An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal consistency between the different measurement times. The strongest consistency can be achieved by reconstructing data directly in phase space, the space of positions and velocities. However, this space is usually too high-dimensional for feasible computations. We introduce a novel dimension reduction technique, based on projections of phase space onto lower-dimensional subspaces, which provably circumvents this curse of dimensionality: Indeed, in the exemplary framework of superresolution we prove that known exact reconstruction results stay true after dimension reduction, and we additionally prove new error estimates of reconstructions from noisy data in optimal transport metrics which are of the same quality as one would obtain in the non-dimension-reduced case.

翻译：现代反问题中的一个重要主题是从有限次测量中重构时间依赖数据。为在此类场景中获得令人满意的重构结果，必须充分利用不同测量时间点间的时间一致性。通过直接在相空间（位置与速度构成的空间）中重构数据可获得最强一致性，然而该空间因维度过高而难以实现可行计算。我们提出一种基于相空间向低维子空间投影的新型降维技术，该技术可严格规避这一维度灾难：事实上，在超分辨率这一典型框架下，我们证明了已知的精确重构结果在降维后依然成立，并进一步在最优传输度量下证明了含噪数据重构的新误差估计，其精度与非降维情形下的结果相当。