The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.
This paper explores an energy-modified leverage sampling strategy for matrix completion in radio map construction. The main goal is to address potential identifiability issues in matrix completion with sparse observations by using a probabilistic sampling approach. Although conventional leverage sampling is commonly employed for designing sampling patterns, it often assigns high sampling probability to locations with low received signal strength (RSS) values, leading to a low sampling efficiency. Theoretical analysis demonstrates that the leverage score produces pseudo images of sources, and in the regions around the source locations, the leverage probability is asymptotically consistent with the RSS. Based on this finding, an energy-modified leverage probability-based sampling strategy is investigated for efficient sampling. Numerical demonstrations indicate that the proposed sampling strategy can decrease the normalized mean squared error (NMSE) of radio map construction by more than 10% for both matrix completion and interpolation-assisted matrix completion schemes, compared to conventional methods.
This paper investigates statistical inference for noisy matrix completion in a semi-supervised model when auxiliary covariates are available. The model consists of two parts. One part is a low-rank matrix induced by unobserved latent factors; the other part models the effects of the observed covariates through a coefficient matrix which is composed of high-dimensional column vectors. We model the observational pattern of the responses through a logistic regression of the covariates, and allow its probability to go to zero as the sample size increases. We apply an iterative least squares (LS) estimation approach in our considered context. The iterative LS methods in general enjoy a low computational cost, but deriving the statistical properties of the resulting estimators is a challenging task. We show that our method only needs a few iterations, and the resulting entry-wise estimators of the low-rank matrix and the coefficient matrix are guaranteed to have asymptotic normal distributions. As a result, individual inference can be conducted for each entry of the unknown matrices. We also propose a simultaneous testing procedure with multiplier bootstrap for the high-dimensional coefficient matrix. This simultaneous inferential tool can help us further investigate the effects of covariates for the prediction of missing entries.
Current spectral compressed sensing methods via Hankel matrix completion employ symmetric factorization to demonstrate the low-rank property of the Hankel matrix. However, previous non-convex gradient methods only utilize asymmetric factorization to achieve spectral compressed sensing. In this paper, we propose a novel nonconvex projected gradient descent method for spectral compressed sensing via symmetric factorization named Symmetric Hankel Projected Gradient Descent (SHGD), which updates only one matrix and avoids a balancing regularization term. SHGD reduces about half of the computation and storage costs compared to the prior gradient method based on asymmetric factorization. {Besides, the symmetric factorization employed in our work is completely novel to the prior low-rank factorization model, introducing a new factorization ambiguity under complex orthogonal transformation}. Novel distance metrics are designed for our factorization method and a linear convergence guarantee to the desired signal is established with $O(r^2\log(n))$ observations. Numerical simulations demonstrate the superior performance of the proposed SHGD method in phase transitions and computation efficiency compared to state-of-the-art methods.
This paper addresses the Sensor Network Localization (SNL) problem using received signal strength. The SNL is formulated as an Euclidean Distance Matrix Completion (EDMC) problem under the unit ball sample model. Using the Burer-Monteiro factorization type cost function, the EDMC is solved by Riemannian conjugate gradient with Hager-Zhang line search method on a quotient manifold. A "rank reduction" preprocess is proposed for proper initialization and to achieve global convergence with high probability. Simulations on a synthetic scene show that our approach attains better localization accuracy and is computationally efficient compared to several baseline methods. Characterization of a small local basin of attraction around the global optima of the s-stress function under Bernoulli sampling rule and incoherence matrix completion framework is conducted for the first time. Theoretical result conjectures that the Euclidean distance problem with a structure-less sample mask can be effectively handled using spectral initialization followed by vanilla first-order methods. This preliminary analysis, along with the aforementioned numerical accomplishments, provides insights into revealing the landscape of the s-stress function and may stimulate the design of simpler algorithms to tackle the non-convex formulation of general EDMC problems.
This paper investigates the effects of coarse quantization with mixed precision on measurements obtained from sparse linear arrays, synthesized by a collaborative automotive radar sensing strategy. The mixed quantization precision significantly reduces the data amount that needs to be shared from radar nodes to the fusion center for coherent processing. We utilize the low-rank properties inherent in the constructed Hankel matrix of the mixed-precision array, to recover azimuth angles from quantized measurements. Our proposed approach addresses the challenge of mixed-quantized Hankel matrix completion, allowing for accurate estimation of the azimuth angles of interest. To evaluate the recovery performance of the proposed scheme, we establish a quasi-isometric embedding with a high probability for mixed-precision quantization. The effectiveness of our proposed scheme is demonstrated through numerical results, highlighting successful reconstruction.
A low-observable distribution system has insufficient measurements for conventional weighted least square state estimators. Matrix completion state estimators have been suggested, but their computational times could be prohibitive. To resolve this problem, a novel and efficient power-flow-embedded projection conic matrix completion method customized for low-observable distribution systems is proposed in this letter. This method can yield more accurate state estimations (2-fold improvement) in a much shorter time (5% or less) than other methods. Case studies on different-scale systems have demonstrated the efficacy of the proposed method when applied to low-observable distribution system state estimation problems.
Matrix completion is an important area of research in recommender systems. Recent methods view a rating matrix as a user-item bi-partite graph with labeled edges denoting observed ratings and predict the edges between the user and item nodes by using the graph neural network (GNN). Despite their effectiveness, they treat each rating type as an independent relation type and thus cannot sufficiently consider the ordinal nature of the ratings. In this paper, we explore a new approach to exploit rating ordinality for GNN, which has not been studied well in the literature. We introduce a new method, called ROGMC, to leverage Rating Ordinality in GNN-based Matrix Completion. It uses cumulative preference propagation to directly incorporate rating ordinality in GNN's message passing, allowing for users' stronger preferences to be more emphasized based on inherent orders of rating types. This process is complemented by interest regularization which facilitates preference learning using the underlying interest information. Our extensive experiments show that ROGMC consistently outperforms the existing strategies of using rating types for GNN. We expect that our attempt to explore the feasibility of utilizing rating ordinality for GNN may stimulate further research in this direction.
We present a novel method for matrix completion, specifically designed for matrices where one dimension significantly exceeds the other. Our Columns Selected Matrix Completion (CSMC) method combines Column Subset Selection and Low-Rank Matrix Completion to efficiently reconstruct incomplete datasets. In each step, CSMC solves a convex optimization problem. We introduce two algorithms to implement CSMC, each tailored to problems of different sizes. A formal analysis is provided, outlining the necessary assumptions and the probability of obtaining a correct solution. To assess the impact of matrix size, rank, and the ratio of missing entries on solution quality and computation time, we conducted experiments on synthetic data. The method was also applied to two real-world problems: recommendation systems and image inpainting. Our results show that CSMC provides solutions of the same quality as state-of-the-art matrix completion algorithms based on convex optimization, while achieving significant reductions in computational runtime.
Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.
Block-wise missing data poses significant challenges in real-world data imputation tasks. Compared to scattered missing data, block-wise gaps exacerbate adverse effects on subsequent analytic and machine learning tasks, as the lack of local neighboring elements significantly reduces the interpolation capability and predictive power. However, this issue has not received adequate attention. Most SOTA matrix completion methods appeared less effective, primarily due to overreliance on neighboring elements for predictions. We systematically analyze the issue and propose a novel matrix completion method ``BlockEcho" for a more comprehensive solution. This method creatively integrates Matrix Factorization (MF) within Generative Adversarial Networks (GAN) to explicitly retain long-distance inter-element relationships in the original matrix. Besides, we incorporate an additional discriminator for GAN, comparing the generator's intermediate progress with pre-trained MF results to constrain high-order feature distributions. Subsequently, we evaluate BlockEcho on public datasets across three domains. Results demonstrate superior performance over both traditional and SOTA methods when imputing block-wise missing data, especially at higher missing rates. The advantage also holds for scattered missing data at high missing rates. We also contribute on the analyses in providing theoretical justification on the optimality and convergence of fusing MF and GAN for missing block data.
We investigate a principled approach for symbolic operation completion (SOC), a minimal task for studying symbolic reasoning. While conceptually similar to matrix completion, SOC poses a unique challenge in modeling abstract relationships between discrete symbols. We demonstrate that SOC can be efficiently solved by a minimal model - a bilinear map - with a novel factorized architecture. Inspired by group representation theory, this architecture leverages matrix embeddings of symbols, modeling each symbol as an operator that dynamically influences others. Our model achieves perfect test accuracy on SOC with comparable or superior sample efficiency to Transformer baselines across most datasets, while boasting significantly faster learning speeds (100-1000$\times$). Crucially, the model exhibits an implicit bias towards learning general group structures, precisely discovering the unitary representations of underlying groups. This remarkable property not only confers interpretability but also significant implications for automatic symmetry discovery in geometric deep learning. Overall, our work establishes group theory as a powerful guiding principle for discovering abstract algebraic structures in deep learning, and showcases matrix representations as a compelling alternative to traditional vector embeddings for modeling symbolic relationships.
We consider the task of causal imputation, where we aim to predict the outcomes of some set of actions across a wide range of possible contexts. As a running example, we consider predicting how different drugs affect cells from different cell types. We study the index-only setting, where the actions and contexts are categorical variables with a finite number of possible values. Even in this simple setting, a practical challenge arises, since often only a small subset of possible action-context pairs have been studied. Thus, models must extrapolate to novel action-context pairs, which can be framed as a form of matrix completion with rows indexed by actions, columns indexed by contexts, and matrix entries corresponding to outcomes. We introduce a novel SCM-based model class, where the outcome is expressed as a counterfactual, actions are expressed as interventions on an instrumental variable, and contexts are defined based on the initial state of the system. We show that, under a linearity assumption, this setup induces a latent factor model over the matrix of outcomes, with an additional fixed effect term. To perform causal prediction based on this model class, we introduce simple extension to the Synthetic Interventions estimator (Agarwal et al., 2020). We evaluate several matrix completion approaches on the PRISM drug repurposing dataset, showing that our method outperforms all other considered matrix completion approaches.
This article introduces a new estimator of average treatment effects under unobserved confounding in modern data-rich environments featuring large numbers of units and outcomes. The proposed estimator is doubly robust, combining outcome imputation, inverse probability weighting, and a novel cross-fitting procedure for matrix completion. We derive finite-sample and asymptotic guarantees, and show that the error of the new estimator converges to a mean-zero Gaussian distribution at a parametric rate. Simulation results demonstrate the relevance of the formal properties of the estimators analyzed in this article.
We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r'\geq r$, we show that, with $\widetilde\Omega(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\| \leq \epsilon$ converges near-linearly to an $\epsilon$-neighborhood of $\rm{X}^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r'$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r'=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $\epsilon>0$, provided the initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis.
This paper studies the principal components (PC) estimator for high dimensional approximate factor models with weak factors in that the factor loading ($\boldsymbol{\Lambda}^0$) scales sublinearly in the number $N$ of cross-section units, i.e., $\boldsymbol{\Lambda}^{0\top} \boldsymbol{\Lambda}^0 / N^\alpha$ is positive definite in the limit for some $\alpha \in (0,1)$. While the consistency and asymptotic normality of these estimates are by now well known when the factors are strong, i.e., $\alpha=1$, the statistical properties for weak factors remain less explored. Here, we show that the PC estimator maintains consistency and asymptotical normality for any $\alpha\in(0,1)$, provided suitable conditions regarding the dependence structure in the noise are met. This complements earlier result by Onatski (2012) that the PC estimator is inconsistent when $\alpha=0$, and the more recent work by Bai and Ng (2023) who established the asymptotic normality of the PC estimator when $\alpha \in (1/2,1)$. Our proof strategy integrates the traditional eigendecomposition-based approach for factor models with leave-one-out analysis similar in spirit to those used in matrix completion and other settings. This combination allows us to deal with factors weaker than the former and at the same time relax the incoherence and independence assumptions often associated with the later.
Modern surveys with large sample sizes and growing mixed-type questionnaires require robust and scalable analysis methods. In this work, we consider recovering a mixed dataframe matrix, obtained by complex survey sampling, with entries following different canonical exponential distributions and subject to heterogeneous missingness. To tackle this challenging task, we propose a two-stage procedure: in the first stage, we model the entry-wise missing mechanism by logistic regression, and in the second stage, we complete the target parameter matrix by maximizing a weighted log-likelihood with a low-rank constraint. We propose a fast and scalable estimation algorithm that achieves sublinear convergence, and the upper bound for the estimation error of the proposed method is rigorously derived. Experimental results support our theoretical claims, and the proposed estimator shows its merits compared to other existing methods. The proposed method is applied to analyze the National Health and Nutrition Examination Survey data.
Matrix completion estimators are employed in causal panel data models to regulate the rank of the underlying factor model using nuclear norm minimization. This convex optimization problem enables concurrent regularization of a potentially high-dimensional set of covariates to shrink the model size. For valid finite sample inference, we adopt a permutation-based approach and prove its validity for any treatment assignment mechanism. Simulations illustrate the consistency of the proposed estimator in parameter estimation and variable selection. An application to public health policies in Germany demonstrates the data-driven model selection feature on empirical data and finds no effect of travel restrictions on the containment of severe Covid-19 infections.
Relationship inference from sparse data is an important task with applications ranging from product recommendation to drug discovery. A recently proposed linear model for sparse matrix completion has demonstrated surprising advantage in speed and accuracy over more sophisticated recommender systems algorithms. Here we extend the linear model to develop a shallow autoencoder for the dual neighborhood-regularized matrix completion problem. We demonstrate the speed and accuracy advantage of our approach over the existing state-of-the-art in predicting drug-target interactions and drug-disease associations.
Matrix completion is one of the crucial tools in modern data science research. Recently, a novel sampling model for matrix completion coined cross-concentrated sampling (CCS) has caught much attention. However, the robustness of the CCS model against sparse outliers remains unclear in the existing studies. In this paper, we aim to answer this question by exploring a novel Robust CCS Completion problem. A highly efficient non-convex iterative algorithm, dubbed Robust CUR Completion (RCURC), is proposed. The empirical performance of the proposed algorithm, in terms of both efficiency and robustness, is verified in synthetic and real datasets.