Relations of the Nuclear Norms of a Tensor and its Matrix Flattenings

2014/12/08 by Shenglong Hu, Hu, Shenglong · 1 citation
Mathematics · Engineering · #Tensor decomposition and applications #Elasticity and Material Modeling #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1412.2443

Abstract

For a 3-tensor of dimensions I₁× I₂× I₃, we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm, and which in turn is upper bounded by √min\I_i : i≠ j\ times the nuclear norm of the matrix flattening in mode j for all j=1,2,3. The results can be generalized to N-tensors with any N≥ 3. Both the lower and upper bounds for the tensor nuclear norm are sharp in the case N=3. A computable criterion for the lower bound being tight is given as well.

Citations

• Most Tensor Problems Are NP-Hard

Cited by

• Causal Inference on Sequential Treatments via Tensor Completion

Related

• Nuclear Norm of Higher-Order Tensors
• Spectral Norm and Nuclear Norm of a Third Order Tensor
• Tensor Norm, Cubic Power and Gelfand Limit
• On the Nuclear Norm and the Singular Value Decomposition of Tensors
• Symmetric Tensor Nuclear Norms