N$^2$: A Unified Python Package and Test Bench for Nearest Neighbor-Based Matrix Completion

Nearest neighbor (NN) methods have re-emerged as competitive tools for matrix completion, offering strong empirical performance and recent theoretical guarantees, including entry-wise error bounds, confidence intervals, and minimax optimality. Despite their simplicity, recent work has shown that NN approaches are robust to a range of missingness patterns and effective across diverse applications. This paper introduces N$^2$, a unified Python package and testbed that consolidates a broad class of NN-based methods through a modular, extensible interface. Built for both researchers and practitioners, N$^2$ supports rapid experimentation and benchmarking. Using this framework, we introduce a new NN variant that achieves state-of-the-art results in several settings. We also release a benchmark suite of real-world datasets, from healthcare and recommender systems to causal inference and LLM evaluation, designed to stress-test matrix completion methods beyond synthetic scenarios. Our experiments demonstrate that while classical methods excel on idealized data, NN-based techniques consistently outperform them in real-world settings.

Adaptively-weighted Nearest Neighbors for Matrix Completion

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.

On adaptivity and minimax optimality of two-sided nearest neighbors

Nearest neighbor (NN) algorithms have been extensively used for missing data problems in recommender systems and sequential decision-making systems. Prior theoretical analysis has established favorable guarantees for NN when the underlying data is sufficiently smooth and the missingness probabilities are lower bounded. Here we analyze NN with non-smooth non-linear functions with vast amounts of missingness. In particular, we consider matrix completion settings where the entries of the underlying matrix follow a latent non-linear factor model, with the non-linearity belonging to a \Holder function class that is less smooth than Lipschitz. Our results establish following favorable properties for a suitable two-sided NN: (1) The mean squared error (MSE) of NN adapts to the smoothness of the non-linearity, (2) under certain regularity conditions, the NN error rate matches the rate obtained by an oracle equipped with the knowledge of both the row and column latent factors, and finally (3) NN's MSE is non-trivial for a wide range of settings even when several matrix entries might be missing deterministically. We support our theoretical findings via extensive numerical simulations and a case study with data from a mobile health study, HeartSteps.