Truncated Matrix Completion - An Empirical Study

TL;DR

Problem: LRMC under data-dependent sampling with missing-not-at-random patterns ($MNAR$). Approach: Evaluate state-of-the-art LRMC algorithms under four data-dependent schemes on rank $r$ matrices $X^* = UV^T$, measuring recovery with the normalized RMSE. Findings: Non-convex methods such as R2RILS and GNMR reliably recover $X^*$ across schemes, with GNMR often outperforming others; convex relaxations such as CVX, FPCA, NNLS struggle under MNAR. Significance: Demonstrates practical viability of LRMC under realistic missingness and informs method choice for sensing, recommender systems, and sequential decision contexts; Future work: develop theoretical guarantees and diagnose why convex relaxations falter under $MNAR$.

Abstract

Low-rank Matrix Completion (LRMC) describes the problem where we wish to recover missing entries of partially observed low-rank matrix. Most existing matrix completion work deals with sampling procedures that are independent of the underlying data values. While this assumption allows the derivation of nice theoretical guarantees, it seldom holds in real-world applications. In this paper, we consider various settings where the sampling mask is dependent on the underlying data values, motivated by applications in sensing, sequential decision-making, and recommender systems. Through a series of experiments, we study and compare the performance of various LRMC algorithms that were originally successful for data-independent sampling patterns.

Figures (4)

• Figure 1: Performance of CVX and NNLS algorithms under ReLU-S and UAR-S schemes.
• Figure 2: Performance of MC algorithms under ReLU-S and UAR-S schemes.
• Figure 3: Performance of MC algorithms under MCT-S and UAR-S schemes.
• Figure 4: Performance of MC algorithms under GS-S and UAR-S schemes.