Convex Tensor Decomposition via Structured Schatten Norm Regularization
This work provides theoretical insights for researchers in machine learning and statistics dealing with tensor decomposition, but it is incremental as it builds on existing norms and focuses on analysis rather than introducing a new method.
The paper tackles the problem of tensor decomposition by analyzing structured Schatten norms, showing theoretically that the 'latent' approach outperforms the 'overlapped' one when the true tensor is low-rank in a specific mode, achieving performance comparable to knowing the mode with the smallest rank, and confirms this with numerical simulations predicting scaling behavior of mean squared error.
We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behavior of the mean squared error.