Adaptively-weighted Nearest Neighbors for Matrix Completion

TL;DR

This paper introduces AWNN, an adaptively weighted nearest neighbor method for matrix completion that automatically balances bias and variance, providing theoretical guarantees and demonstrating effectiveness through synthetic experiments.

Contribution

The paper proposes AWNN, a novel adaptive weighting scheme for nearest neighbor methods in matrix completion, with theoretical analysis and minimal assumption guarantees.

Findings

AWNN effectively balances bias and variance in matrix completion.

Theoretical guarantees are established under minimal assumptions.

Synthetic experiments support the method's performance and robustness.

Abstract

In this technical note, we introduce and analyze AWNN: an adaptively weighted nearest neighbor method for performing matrix completion. Nearest neighbor (NN) methods are widely used in missing data problems across multiple disciplines such as in recommender systems and for performing counterfactual inference in panel data settings. Prior works have shown that in addition to being very intuitive and easy to implement, NN methods enjoy nice theoretical guarantees. However, the performance of majority of the NN methods rely on the appropriate choice of the radii and the weights assigned to each member in the nearest neighbor set and despite several works on nearest neighbor methods in the past two decades, there does not exist a systematic approach of choosing the radii and the weights without relying on methods like cross-validation. AWNN addresses this challenge by judiciously balancing the bias variance trade off inherent in weighted nearest-neighbor regression. We provide theoretical guarantees for the proposed method under minimal assumptions and support the theory via synthetic experiments.