and characterize stationary points through first- and second-order conditions. We prove the existence and local uniqueness of a smooth branch of entropy-maximizing configurations emanating from the ordinary least squares solution and establish its global continuation under suitable nondegeneracy conditions. Furthermore, we investigate the asymptotic regime as the prescribed MSE tends to zero and show that, under appropriate assumptions, the limiting configuration concentrates on a largest subset of data consistent with the linear model, thus suppressing the influence of outliers. Two numerical experiments illustrate the theoretical findings and confirm the robustness properties of the method.

翻译：我们研究了一种最近提出的基于熵的加权最小二乘公式的理论基础，该方法用于超定线性系统的逼近，其动机是在存在稀疏粗大误差情况下的鲁棒数据拟合。权重向量被解释为离散概率分布，并通过在归一化和给定均方误差约束下最大化香农熵来确定。与经典普通最小二乘法（其中误差水平是极小化过程的输出）不同，此处均方误差值扮演控制参数的角色，而熵则选择实现规定精度的最无偏权重分布。由于残差约束引起的权重与解之间的非线性耦合，所得优化问题是非凸的。我们分析了相关的优化系统，并通过一阶和二阶条件刻画了驻点特性。我们证明了从普通最小二乘解出发存在一条熵最大化配置的光滑分支，并建立了其在适当非退化条件下的全局延拓。此外，我们研究了当规定均方误差趋于零时的渐近状态，并证明在适当假设下，极限配置会集中于与线性模型一致的最大数据子集，从而抑制异常值的影响。两个数值实验说明了理论结果，并验证了该方法的鲁棒性。