  Spectral clustering and co-clustering are well-known techniques in data
analysis, and recent work has extended spectral clustering to square, symmetric
tensors and hypermatrices derived from a network. We develop a new tensor
spectral co-clustering method that applies to any non-negative tensor of data.
The result of applying our method is a simultaneous clustering of the rows,
columns, and slices of a three-mode tensor, and the idea generalizes to any
number of modes. The algorithm we design works by recursively bisecting the
tensor into two pieces. We also design a new measure to understand the role of
each cluster in the tensor. Our new algorithm and pipeline are demonstrated in
both synthetic and real-world problems. On synthetic problems with a planted
higher-order cluster structure, our method is the only one that can reliably
identify the planted structure in all cases. On tensors based on n-gram text
data, we identify stop-words and semantically independent sets; on tensors from
an airline-airport multimodal network, we find worldwide and regional
co-clusters of airlines and airports; and on tensors from an email network, we
identify daily-spam and focused-topic sets.
