31 research outputs found
Convex Tensor Decomposition via Structured Schatten Norm Regularization
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.Comment: 12 pages, 3 figure
A dual framework for low-rank tensor completion
One of the popular approaches for low-rank tensor completion is to use the
latent trace norm regularization. However, most existing works in this
direction learn a sparse combination of tensors. In this work, we fill this gap
by proposing a variant of the latent trace norm that helps in learning a
non-sparse combination of tensors. We develop a dual framework for solving the
low-rank tensor completion problem. We first show a novel characterization of
the dual solution space with an interesting factorization of the optimal
solution. Overall, the optimal solution is shown to lie on a Cartesian product
of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian
optimization framework for proposing computationally efficient trust region
algorithm. The experiments illustrate the efficacy of the proposed algorithm on
several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing
Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on
Synergies in Geometric Data Analysis 201