[Paper Review] Provable Inductive Matrix Completion

Prateek Jain, Inderjit S. Dhillon|arXiv (Cornell University)|Jun 4, 2013

Sparse and Compressive Sensing Techniques9 references124 citations

TL;DR

This paper proposes a provably convergent alternating minimization method for inductive matrix completion and general low-rank matrix estimation using rank-1 measurements. It establishes that under mild conditions on feature vectors, alternating minimization converges linearly to the global optimum, significantly reducing computational and storage costs compared to RIP-based methods, with theoretical guarantees for matrix sensing, inductive matrix completion, and multi-label regression with missing labels.

ABSTRACT

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimation using rank-1 measurements. We study this generic problem and provide conditions that the set of measurements should satisfy so that the alternating minimization method (which otherwise is a non-convex method with no convergence guarantees) is able to recover back the {\em exact} underlying low-rank matrix. In addition to inductive matrix completion, we show that two other low-rank estimation problems can be studied in our framework: a) general low-rank matrix sensing using rank-1 measurements, and b) multi-label regression with missing labels. For both the problems, we provide novel and interesting bounds on the number of measurements required by alternating minimization to provably converges to the {\em exact} low-rank matrix. In particular, our analysis for the general low rank matrix sensing problem significantly improves the required storage and computational cost than that required by the RIP-based matrix sensing methods \cite{RechtFP2007}. Finally, we provide empirical validation of our approach and demonstrate that alternating minimization is able to recover the true matrix for the above mentioned problems using a small number of measurements.

Motivation & Objective

• To address the limitation of standard matrix completion, which cannot generalize to new users or items, by incorporating side information (feature vectors) for inductive prediction.
• To formalize a unified framework—Low-Rank Matrix Estimation using Rank-One Measurements (LRROM)—that captures inductive matrix completion, general matrix sensing, and multi-label regression with missing labels.
• To provide theoretical guarantees for alternating minimization in the LRROM setting, showing linear convergence to the global optimum under mild conditions on the measurement operator.
• To demonstrate that the proposed method achieves comparable or better recovery accuracy than RIP-based methods while requiring significantly less storage and computation.

Proposed method

• Formulates the inductive matrix completion problem as recovering a low-rank matrix $W_*$ from measurements $\bm{b}_i = \bm{x}_i^T W_* \bm{y}_j$, where $\bm{x}_i, \bm{y}_j$ are user and item feature vectors.
• Introduces a general LRROM framework where the measurement operator maps $W_*$ to a vector of rank-1 inner products, enabling a unified treatment of multiple low-rank estimation problems.
• Proposes alternating minimization with spectral initialization to recover $W_*$, proving linear convergence under three key properties of the measurement operator: incoherence, bounded variance, and concentration of measure.
• Establishes theoretical bounds on the number of measurements required for exact recovery, showing $m = \Omega(k^4 \beta^2 (d_1 + d_2) \log(d_1 + d_2))$ suffices for Gaussian features, where $\beta$ is the condition number of $W_*$.
• Uses random matrix theory and matrix concentration inequalities (e.g., matrix Chernoff bounds) to analyze the behavior of measurement operators and prove convergence guarantees.
• Applies the general analysis to three specific problems: (1) Gaussian matrix sensing, (2) inductive matrix completion with uniformly sampled entries, and (3) multi-label regression with missing labels, showing all satisfy the required conditions.

Experimental results

Research questions

• RQ1 Can alternating minimization be provably guaranteed to converge to the global optimum in inductive matrix completion with side information?
• RQ2 What conditions on the feature vectors and measurement operator ensure linear convergence of alternating minimization in low-rank matrix estimation?
• RQ3 How does the number of required measurements scale for exact recovery in the LRROM framework, especially compared to RIP-based methods?
• RQ4 Can the proposed framework unify and generalize existing low-rank estimation problems such as matrix sensing and multi-label regression?
• RQ5 What is the trade-off between recovery accuracy and computational efficiency when using rank-1 measurements versus RIP-based measurements?

Key findings

• For Gaussian rank-1 measurements, alternating minimization recovers the true low-rank matrix $W_*$ with high probability when $m = \Omega(k^4 \beta^2 (d_1 + d_2) \log(d_1 + d_2))$ measurements are used.
• The proposed method achieves recovery accuracy comparable to RIP-based matrix sensing but with two orders of magnitude lower running time, making it significantly more efficient.
• In inductive matrix completion with uniformly sampled entries, the method recovers the underlying matrix accurately even with a small number of observations, as shown by empirical results with low test error.
• For multi-label regression with missing labels, the method maintains low test error even when the number of labels $L=50$ and feature dimension $d$ is small, with error degrading gracefully as $k$ or $d$ increases.
• Theoretical analysis confirms that the measurement operator satisfies the required concentration and incoherence properties under mild assumptions, enabling linear convergence of alternating minimization.
• The framework improves upon existing RIP-based methods by reducing storage and computational costs, particularly in large-scale settings.