Online Robust PCA via Stochastic Optimization 论文

摘要

For the data projection r and noise estimation e, we can get the closed-form solutions for them respectively, as shown in Algorithm 1.1. In particular, the closed-form solution to a projection to `1-ball in updating e involves a soft thresholding operator Sλ[·] Hale et al. (2008), which is defined as: Sλ[x], x − λ, if x> λ, x+ λ, if x < −λ, 0, otherwise. And it is conducted element-wisely on the involved vectors. The optimization iteration is terminated when the following convergence criterion is met: max(‖rk+1 − rk‖/‖z‖, ‖ek+1 − ek‖/‖z‖) < ε. Here ε is set as 1 × 10−6 throughout the simulations. The details of the algorithm are summarized as follows, Algorithm 1.1 Data Projection Input: L = [l1,..., lr] ∈ Rp×r (input basis), z ∈ Rp, parameters λ1 and λ2. e ← 0. while not converged do Update the coefficient r: r ← (LTL+ λ1I)−1LT (z − e). Update the sparse error e: e ← Sλ2 [z − Lr]. end while