BRIDGING CONVEX AND NONCONVEX OPTIMIZATION IN ROBUST PCA: NOISE, OUTLIERS, AND MISSING DATA

Area of Science:

• * Optimization Theory
• * Machine Learning
• * Data Science

Background:

• * Robust Principal Component Analysis (robust PCA) is crucial for low-rank matrix estimation with noisy, outlier-corrupted, or incomplete data.
• * Existing convex programming methods for robust PCA have suboptimal statistical guarantees, especially concerning random noise.
• * Improved theoretical support is needed to fully leverage convex relaxation in robust PCA applications.

Purpose of the Study:

• * To strengthen the theoretical guarantees of convex programming for robust PCA.
• * To analyze the stability of convex relaxation methods in the presence of random noise.
• * To achieve near-optimal statistical accuracy in low-rank matrix estimation despite significant data corruption.

Main Methods:

• * Development and analysis of a principled convex programming approach for robust PCA.
• * Bridging theoretical analysis between convex and nonconvex optimization algorithms.
• * Derivation of new statistical guarantees for Euclidean and ℓ∞ loss.

Main Results:

• * Improved theoretical guarantees for convex programming in robust PCA.
• * Demonstration of near-optimal statistical accuracy even with a constant fraction of arbitrary outliers.
• * Enhanced stability analysis concerning random noise in matrix estimation.

Conclusions:

• * The proposed convex program offers robust and accurate low-rank matrix estimation.
• * The theoretical advancements strengthen the foundation for applying robust PCA in diverse fields.
• * Bridging convex and nonconvex optimization provides novel analytical insights.