Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space
This addresses a specific tensor completion issue for applications like image and medical data analysis, but appears incremental as it builds on existing embedding and factorization techniques.
The paper tackled the problem of recovering missing continuous slices in tensor data, where existing methods like nuclear-norm and total-variation regularization often fail, by proposing a low-rank model in an embedded space using a multi-way delay-embedding transform and Tucker-based factorization, resulting in successful recovery for color images and fMRI data.
Let us consider a case where all of the elements in some continuous slices are missing in tensor data. In this case, the nuclear-norm and total variation regularization methods usually fail to recover the missing elements. The key problem is capturing some delay/shift-invariant structure. In this study, we consider a low-rank model in an embedded space of a tensor. For this purpose, we extend a delay embedding for a time series to a "multi-way delay-embedding transform" for a tensor, which takes a given incomplete tensor as the input and outputs a higher-order incomplete Hankel tensor. The higher-order tensor is then recovered by Tucker-based low-rank tensor factorization. Finally, an estimated tensor can be obtained by using the inverse multi-way delay embedding transform of the recovered higher-order tensor. Our experiments showed that the proposed method successfully recovered missing slices for some color images and functional magnetic resonance images.