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Towards Multi-Mode Outlier Robust Tensor Ring Decomposition
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Proposes ORTRD, a convex tensor-ring robust decomposition with a multi-mode tensor group sparsity norm to recover low-rank tensors corrupted by outliers spanning multiple modes, with ADMM solver and error bounds.
Problem
Robust tensor decomposition typically assumes outliers lie in a single mode, which fails when outliers corrupt multiple modes of high-order tensors simultaneously.
Approach
Model the outlier tensor as a sum of latent components, each group-sparse along one mode, penalized by a new multi-mode tensor group sparsity (MTGS) norm, combined with a tensor ring nuclear norm for the low-rank part; includes non-asymptotic estimation error bounds.
Key results
- On four hyperspectral videos with outliers in up to four modes, the proposed method achieves RSE around 0.045 while ORPCA rises to 0.6385 and RTRC to 0.0603 on the bus video.
- On light field images, the proposed method has the lowest RSE in all tested images and settings, and reaches about 99.6-99.8% outlier detection accuracy when outliers span multiple modes versus about 72-82% for ORPCA, RTRC and ORTPCA.
- Synthetic experiments show the reconstruction error grows linearly with the squared TR rank and with the number of corrupted columns, consistent with the theoretical bound.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ORTRD instead of the strongest listed baseline | HSV recovery · bus HSV, outlier modes 1,2,3,4 | RSE: 0.0603 -> 0.0452 | Helped |
| Use ORTRD instead of the strongest listed baseline | HSV recovery · coin HSV, outlier modes 1,2,3,4 | RSE: 0.0934 -> 0.0416 | Helped |
| Use ORTRD instead of the strongest listed baseline | HSV recovery · face HSV, outlier mode 1 | RSE: 0.0438 -> 0.0448 | Hurt |
| Use ORTRD instead of the strongest listed baseline | HSV recovery · face HSV, outlier modes 1,2,3,4 | RSE: 0.0583 -> 0.0445 | Helped |
| Use ORTRD instead of the strongest listed baseline | HSV recovery · hand HSV, outlier mode 1 | RSE: 0.0407 -> 0.0456 | Hurt |
| Use ORTRD instead of the strongest listed baseline | HSV recovery · hand HSV, outlier modes 1,2,3,4 | RSE: 0.0559 -> 0.0453 | Helped |
| Distribute outliers across multiple modes rather than only mode 1 | ORPCA, ORTPCA and other compared methods · HSV recovery | RSE | Mixed |
| Use ORTRD for outlier detection instead of compared methods | Light field images · Light field images, outlier detection | clustering accuracy: 100% ... 99.70% | Helped |
Surprising findings
- Mode-specific robust methods such as ORPCA and ORTPCA perform comparably only when outliers lie in one mode, and degrade sharply as outlier modes increase.
Limitations
- The TRNN solver relies on repeated SVDs and may not scale to large tensors.
- The convex relaxation may not be tight compared with non-convex rank and sparsity functions.
Experimental setup
Datasets: synthetic low-rank tensors hyperspectral videos (4 HSV datasets) light field images (6 images from public dataset)
Benchmarks: hyperspectral videos light field images
Metrics: relative squared error (RSE) reconstruction error outlier detection accuracy (%)
Baselines: RPCA ORPCA BRTF HoRPCA RTRC TTRPCA ORTPCA
Builds on: Tensor Ring decomposition TRNN (tensor ring nuclear norm) ORPCA OTRPCA
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