Robust Matrix Completion

Accurate and reliable collection of electricity consumption data through Advanced Metering Infrastructure (AMI) is of great importance for the operation of smart grids, especially for the detection of non-technical losses (NTL). However, real-world datasets frequently suffer from missing values due to communication failures. This paper presents an empirical evaluation of three advanced algorithms for large-scale data imputation: the Optimally Weighted Average (OWA) method, Low-Rank Matrix Completion via SoftImpute, and a Shape-Modeling Autoencoder. Existing studies on missing value imputation in electricity consumption data often lack validation on larger datasets. Therefore, the goal of this paper is to validate the selected algorithms on a large-scale real-world electricity consumption dataset from North Macedonia that includes 17,428 commercial smart meters over two years. The robustness of each algorithm is evaluated by simulating continuous gaps in the data ranging from 1 to 168 hours. The results indicate that OWA provides the lowest overall reconstruction error across the evaluated gap sizes and strong stability in worst-case scenarios for gaps of up to one week. In contrast, the autoencoder exhibits higher variance, while SoftImpute has stable but inferior accuracy. These findings suggest that imputation methods should be selected based on the characteristics of load curve data and highlight the potential for hybrid algorithmic architectures in future grid management systems.

A modern model release reports scores on 40+ benchmarks and the same evaluations were run many more times before it: to track training progress, compare design choices, and select the checkpoint for the release. But do we need to run every eval? We compile a public score matrix of 84 frontier models on 133 benchmarks (2,604 cells, 23.3% filled) and find it is approximately rank-2: a model's scores across all 133 benchmarks are largely determined by just two numbers. We confirm this in two ways: scores hidden from the matrix are best recovered using two factors, and two factors already explain over 90% of the variation among models on the benchmarks they share. Building on this, we design BenchPress: a logit-space rank-2 matrix completion method that recovers held-out scores to within 4.6 points, and a confidence layer that says when each prediction can be trusted. Using BenchPress, we find a subset of five benchmarks {GPQA-D, HLE, Codeforces, MMLU-Pro, ARC-AGI-1} that can recover the rest of a model's public scorecard to within 3.93 points. For a tighter inference budget, a cheaper set {GPQA-D, MMLU-Pro, Aider Polyglot, MATH-500, AIME 2026} can predict a model's evals to within 4.55. We release the score matrix, the BenchPress code, and an interactive tool that predicts any model's score on any benchmark.

We study a nonlinear factor model in which observed responses depend on low-rank latent factors through an unknown monotone link function. This setting is challenging and largely underexplored due to severe nonconvexity and identifiability issues. The link function is assumed to lie in a reproducing kernel Hilbert space (RKHS), enabling flexible nonparametric modeling while preserving identifiability. We formulate the problem as the joint recovery of the low-rank factors, loadings, and the nonlinear link function from possibly incomplete and noisy observations and propose a projected block coordinate descent (BCD) algorithm with explicit regularization to address scale and rotational ambiguities. Under mild incoherence of factors and standard sampling conditions, we establish convergence guarantees in both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. Our results extend classical linear factor models to a broad nonlinear regime and provide a principled framework for learning nonlinear latent structures. We evaluate the proposed approach using controlled synthetic experiments, indicating promising performance.

Multi-robot task allocation usually assumes some combination of communication, known task models, or a coordinator. We study the opposite extreme, a regime common in practice but overlooked in theory, which we name Zero-Knowledge MRTA (ZK-MRTA): a robot team with no prior knowledge (no task models, not even the latent rank), no communication (no messages, no parameter sharing, no coordinator), and only a partial and privately-noisy view of a public stream of teammates' outcomes. A hidden low-rank structure governs which robot suits which task, and there are far more tasks than rounds, so most (robot, task) pairs are never attempted. Yet each robot can act well on tasks it never attempted, and onboard new tasks, by running online low-rank collaborative filtering over the broadcast (SwarmCF). The advantage over any structure-free learner is categorical, not a constant factor: a structure-free learner is provably at the prior-mean error floor on unseen pairs. We prove a matching per-robot sample complexity (Θ(d) versus Θ(n), in the rank d and the task count n), an anytime (cumulative-reward) separation under task scarcity, and a deterministic condition under which decentralized recovery from the masked broadcast is exact (validated empirically). Experiments quantify the value of the broadcast, a positive scaling law (per-robot unseen-pair skill rises with team size), and the strongest masking-robustness and anytime profile among low-rank methods, recovering most (about 80% on earned skill) of a centralized full-communication ceiling, and holding under capacity-1 contention and in a robotics-grounded sensing instance.

Inductive matrix completion (IMC) is a variant of low-rank matrix completion that incorporates row and column side-information. In principle, it can reduce the effective dimension of the recovery problem from the ambient matrix size to the dimension of the side-information features. Existing theory, however, does not fully realize this advantage in the noisy setting: sample-efficient guarantees only apply to noiseless recovery, while noisy guarantees require sample sizes comparable to ordinary matrix completion. This paper closes this gap for noisy IMC. We analyze a nonconvex projected gradient descent algorithm with spectral initialization and prove that, under exact side-information, it achieves linear convergence and stable recovery at a sample complexity governed by the effective side-information dimension rather than the ambient matrix dimension. The key technical ingredient is a local regularity condition for the IMC loss that holds at this reduced sample size, despite the mismatch between the observation pattern and the side-information subspaces. We further extend the analysis to inexact side-information, showing that the same reduced sample complexity is preserved and that the estimation error degrades optimally with the level of subspace misspecification. Motivated by this trade-off, we also propose a penalized interpolation between IMC and ordinary matrix completion that balances sample efficiency against robustness to imperfect side-information. Simulations and experiments on the MovieLens dataset support the theoretical findings and illustrate the practical benefits of exploiting side-information in low-sample regimes.

Selecting the best large language model (LLM) for a fixed benchmark is often expensive, since exhaustive evaluation requires running every model on every example. Multi-armed bandit (MAB) algorithms can reduce the number of LLM calls by sequentially selecting the next model-example pair to evaluate, thereby avoiding wasted evaluations on clearly underperforming models. Further savings can be achieved by predicting model scores from the partially observed model-example score matrix using low-rank factorization. However, such predictions are not ground truth: they can be biased and may therefore lead to incorrect identification of the best model. In this work, we propose a principled framework that combines MAB with cheap predicted scores without compromising statistical validity. Specifically, we derive doubly robust estimators of each model's performance that use the low-rank predictions to reduce variance. This enables the construction of valid finite-sample confidence intervals in our setting, where models are selected adaptively and examples are sampled without replacement. Empirical results on real-world benchmarks show that our approach reduces the number of required evaluations, yielding meaningful savings in compute and cost while accurately identifying the best-performing model.

Recent advances in drug discovery have demonstrated that incorporating side information (e.g., chemical properties about drugs and genomic information about diseases) often greatly improves prediction performance. However, these side features can vary widely in relevance and are often noisy and high-dimensional. We propose Bayesian Variable Selection-Guided Inductive Matrix Completion (BVSIMC), a new Bayesian model that enables variable selection from side features in drug discovery. By learning sparse latent embeddings, BVSIMC improves both predictive accuracy and interpretability. We validate our method through simulation studies and two drug discovery applications: 1) prediction of drug resistance in Mycobacterium tuberculosis, and 2) prediction of new drug-disease associations in computational drug repositioning. On both synthetic and real data, BVSIMC outperforms several other state-of-the-art methods in terms of prediction. In our two real examples, BVSIMC further reveals the most clinically meaningful side features.

Recommender systems are widely used in the digital landscape to match users with content fitting their preferences. However, growing concerns about fake accounts, strategic manipulation, and other deceptive online behavior place increasing pressure on the reliability of these systems. A common statistical approach behind recommender systems is so-called matrix completion, which predicts how users would rate items they have not yet consumed based on patterns in observed ratings. Realistically applying matrix completion methods requires jointly addressing several overlooked challenges: (i) ratings on discrete scales (such as 1--5 stars); (ii) the presence of malicious users who deliberately manipulate the system to their advantage through fake profiles; (iii) ratings missing not at random since users are more likely to consume items they expect to like; and (iv) fostering transparency, reproducibility, and stability. We jointly address these challenges by proposing a novel method, Robust Discrete Matrix Completion (RDMC), designed to capture the key characteristics of sparse rating data while remaining reliable in the presence of manipulation. We evaluate RDMC through two case studies and carefully designed simulation experiments. Our work thereby offers a statistically-sound blueprint for future studies on how to evaluate recommender systems under realistic scenarios.