Certifying optimality in nonconvex robust PCA

Robust principal component analysis seeks to recover a low-rank matrix from fully observed data with sparse corruptions. A scalable approach fits a low-rank factorization by minimizing the sum of entrywise absolute residuals, leading to a nonsmooth and nonconvex objective. Under standard incoherence conditions and a random model for the corruption support, we study factorizations of the ground-truth rank-$r$ matrix with both factors of rank $r$. With high probability, every such factorization is a Clarke critical point. We also characterize the local geometry: when the factorization rank equals $r$, these solutions are sharp local minima; when it exceeds $r$, they are strict saddle points.

💡 Research Summary

This paper investigates a non‑convex, non‑smooth formulation of robust principal component analysis (RPCA) in which the low‑rank component is represented by a factorization X Yᵀ and the loss function is the entry‑wise ℓ₁ norm ‖X Yᵀ − M‖₁, where M = L + S with L low‑rank and S sparse. The authors aim to answer two fundamental questions: (1) Are the true factorizations of the ground‑truth low‑rank matrix Clarke critical points of the objective? (2) If they are critical, are they local minima or saddle points?

The setting assumes the standard incoherence condition on L (µ‑incoherence) and a random Bernoulli model for the support of S: each entry of S is non‑zero independently with probability p. Let the true rank be r and the factorization rank be k (k ≥ r). The Clarke sub‑differential of the objective can be expressed as
∂f(X,Y) = { (Λ Y, Λᵀ X) : Λ ∈ sign(X Yᵀ − M) },
where sign denotes the element‑wise sign set. Hence 0 ∈ ∂f(X*,Y*) is equivalent to the existence of a matrix Λ ∈ sign(−S) satisfying Λ V = 0 and Λᵀ U = 0, where U Σ Vᵀ is the SVD of L. This condition can be reduced to a restricted ℓ₁ operator‑norm bound on the projection onto the support Ω of S: for every matrix W in the linear subspace T = { H Vᵀ + U Kᵀ : H ∈ ℝ^{m×r}, K ∈ ℝ^{n×r} }, we need
‖P_Ω(W)‖₁ ≤ (1 − ε²)‖W‖₁.

The core technical contribution is a high‑probability bound on this restricted norm. By defining a centered random process X_W = ∑{i,j}(Ω{ij} − p)|W_{ij}| and showing it satisfies a mixed sub‑Gaussian/sub‑exponential increment condition, the authors apply generic chaining to obtain
sup_{W∈T,‖W‖₁=1} X_W ≤ C