[Paper Review] Learning Latent Features with Pairwise Penalties in Matrix Completion.

Kaiyi Ji, Jian Tan+2 more|arXiv (Cornell University)|Feb 16, 2018

Sparse and Compressive Sensing Techniques24 references3 citations

TL;DR

This paper proposes a general optimization framework for matrix completion that uses non-convex pairwise penalties to improve latent feature learning, overcoming limitations of traditional L2-based graph regularization. The method achieves better prediction accuracy and provides theoretical convergence and statistical error bounds, outperforming standard trace-norm regularization in both synthetic and real-world experiments.

ABSTRACT

Low-rank matrix completion (MC) has achieved great success in many real-world data applications. A latent feature model formulation is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning, e.g., the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use a squared L2 norm to measure the pairwise difference, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is further developed to uniformly solve the optimization problem, with a theoretical convergence guarantee. In an important situation where the latent variables form a small number of subgroups, its statistical guarantee is also fully characterized. In particular, we theoretically characterize the complexity-regularized maximum likelihood estimator, as a special case of our framework. It has a better error bound when compared to the standard trace-norm regularized matrix completion. We conduct extensive experiments on both synthetic and real datasets to demonstrate the superior performance of this general framework.

Motivation & Objective

• To address the limitation of existing graph regularized matrix factorization (GRMF) methods that rely on squared L2 norms, which are sensitive to dissimilar pairs and degrade performance.
• To develop a general optimization framework that supports a rich class of (non-)convex pairwise penalty functions for more robust latent feature learning.
• To provide a theoretically grounded, efficient algorithm with convergence guarantees for solving the proposed optimization problem.
• To establish a statistical error bound for the framework in the case of latent variables forming a small number of subgroups.
• To demonstrate superior empirical performance on both synthetic and real-world datasets compared to standard trace-norm regularized matrix completion.

Proposed method

• Proposes a general optimization framework that replaces the standard L2 pairwise penalty with a flexible class of (non-)convex pairwise penalty functions to better model latent variable similarities.
• Develops an efficient algorithm based on alternating direction method of multipliers (ADMM) to solve the non-convex optimization problem, with a theoretical convergence guarantee.
• Incorporates a complexity-regularized maximum likelihood estimator as a special case of the framework, enabling statistical error analysis.
• Models the latent features as forming a small number of subgroups, allowing for a refined theoretical characterization of the estimator’s performance.
• Uses a unified algorithmic approach that can handle various pairwise penalties, including non-convex ones, without requiring problem-specific derivations.
• Employs a Schur complement-based reformulation to enable efficient computation within the ADMM framework.

Experimental results

Research questions

• RQ1 Can a general framework with flexible pairwise penalties improve matrix completion performance beyond standard L2-based regularization?
• RQ2 How does the proposed method perform in comparison to trace-norm regularized matrix completion in terms of prediction error?
• RQ3 What is the theoretical convergence behavior of the proposed algorithm under non-convex pairwise penalties?
• RQ4 Does the framework provide a tighter statistical error bound when latent features form a small number of subgroups?
• RQ5 Can the method effectively exploit subgroup structures in latent variables to improve prediction accuracy?

Key findings

• The proposed framework achieves superior prediction performance on both synthetic and real-world datasets compared to standard trace-norm regularized matrix completion.
• The framework provides a better statistical error bound than standard trace-norm regularization when latent variables form a small number of subgroups.
• The algorithm converges to a stationary point with a theoretical convergence guarantee, even under non-convex pairwise penalties.
• The complexity-regularized maximum likelihood estimator, as a special case, achieves improved error bounds over standard trace-norm regularization.
• Empirical results show consistent gains in prediction accuracy across multiple real-world datasets, particularly when pairwise similarities are well-structured.
• The use of non-convex pairwise penalties reduces the influence of dissimilar pairs, leading to more robust latent feature estimation.