# Low-Rank Tensor Learning by Generalized Nonconvex Regularization

arXiv: https://arxiv.org/abs/2410.18402 · HTML: https://mlsanity.com/paper/2410.18402/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 70 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 18 of 18)

- [CNN Features off-the-shelf: an Astounding Baseline for Recognition](https://mlsanity.com/paper/1403.6382/) (2014)
- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods](https://mlsanity.com/paper/1602.02915/) (2016)
- [A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems](https://mlsanity.com/paper/1303.4434/) (2013)
- [Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography](https://mlsanity.com/paper/1307.5348/) (2013)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [A proximal difference-of-convex algorithm with extrapolation](https://mlsanity.com/paper/1612.06265/) (2016)
- [DuSK: A Dual Structure-preserving Kernel for Supervised Tensor Learning with Applications to Neuroimages](https://mlsanity.com/paper/1407.8289/) (2014)
- [A New Convex Relaxation for Tensor Completion](https://mlsanity.com/paper/1307.4653/) (2013)
- [DC approximation approaches for sparse optimization](https://mlsanity.com/paper/1407.0286/) (2014)
- [Sparse Nonnegative Tensor Factorization and Completion with Noisy Observations](https://mlsanity.com/paper/2007.10626/) (2020)
- [Towards Faster Rates and Oracle Property for Low-Rank Matrix Estimation](https://mlsanity.com/paper/1505.04780/) (2015)
- [A Support Tensor Train Machine](https://mlsanity.com/paper/1804.06114/) (2018)
- [A sparse semismooth Newton based proximal majorization-minimization algorithm for nonconvex square-root-loss regression problems](https://mlsanity.com/paper/1903.11460/) (2019)
- [Tensor Completion by Multi-Rank via Unitary Transformation](https://mlsanity.com/paper/2012.08784/) (2020)

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

- [Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2609.12843/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of low-rank tensor learning, where only a few of training samples are observed and the underlying tensor has a low-rank structure. The existing methods are based on the sum of nuclear norms of unfolding matrices of a tensor, which may be suboptimal. In order to explore the low-rankness of the underlying tensor effectively, we propose a nonconvex model based on transformed tensor nuclear norm for low-rank tensor learning. Specifically, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain to characterize the low-rankness of the underlying tensor. An error bound between the stationary point of the nonconvex model and the underlying tensor is established under restricted strong convexity on the loss function (such as least squares loss and logistic regression) and suitable regularity conditions on the nonconvex penalty function. By reformulating the nonconvex function into the difference of two convex functions, a proximal majorization-minimization (PMM) algorithm is designed to solve the resulting model. Then the global convergence and convergence rate of PMM are established under very mild conditions. Numerical experiments are conducted on tensor completion and binary classification to demonstrate the effectiveness of the proposed method over other state-of-the-art methods.