lasso最小角回归算法推导
Title: Derivation of the Lasso Least Angle Regression Algorithm
The Lasso Least Angle Regression (LARS) algorithm is a powerful tool in statistical learning, combining the principles of both least squares regression and the Lasso regularization technique. This algorithm provides a computationally efficient means to estimate regression coefficients, especially in high-dimensional datasets.
拉索最小角回归（LARS）算法是统计学习中的一个强大工具，它结合了最小二乘回归和拉索正则化技术的原理。该算法提供了一种计算效率高的方法，用于估计回归系数，特别是在高维数据集中。
At its core, the LARS algorithm aims to find a set of coefficients that minimize the residual sum of squares, subject to a constraint on the sum of the absolute values of the coefficients. This constraint introduces sparsity, allowing for the selection of only the most relevant predictors.
在核心上，LARS算法旨在到一组系数，这些系数在最小化残差平方和的同时，还要满足系数绝对值之和的约束。这种约束引入了稀疏性，使得只有最重要的预测因子被选中。
正则化最小二乘问题
The algorithm proceeds iteratively, at each step updating the coefficients by moving along the direction that most closely aligns with the residuals. This direction is determined by the least angle between the current residual and the predictors.
该算法进行迭代处理，每一步都通过沿着与残差最对齐的方向更新系数。这个方向由当前残差和预测因子之间的最小角度决定。
As the algorithm progresses, it adds predictors to the model one at a time, in the order of their correlation with the residual. This process continues until all predictors have been considered or until the desired level of regularization is achieved.
随着算法的进展，它按预测因子与残差的相关性顺序，一次添加一个预测因子到模型中。这个过程一直持续到所有预测因子都被考虑，或者达到所需的正则化水平为止。
The LARS algorithm offers a balance between bias and variance, making it suitable for a wi
de range of regression tasks, especially those involving collinear predictors or where the number of predictors exceeds the number of observations.
LARS算法在偏差和方差之间到了平衡，使其适用于广泛的回归任务，尤其是那些涉及共线性预测因子或预测因子数量超过观测值数量的情况。
In conclusion, the Lasso Least Angle Regression algorithm provides a computationally efficient and robust method for estimating regression coefficients in high-dimensional settings, leveraging the benefits of both least squares regression and Lasso regularization.
综上所述，拉索最小角回归算法在高维设置中提供了一种计算效率高且稳健的回归系数估计方法，它充分利用了最小二乘回归和拉索正则化的优点。

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