# Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

arXiv: https://arxiv.org/abs/1511.01245 · HTML: https://mlsanity.com/paper/1511.01245/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2015.

## Built on by (most cited first, 13 of 13)

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- [Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guarantee](https://mlsanity.com/paper/2203.16328/) (2022)
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- [Learning Temporal Distribution and Spatial Correlation Towards Universal Moving Object Segmentation](https://mlsanity.com/paper/2304.09949/) (2023)
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- [Working memory inspired hierarchical video decomposition with transformative representations](https://mlsanity.com/paper/2204.10105/) (2022)
- [Robust Foreground-Background Separation for Severely-Degraded Videos Using Convolutional Sparse Representation Modeling](https://mlsanity.com/paper/2506.17838/) (2025)
- [Physics-Guided Dual Implicit Neural Representations for Source Separation](https://mlsanity.com/paper/2507.05249/) (2025)
- [Robust Dual-Graph Regularized Moving Object Detection](https://mlsanity.com/paper/2204.11939/) (2022)
- [Human Motion Detection Based on Dual-Graph and Weighted Nuclear Norm Regularizations](https://mlsanity.com/paper/2304.04879/) (2023)
- [Optimal Bound for PCA with Outliers using Higher-Degree Voronoi Diagrams](https://mlsanity.com/paper/2408.06867/) (2024)

## Abstract (arXiv)

Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.