Non-convex Robust PCA (1410.7660v1)

Abstract: We propose a new method for robust PCA -- the task of recovering a low-rank matrix from sparse corruptions that are of unknown value and support. Our method involves alternating between projecting appropriate residuals onto the set of low-rank matrices, and the set of sparse matrices; each projection is {\em non-convex} but easy to compute. In spite of this non-convexity, we establish exact recovery of the low-rank matrix, under the same conditions that are required by existing methods (which are based on convex optimization). For an $m \times n$ input matrix ($m \leq n)$, our method has a running time of $O(r^2mn)$ per iteration, and needs $O(\log(1/\epsilon))$ iterations to reach an accuracy of $\epsilon$. This is close to the running time of simple PCA via the power method, which requires $O(rmn)$ per iteration, and $O(\log(1/\epsilon))$ iterations. In contrast, existing methods for robust PCA, which are based on convex optimization, have $O(m^2n)$ complexity per iteration, and take $O(1/\epsilon)$ iterations, i.e., exponentially more iterations for the same accuracy. Experiments on both synthetic and real data establishes the improved speed and accuracy of our method over existing convex implementations.

• The paper introduces an iterative algorithm using alternating projections to separate low-rank and sparse components efficiently.
• It achieves robust theoretical recovery guarantees with computational complexity approaching that of classical PCA.
• Empirical results on synthetic and real datasets demonstrate faster convergence and superior performance compared to state-of-the-art methods.

Non-convex Robust PCA

The paper "Non-convex Robust PCA" explores a non-convex approach to the problem of Robust Principal Component Analysis (RPCA), aiming to improve computational efficiency without sacrificing the strong theoretical recovery guarantees offered by convex methods. The authors propose an iterative algorithm involving alternating projections onto the set of low-rank matrices and the set of sparse matrices. While each projection is non-convex, they are computationally efficient and, under specific conditions, guarantee exact recovery of the low-rank matrix from sparse corruptions.

Summary of Contributions

1. Algorithm Development: The authors propose a method utilizing alternating projections that remarkably retains the robust recovery guarantees of convex optimization techniques while significantly improving computational efficiency. For an input matrix of dimensions $m \times n$ (with $m \leq n$), the algorithm requires $O(r^2mn)$ operations per iteration and $O(\log(1/\epsilon))$ iterations to reach a desired accuracy $\epsilon$. This efficiency brings it closer to the complexity of traditional PCA, which operates with $O(rmn)$ complexity per iteration.
2. Theoretical Guarantees: Under the deterministic sparsity model, the method requires that each row and column of the sparse matrix contain at most $\alpha = O(1/(\mu^2 r))$ non-zero entries, an incoherence condition similar to those required by convex RPCA approaches but achieving this through non-convex methods.
3. Empirical Validation: Experiments conducted using both synthetic and real-world datasets show that the proposed method outperforms the state-of-the-art inexact augmented Lagrangian multiplier (IALM) method. The non-convex approach demonstrated faster convergence and more accurate separation of low-rank and sparse components under various experimental settings.

Implications and Future Directions

Practical Implications: The significant reduction in computation time makes this approach highly practical for large-scale applications such as video background modeling, 3D reconstruction, robust topic modeling, and community detection. By extending traditional PCA to be robust against sparse corruptions, this method potentially broadens the usability in practical scenarios where data may be incomplete or corrupted.

Theoretical Implications: This work highlights an intriguing aspect of non-convex optimization—not only can it efficiently converge under certain conditions, but it can also maintain robustness and provide guarantees akin to convex optimization methods. This challenges the conventional preference for convexity in statistically robust algorithms, suggesting potential unexplored avenues in non-convex optimization.

Future Directions: This non-convex approach paves the way for further research into other non-convex methodologies in machine learning and data analysis. Possible expansions could include exploring the effects of various noise models, further reduction of computational complexity, and extending the approach to more generalized data structures such as tensors. Additionally, future research could focus on tackling other matrix decomposition problems by leveraging non-convex methodologies.

In conclusion, "Non-convex Robust PCA" provides a substantial step forward in the field of efficient matrix decomposition, balancing robustness and computational demands. Its implications could expand to more complex datasets and pave the way for advanced studies and applications in modern data-driven disciplines.