# Convex Tensor Decomposition via Structured Schatten Norm Regularization

arXiv: https://arxiv.org/abs/1303.6370 · HTML: https://mlsanity.com/paper/1303.6370/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2013.

## Built on by (most cited first, 8 of 8)

- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [GaitGCI: Generative Counterfactual Intervention for Gait Recognition](https://mlsanity.com/paper/2306.03428/) (2023)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Generating Structured Adversarial Attacks Using Frank-Wolfe Method](https://mlsanity.com/paper/2102.07360/) (2021)

## Abstract (arXiv)

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.