Adaptively-weighted Nearest Neighbors for Matrix Completion

2025-05-14

AI Summary

Existing nearest-neighbor matrix completion methods rely on manually specified neighborhood radii and weighting schemes, lacking a systematic, cross-validation-free adaptive mechanism. This work proposes Adaptive Weighted Nearest Neighbors (AWNN), a data-driven framework that dynamically determines both the neighborhood radius and kernel weights to automatically balance the bias–variance trade-off. To our knowledge, this is the first method to establish a joint adaptive selection framework for radius and weights under mild assumptions—without requiring cross-validation—and to provide rigorous theoretical convergence guarantees. By integrating weighted nearest-neighbor regression, adaptive kernel design, and tight error bound analysis, we prove that the estimation error converges at the minimax-optimal rate. Empirical evaluation on synthetic data demonstrates that AWNN significantly outperforms baseline methods employing fixed radii or weights.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior
• Multiagent Systems: Adversarial Agents

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• AWNN addresses adaptive weighting in nearest neighbor matrix completion
• Lack of systematic radius and weight selection in NN methods
• Balancing bias-variance trade-off in weighted nearest-neighbor regression

Innovation

• Adaptively-weighted Nearest Neighbors for matrix completion
• Systematic radii and weights selection without cross-validation
• Balances bias-variance trade-off in weighted regression