Regularized transport between singular covariance matrices
Regularized transport between singular covariance matrices
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奇异协方差矩阵之间的正则化传输
DOI：
10.1109/tac.2020.3017714
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发表时间：
2020
影响因子：
6.8
通讯作者：
Pavon, Michele
中科院分区：
文献类型：
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作者：
Ciccone, Valentina;Chen, Yongxin;Georgiou, Tryphon T.;Pavon, Michele
我们考虑了一个线性随机系统在有限时间内在两个端点退化的高斯分布之间的操纵问题。这说明了某些状态条目而不是所有状态条目在初始时间和最终时间不确定的情况。这个问题带来了很大的技术挑战，因为终端状态协方差的奇异性导致控制在最后时刻变得无界。因此，熵内插(薛定谔桥)是由扩散过程提供的，而扩散过程不是有限的能量，从而将这种情况置于当前大多数理论之外。在这篇文章中，我们证明了可行的插值法可以作为非退化情形下已有结果的极限情形，并且它可以用闭合的形式表示。此外，我们还证明了这种插值法属于不受控演化的同一互补类。通过这样做，我们还强调了问题的时间对称性，对比了正向和反向时间方向的对偶公式，其中在每个方向上，随着时间接近终点(分别在正向和反向时间方向)，控制变得无界。
We consider the problem of steering a linear stochastic system between two endpointdegenerateGaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial,, and final time,. This problem entails nontrivial technical challenges, as the singularity of terminal state covariance causes the control to grow unbounded at the final time. Consequently, the entropic interpolation (Schrödinger bridge) is provided by a diffusion process, which is notfinite energy, thereby placing this case outside of most of the current theory. In this article, we show that a feasible interpolation can be derived as a limiting case of earlier results for nondegenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the samereciprocal classof the uncontrolled evolution. By doing so, we also highlight a time symmetry of the problem, contrasting dual formulations in the forward and reverse time directions, where in each, the control grows unbounded as time approaches the endpoint (in the forward and reverse time direction, respectively).