[Paper Review] Online Decomposition of Compressive Streaming Data Using $n$-$\ell_1$ Cluster-Weighted Minimization

Huynh Van Luong, Nikos Deligiannis+2 more|arXiv (Cornell University)|Feb 8, 2018

Sparse and Compressive Sensing Techniques21 references3 citations

TL;DR

This paper proposes CODA, an online compressive streaming data decomposition method that uses $n$-$\ell_1$ cluster-weighted minimization to jointly recover sparse and low-rank components from compressive measurements. By leveraging multiple prior sparse vectors through iterative clustering and re-weighting, CODA achieves superior foreground-background separation in video streams with minimal measurements, outperforming existing online and compressive RPCA methods at low measurement rates.

ABSTRACT

We consider a decomposition method for compressive streaming data in the context of online compressive Robust Principle Component Analysis (RPCA). The proposed decomposition solves an $n$-$\ell_1$ cluster-weighted minimization to decompose a sequence of frames (or vectors), into sparse and low-rank components, from compressive measurements. Our method processes a data vector of the stream per time instance from a small number of measurements in contrast to conventional batch RPCA, which needs to access full data. The $n$-$\ell_1$ cluster-weighted minimization leverages the sparse components along with their correlations with multiple previously-recovered sparse vectors. Moreover, the proposed minimization can exploit the structures of sparse components via clustering and re-weighting iteratively. The method outperforms the existing methods for both numerical data and actual video data.

Motivation & Objective

• To address the limitations of batch and online Robust PCA (RPCA) in long video streams with background variations and high data volume.
• To enable real-time decomposition of streaming data from compressive measurements without requiring full data access.
• To exploit multiple prior sparse components and their structural correlations via clustering and re-weighting to improve recovery accuracy.
• To develop a method that maintains high performance under low measurement rates, reducing storage and communication costs.

Proposed method

• CODA formulates an online optimization problem that minimizes the nuclear norm of the low-rank component and the $\ell_1$-norm of the current sparse component, augmented with a prior-based regularization term.
• The method incorporates three side information signals: the immediately prior reconstructed foreground, and two motion-compensated extrapolated versions from earlier frames.
• It applies iterative clustering of sparse components across time to identify structural patterns and re-weight the $\ell_1$-norm based on cluster membership to enhance sparsity.
• The optimization is solved sequentially per time frame using compressive measurements $\mathbf{y}_t = \mathbf{\Phi}(\mathbf{x}_t + \mathbf{v}_t)$, where $\mathbf{\Phi}$ is a random projection matrix with $m \ll n$.
• The algorithm uses a re-weighting scheme that adaptively increases the penalty on less likely sparse components based on cluster structure and prior recovery quality.
• CODA is evaluated using Monte Carlo simulations and real video sequences, with performance measured by recovery success rates under varying measurement rates.

Experimental results

Research questions

• RQ1 Can multiple prior sparse components be effectively leveraged to improve online decomposition of compressive streaming data?
• RQ2 Does iterative clustering and re-weighting of sparse components enhance recovery accuracy in compressive online RPCA?
• RQ3 How does the performance of CODA compare to state-of-the-art methods like CORPCA, ReProCS, and GRASTA under low measurement rates?
• RQ4 To what extent can CODA recover structured sparse components (e.g., foreground in video) from compressive measurements with minimal data?

Key findings

• CODA successfully recovers 500-dimensional data from measurement rates as low as $m/n = 0.25$, with $100\%$ success rate in sparse and low-rank recovery under optimal conditions.
• For $s_0 > 70$, CORPCA-$n$-$\ell_1$ fails to recover sparse components, while CODA maintains $100\%$ success rate, demonstrating robustness to high sparsity.
• At $m/n = 0.5$, CODA recovers foreground structures in the Bootstrap video sequence with high fidelity, outperforming CORPCA and ReProCS even at $m/n = 0.8$.
• ReProCS shows significantly worse performance than CODA, especially in low-rank recovery, with visible artifacts even at high measurement rates.
• GRASTA delivers lower low-rank recovery performance than CODA, particularly at $m/n = 0.4$, highlighting CODA's efficiency in preserving background structure.
• The bone-colored success rate maps in Fig. 1 show that CODA achieves white (100% success) regions at $m/n = 0.6$, while ReProCS and CORPCA show grey (failure) regions at the same rates.