Riccati 类矩阵微分方程的低秩加对角近似
Low-rank plus diagonal approximations for Riccati-like matrix differential equations
AI总结：
本文提出用“低秩加对角”矩阵近似 Riccati 类矩阵微分方程的解，通过动态低秩投影实现线性存储与计算成本，并保证矩阵可逆，成功应用于 Wasserstein 梯度流和 Kalman 滤波。
摘要：
我们考虑计算随时间变化的 d×d 大型半正定（PSD）矩阵的可处理近似问题，这些矩阵定义为矩阵微分方程的解。我们提出使用“低秩加对角”PSD 矩阵作为近似，其存储成本与高维数 d 呈线性关系。为了约束微分方程的解保持在该子集内，我们遵循动态低秩近似的方法，在每一时刻将导数投影到该子集的切空间上。我们推导了投影的闭式公式，并证明经过一些处理后，它可以以与 d 呈线性关系的数值成本进行计算，从而实现可处理的实现。与之前基于纯低秩近似的方法相比，对角项的加入使得我们的近似可以成为可逆矩阵，并且可以以与 d 呈线性关系的成本进行求逆。我们将该技术应用于 Riccati 类方程，然后应用于两个具体问题。首先是对我们近期提出的用于近似贝叶斯推断中后验分布高斯近似的 Wasserstein 梯度流进行低秩近似，其次是针对高维系统的一种新型 Kalman 滤波低秩近似。数值模拟验证了这些结果。
英文摘要：
We consider the problem of computing tractable approximations of time-dependent d x d large positive semi-definite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use "low-rank plus diagonal" PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension d. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closed-form formula for the projection, and show that after some manipulations it can be computed with a numerical cost being linear in d, allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices, that can moreover be inverted with linear cost in d. We apply the technique to Riccati-like equations, then to two particular problems. Firstly a low-rank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference, and secondly a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results.
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