# Complexity and computation for the spectral norm and nuclear norm of order three tensors with one fixed dimension

arXiv: https://arxiv.org/abs/2212.14775 · HTML: https://mlsanity.com/paper/2212.14775/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Dec 2022.

## Built on by (most cited first, 1 of 1)

- [Tensorial template matching for fast cross-correlation with rotations and its application for tomography](https://mlsanity.com/paper/2408.02398/) (2024)

## Abstract (arXiv)

The recent decade has witnessed a surge of research in modelling and computing from two-way data (matrices) to multiway data (tensors). However, there is a drastic phase transition for most tensor optimization problems when the order of a tensor increases from two (a matrix) to three: Most tensor problems are NP-hard while that for matrices are easy. It triggers a question on where exactly the transition occurs. The paper aims to study this kind of question for the spectral norm and the nuclear norm. Although computing the spectral norm for a general ℓ× m× n tensor is NP-hard, we show that it can be computed in polynomial time if ℓ is fixed. This is the same for the nuclear norm. While these polynomial-time methods are not implementable in practice, we propose fully polynomial-time approximation schemes (FPTAS) for the spectral norm based on spherical grids and for the nuclear norm with further help of duality theory and semidefinite optimization. Numerical experiments on simulated data show that our FPTAS can compute these tensor norms for small ℓ ≤ 6 but large m, n≥50. To the best of our knowledge, this is the first method that can compute the nuclear norm of general asymmetric tensors. Both our polynomial-time algorithms and FPTAS can be extended to higher-order tensors as well.