Smooth tensor estimation with unknown permutations

Higher-order tensor datasets are rising ubiquitously in modern data science applications, for instance, recommendation systems (Baltrunas et al., 2011; Bi et al., 2018), social networks (Bickel and Chen, 2009), genomics (Hore et al., 2016), and neuroimaging (Zhou et al., 2013). Tensor provides effective representation of data structure that classical vector-and matrix-based methods fail to capture. One example is music recommendation system (Baltrunas et al., 2011) that records ratings of songs from users on various contexts. This three-way tensor of user song context allows us to investigate interactions of users and songs in a context-specific manner. Another example is network dataset that records the connections among a set of nodes. Pairwise interactions are often insufficient to capture the complex relationships, whereas multi-way interactions improve the understanding of networks in molecular system (Young et al., 2018) and social networks (Han et al., 2020). In both examples, higher-order tensors represent multi-way interactions in an efficient way. Tensor estimation problem cannot be solved without imposing structures. An appropriate reordering of tensor entries often provides effective representation of the hidden salient structure.