# Subspace Learning and Imputation for Streaming Big Data Matrices and Tensors

arXiv: https://arxiv.org/abs/1404.4667 · HTML: https://mlsanity.com/paper/1404.4667/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2014.

## Built on by (most cited first, 15 of 15)

- [Unraveling Attention via Convex Duality: Analysis and Interpretations of Vision Transformers](https://mlsanity.com/paper/2205.08078/) (2022)
- [An AO-ADMM approach to constraining PARAFAC2 on all modes](https://mlsanity.com/paper/2110.01278/) (2021)
- [Robust Factorization of Real-world Tensor Streams with Patterns, Missing Values, and Outliers](https://mlsanity.com/paper/2102.08466/) (2021)
- [Learning Spatial-Temporal Regularized Tensor Sparse RPCA for Background Subtraction](https://mlsanity.com/paper/2309.15576/) (2023)
- [Multi-version Tensor Completion for Time-delayed Spatio-temporal Data](https://mlsanity.com/paper/2105.05326/) (2021)
- [Streaming data preprocessing via online tensor recovery for large environmental sensor networks](https://mlsanity.com/paper/2109.00596/) (2021)
- [Application Performance Modeling via Tensor Completion](https://mlsanity.com/paper/2210.10184/) (2022)
- [Low-Rank Adaptation Redux for Large Models](https://mlsanity.com/paper/2604.21905/) (2026)
- [Patch Tracking-based Streaming Tensor Ring Completion for Visual Data Recovery](https://mlsanity.com/paper/2105.14620/) (2021)
- [Streaming Generalized Canonical Polyadic Tensor Decompositions](https://mlsanity.com/paper/2110.14514/) (2021)
- [GOCPT: Generalized Online Canonical Polyadic Tensor Factorization and Completion](https://mlsanity.com/paper/2205.03749/) (2022)
- [Robust Data Clustering with Outliers via Transformed Tensor Low-Rank Representation](https://mlsanity.com/paper/2307.09055/) (2023)
- [ACT-Tensor: Tensor Completion Framework for Financial Dataset Imputation](https://mlsanity.com/paper/2508.01861/) (2025)
- [A Spatio-Temporal Online Robust Tensor Recovery Approach for Streaming Traffic Data Imputation](https://mlsanity.com/paper/2511.01267/) (2025)
- [Online TT-ALS for Streaming Tensor Decomposition with Incremental Orthogonalization](https://mlsanity.com/paper/2606.31061/) (2026)

## Abstract (arXiv)

Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for em real-time processing of streaming data pose major challenges to this end. In this context, the present paper permeates benefits from rank minimization to scalable imputation of missing data, via tracking low-dimensional subspaces and unraveling latent (possibly multi-way) structure from incomplete streaming data. For low-rank matrix data, a subspace estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the nuclear norm. After recasting the non-separable nuclear norm into a form amenable to online optimization, real-time algorithms with complementary strengths are developed and their convergence is established under simplifying technical assumptions. In a stationary setting, the asymptotic estimates obtained offer the well-documented performance guarantees of the em batch nuclear-norm regularized estimator. Under the same unifying framework, a novel online (adaptive) algorithm is developed to obtain multi-way decompositions of low-rank tensors with missing entries, and perform imputation as a byproduct. Simulated tests with both synthetic as well as real Internet and cardiac magnetic resonance imagery (MRI) data confirm the efficacy of the proposed algorithms, and their superior performance relative to state-of-the-art alternatives.