# Robust Low-rank Tensor Train Recovery arXiv: https://arxiv.org/abs/2410.15224 · HTML: https://mlsanity.com/paper/2410.15224/ ## Key facts - Cited by 2 papers in the map (data through 14 Oct 2026). - Builds on 39 of its 90 references that are in the map. - First posted on arXiv 19 Oct 2024. ## Builds on (most cited first, 30 of 39) - [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016) - [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014) - [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015) - [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021) - [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020) - [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017) - [Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization: A Survey](https://mlsanity.com/paper/1507.01030/) (2015) - [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020) - [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019) - [Algorithmic Regularization in Model-free Overparametrized Asymmetric Matrix Factorization](https://mlsanity.com/paper/2203.02839/) (2022) - [The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing](https://mlsanity.com/paper/2302.01186/) (2023) - [Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence](https://mlsanity.com/paper/1904.10020/) (2019) - [Multilinear tensor regression for longitudinal relational data](https://mlsanity.com/paper/1412.0048/) (2014) - [Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number](https://mlsanity.com/paper/2010.13364/) (2020) - [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021) - [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017) - [Non-Convex Projected Gradient Descent for Generalized Low-Rank Tensor Regression](https://mlsanity.com/paper/1611.10349/) (2016) - [Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery](https://mlsanity.com/paper/2109.11154/) (2021) - [Global Convergence of Sub-gradient Method for Robust Matrix Recovery: Small Initialization, Noisy Measurements, and Over-parameterization](https://mlsanity.com/paper/2202.08788/) (2022) - [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013) - [Parsimonious Tensor Response Regression](https://mlsanity.com/paper/1501.07815/) (2015) - [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021) - [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022) - [Sparse and Low-rank Tensor Estimation via Cubic Sketchings](https://mlsanity.com/paper/1801.09326/) (2018) - [Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption](https://mlsanity.com/paper/2305.04080/) (2023) - [Nonconvex Low-Rank Matrix Recovery with Arbitrary Outliers via Median-Truncated Gradient Descent](https://mlsanity.com/paper/1709.08114/) (2017) - [Fast and robust quantum state tomography from few basis measurements](https://mlsanity.com/paper/2009.08216/) (2020) - [Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance](https://mlsanity.com/paper/2012.10249/) (2020) - [Quantum State Tomography for Matrix Product Density Operators](https://mlsanity.com/paper/2306.09432/) (2023) - [Iterative Hard Thresholding for Low CP-rank Tensor Models](https://mlsanity.com/paper/1908.08479/) (2019) ## Built on by (most cited first, 2 of 2) - [Efficient Low-Tubal-Rank Tensor Estimation via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2512.07490/) (2025) - [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026) ## Abstract (arXiv) Tensor train (TT) decomposition represents an N-order tensor using O(N) matrices (i.e., factors) of small dimensions, achieved through products among these factors. Due to its compact representation, TT decomposition has found wide applications, including various tensor recovery problems in signal processing and quantum information. In this paper, we study the problem of reconstructing a TT format tensor from measurements that are contaminated by outliers with arbitrary values. Given the vulnerability of smooth formulations to corruptions, we use an ℓ1 loss function to enhance robustness against outliers. We first establish the ℓ1/ℓ2-restricted isometry property (RIP) for Gaussian measurement operators, demonstrating that the information in the TT format tensor can be preserved using a number of measurements that grows linearly with N. We also prove the sharpness property for the ℓ1 loss function optimized over TT format tensors. Building on the ℓ1/ℓ2-RIP and sharpness property, we then propose two complementary methods to recover the TT format tensor from the corrupted measurements: the projected subgradient method (PSubGM), which optimizes over the entire tensor, and the factorized Riemannian subgradient method (FRSubGM), which optimizes directly over the factors. Compared to PSubGM, the factorized approach FRSubGM significantly reduces the memory cost at the expense of a slightly slower convergence rate. Nevertheless, we show that both methods, with diminishing step sizes, converge linearly to the ground-truth tensor given an appropriate initialization, which can be obtained by a truncated spectral method.