DLAT: A Unified View of Robust Decomposition in Low-rank + Additive Tensors

1. Introduction

Problem formulations of robust subspace learning/tracking by decomposition into low-rank plus additive tensors show a suitable framework for image and video analysis.

2. Robust Decomposition in Low-rank + Additive Matrices: Principle

All the decompositions in these different problem formulations of robust subspace learning/tracking can be considered in a unified view that we called Decomposition into Low-rank plus Additive Tensors (DLAT). Thus, all the decompositions can be written in a general formulation as follows:

• The first tensor M₁ is a low-rank tensor L.

• The second tensor M₂ is an unconstrained (residual) tensor S in RLRM and RMC, and a sparse tensor S in RPCA, RNMF, RSR and ST.

• The third tensor M₃ is generally the noise E. The noise can be modeled by a Gaussian, a mixture of Gaussians (MoG) or a Laplacian distribution. E can capture turbulences in the case of background/foreground separation .

Practically, the decomposition is implicit when K=1. It is the degenerated case for problem formulations in their basic formulation which are not robust because there are no constraints on the tensor S=A-L.

The decomposition is explicit when K =2 and we have A=L+S. It is the for problem formulations in their robust formulation.

In the case of K=3, the decomposition is A = L+S +E. This explicit decomposition is called ”stable” decomposition as it separates the outliers in S and the noise in E.

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As this website gives many information that come from my research, please cite my following survey papers:

T. Bouwmans, A. Sobral, S. Javed, S. Jung, E. Zahzah, "Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset", Computer Science Review, Volume 23, pages 1-71, February 2017. [pdf]

A. Sobral, C. Baker, T. Bouwmans, E. Zahzah, “Incremental and Multi-feature Tensor Subspace Learning applied for Background Modeling and Subtraction”, International Conference on Image Analysis and Recognition, ICIAR 2014, October 2014.

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