Robust PCA With Partial Subspace Knowledge

@article{Zhan2015RobustPW,
  title={Robust PCA With Partial Subspace Knowledge},
  author={Jinchun Zhan and Namrata Vaswani},
  journal={IEEE Transactions on Signal Processing},
  year={2015},
  volume={63},
  pages={3332-3347}
}
In recent work, robust Principal Components Analysis (PCA) has been posed as a problem of recovering a low-rank matrix L and a sparse matrix S from their sum, M: = L + S and a provably exact convex optimization solution called PCP has been proposed. This work studies the following problem. Suppose that we have partial knowledge about the column space of the low rank matrix L. Can we use this information to improve the PCP solution, i.e., allow recovery under weaker assumptions? We propose here… 

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References

SHOWING 1-10 OF 64 REFERENCES

Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations

This paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and Gaussian noise, and shows that the resulting model falls into the applicable scope of the classical augmented Lagrangian method.

Stable Principal Component Pursuit

This result shows that the proposed convex program recovers the low-rank matrix even though a positive fraction of its entries are arbitrarily corrupted, with an error bound proportional to the noise level, the first result that shows the classical Principal Component Analysis, optimal for small i.i.d. noise, can be made robust to gross sparse errors.

Robust principal component analysis?

It is proved that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, this suggests the possibility of a principled approach to robust principal component analysis.

A novel M-estimator for robust PCA

The minimizer and its subspace are interpreted as robust versions of the empirical inverse covariance and the PCA subspace respectively and compared with many other algorithms for robust PCA on synthetic and real data sets and demonstrate state-of-the-art speed and accuracy.

Robust Matrix Decomposition with Outliers

This work studies conditions under which recovering a given observation matrix as the sum of a low-rank matrix and a sparse matrix is possible via a combination of $\ell_1$ norm and trace norm minimization and obtains stronger recovery guarantees than previous studies.

Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

A simple modification of the original ReProCS idea, which assumes knowledge of a subspace change model on the Lt's, and shows that the proposed approach can exactly recover the support set of St at all times, and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value.

Robust Matrix Decomposition With Sparse Corruptions

This work studies conditions under which recovering a given observation matrix as the sum of a low-rank matrix and a sparse matrix is possible via a combination of ℓ1 norm and trace norm minimization and obtains stronger recovery guarantees than previous studies.

Exact Matrix Completion via Convex Optimization

It is proved that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries, and that objects other than signals and images can be perfectly reconstructed from very limited information.

Robust PCA and subspace tracking from incomplete observations using $$\ell _0$$ℓ0-surrogates

This work proposes a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations and can cope with more outliers and with an underlying matrix of higher rank than other state-of-the-art methods.
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