This paper presents mixture matrix completion (MMC) as a novel machine learning tool for learning from low-rank and incomplete data. MMC is a problem that is similar to the problem of subspace clustering with missing data, but is more difficult. Specifically, in MMC the data is assumed to lie in a union of (unknown) low-dimensional subspaces, but the data is not fully observed: only a few entries of each data point are observed, and (unlike subspace clustering with missing data) there is no information as to which entries correspond to the same point. Therefore, one would need to estimate the assignment of entries to data points, the assignment of data points to subspaces, the missing entries, and the subspaces altogether. The major contribution of this paper is the introduction of the MMC problem, a theoretical analysis for when the problem of MMC is well-defined, and an alternating estimation algorithm for solving the MMC problem.

Human intelligence seems so mysterious that we have not successfully understood its foundation until now. Here, I want to present a basic cognitive process, motion mapping cognition (MMC), which should be a nondecomposable primary function in human vision. Wherein, I point out that, MMC process can be used to explain most of human visual functions in fundamental, but can not be effectively modelled by traditional visual processing ways including image segmentation, object recognition, object tracking etc. Furthermore, I state that MMC may be looked as an extension of Chen's theory of topological perception on human vision, and seems to be unsolvable using existing intelligent algorithm skills. Finally, along with the requirements of MMC problem, an interesting computational model, quantized topological matching principle can be derived by developing the idea of optimal transport theory. Above results may give us huge inspiration to develop more robust and interpretable machine vision models.