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2) Low-rank tensor completion
Like Sudokou? Try doing it in 3 or 4-D\! The problem of reconstructing a multi-dimensional array (or tensor) from a set of incomplete data arises in many areas of signal and image processing, communications, control theory, systems biology, etc. Examples include video signals (3-D array), EEG signals (3-D array), social networks, etc. In many applications the incomplete data are (possibly noisy) random samples of the entries of the array. Without any other restriction on the data, such problems do not have a unique answer; however, if there are only a few factors that influence the relationship between the entries, the tensor is low-rank and under some condition unique solutions can be obtained. This situation is very common in many various applications including the ones mentioned above. Many theoretical results and practical algorithms have been introduced for 2-D arrays (matrices). The goal of this project is to generalize these results to higher dimensional arrays.