Generalized Canonical Polyadic Tensor Decompositions with General Symmetry

Canonical Polyadic (CP) tensor decomposition is a workhorse algorithm for discovering underlying low-dimensional structure in tensor data. This is accomplished in conventional CP decomposition by fitting a low-rank tensor to data with respect to the least-squares loss. Generalized CP (GCP) decompositions generalize this approach by allowing general loss functions that can be more appropriate, e.g., to model binary and count data or to improve robustness to outliers. However, GCP decompositions do not explicitly account for any symmetry in the tensors, which commonly arises in modern applications. For example, a tensor formed by stacking the adjacency matrices of a dynamic graph over time will naturally exhibit symmetry along the two modes corresponding to the graph nodes. In this paper, we develop a symmetric GCP (SymGCP) decomposition that allows for general forms of symmetry, i.e., symmetry along any subset of the modes. SymGCP accounts for symmetry by enforcing the corresponding symmetry in the decomposition. We derive gradients for SymGCP that enable its efficient computation via all-at-once optimization with existing tensor kernels. The form of the gradients also leads to various stochastic approximations that enable us to develop stochastic SymGCP algorithms that can scale to large tensors. We demonstrate the utility of the proposed SymGCP algorithms with a variety of experiments on both synthetic and real data.