In this paper, we propose an information geometry (IG) framework to solve the standard linear regression problem. The proposed framework is an extension of the one for computing the mean of complex multivariate Gaussian distribution. By applying the proposed framework, the information geometry approach (IGA) and the approximate information geometry approach (AIGA) for basis pursuit de-noising (BPDN) in standard linear regression are derived. The framework can also be applied to other standard linear regression problems. With the transformations of natural and expectation parameters of Gaussian distributions, we then show the relationship between the IGA and the message passing (MP) algorithm. Finally, we prove that the AIGA is equivalent to the approximate message passing (AMP) algorithm. These intrinsic results offer a new perspective for the AMP algorithm, and clues for understanding and improving stochastic reasoning methods.

翻译：本文提出了一种信息几何（IG）框架来解决标准线性回归问题。该框架是对计算复多元高斯分布均值的现有框架的扩展。通过应用所提出的框架，推导出了用于标准线性回归中基追踪去噪（BPDN）的信息几何方法（IGA）和近似信息几何方法（AIGA）。该框架也可应用于其他标准线性回归问题。借助高斯分布的自然参数和期望参数的变换，我们揭示了IGA与消息传递（MP）算法之间的关系。最后，我们证明了AIGA等价于近似消息传递（AMP）算法。这些本质结果为AMP算法提供了新的视角，并为理解和改进随机推理方法提供了线索。