A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like "how does an intervention affect each unit," rather than only on average. We study this problem with panel-data where we observe $n$ units across $m$ times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit--time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$. Technically, our analysis establishes the first sharp row-wise $\ell_2$-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.

Modern matrix completion problems often involve heterogeneous data whose rows simultaneously belong to many meta-categories, such as demographic and age groups in recommendation systems, or region and recording session labels in neural electrophysiological experiments. Standard low-rank estimators impose a single global latent geometry, which can recover average structure but may smooth away subgroup-specific variation, especially when observations are unevenly distributed across groups. We introduce Group-Aware Matrix Estimation (GAME), a convex estimator for overlapping subgroup-wise low-rank matrix estimation. GAME regularizes category-specific submatrices through overlapping nuclear-norm penalties, allowing related groups to borrow information while preserving local latent structure in a shared coordinate system. We provide finite-sample guarantees for both reconstruction error and subgroup-specific subspace recovery, showing how performance depends on sampling density, subgroup rank, and overlap structure. Experiments on synthetic, recommendation, ecological, and neuroscience datasets show that GAME is most beneficial in structured missingness regimes, where subgroup-aware regularization improves both reconstruction accuracy and latent subspace fidelity. Across these benchmarks, GAME is competitive or best among global low-rank, side-information, and modern imputation baselines, with the largest gains when subgroups exhibit distinct low-rank structure.

Low-rank matrix completion is a widely studied problem with many variants. Inductive matrix completion (IMC) incorporates row and column side information to significantly narrow the search space. Prior work falls into two regimes: methods that exploit this structure to achieve reduced sample complexity but only in noiseless settings, and methods that handle noise but require sample complexity matching the ambient matrix dimension, forfeiting the sample efficiency that side information should provide. In this paper, we close this gap by studying noisy IMC with a nonconvex projected gradient descent algorithm with spectral initialization. Our main technical contribution is establishing a regularity condition for the IMC loss function that holds at the reduced sample complexity determined by the effective problem size, scaling with the side information dimension a rather than the ambient dimension n. This directly yields linear convergence and an estimation error that both depend only on the effective problem size rather than the ambient matrix dimension. We further extend our analysis to the inexact side information setting, demonstrating that the reduced sample complexity is maintained and the estimation error is order-optimal with respect to the inexactness of the side information. Extensive simulations and real-world experiments on the MovieLens dataset validate our theoretical findings.

Matrix completion is a fundamental problem in machine learning, where the objective is to recover missing entries of a partially observed matrix. A prominent example is the Netflix Prize (Bennett and Lanning, 2007), which involved predicting a matrix of movie ratings by users for recommendation purposes. Beyond recommendation, matrix completion has recently found applications in causal inference for panel data (Athey et al., 2021). A standard assumption in matrix completion is that the underlying matrix is approximately low-rank, reflecting a few latent factors that govern interactions between rows and columns. A substantial body of work has established theoretical guarantees and developed efficient algorithms for matrix completion (Cai, Cand`es and Shen, 2010; Cand`es and Recht, 2008; Keshavan, Montanari, and Oh, 2010; Mazumder, Hastie and Tibshirani, 2010; Recht, 2011), predominantly focusing on cases where the observed entries are continuous-valued. In many applications, however, observations are not continuous-valued but binary.

We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.

In this work, we formulate a new multi-task active learning setting in which the learner's goal is to solve multiple matrix completion problems simultaneously. At each round, the learner can choose from which matrix it receives a sample from an entry drawn uniformly at random. Our main practical motivation is market segmentation, where the matrices represent different regions with different preferences of the customers. The challenge in this setting is that each of the matrices can be of a different size and also of a different rank which is unknown. We provide and analyze a new algorithm, MAlocate that is able to adapt to the unknown ranks of the different matrices. We then give a lower-bound showing that our strategy is minimax-optimal and demonstrate its performance with synthetic experiments.

Aisasetofweightedlinear combinations of the entries ofX, this problem is often referred to as thematrix sensingproblem.