Low-rank Matrix Completion with Noisy Observations: a Quantitative Comparison

We consider a problem of significant practical importance, namely, the reconstruction of a low-rank data matrix from a small subset of its entries. This problem appears in many areas such as collaborative filtering, computer vision and wireless sensor networks. In this paper, we focus on the matrix completion problem in the case when the observed samples are corrupted by noise. We compare the performance of three state-of-the-art matrix completion algorithms (OptSpace, ADMiRA and FPCA) on a single simulation platform and present numerical results. We show that in practice these efficient algorithms can be used to reconstruct real data matrices, as well as randomly generated matrices, accurately.

💡 Research Summary

The paper addresses the practically important problem of recovering a low‑rank matrix from a subset of its entries when those observations are corrupted by noise. This scenario arises in collaborative filtering, computer vision, sensor networks, and many other domains where data are incomplete and noisy. The authors focus on a quantitative comparison of three state‑of‑the‑art matrix completion algorithms—OptSpace, ADMiRA, and FPCA—by implementing them on a unified simulation platform and evaluating their performance on both synthetic and real‑world data sets.

Problem formulation
Let (M\in\mathbb{R}^{n_1\times n_2}) be a rank‑(r) matrix. A random subset (\Omega) of its entries is observed, but each observed entry is contaminated by additive Gaussian noise: (Y_{ij}=M_{ij}+Z_{ij}) for ((i,j)\in\Omega), where (Z_{ij}\sim\mathcal{N}(0,\sigma^2)). The goal is to reconstruct (M) using only (\Omega) and the noisy measurements (Y).

Algorithms under study

1. OptSpace – The method first rescales the observed matrix and computes a truncated SVD to obtain an initial low‑rank estimate. It then refines this estimate by solving a non‑linear least‑squares problem on the Grassmann manifold using gradient descent. The algorithm incorporates step‑size scheduling and regularization to improve robustness against noise.

2. ADMiRA – Inspired by compressed sensing, ADMiRA iteratively builds a low‑rank approximation. At each iteration it selects the top‑(r) singular vectors of the current residual matrix, updates the estimate, and repeats. A thresholding rule is employed to suppress singular components that are likely dominated by noise, making the method sensitive to the choice of the threshold.

3. FPCA – This approach formulates matrix completion as a convex optimization problem with a nuclear‑norm regularizer:
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