We consider the problem of matrix completion with graphs as side information depicting the interrelations between variables. The key challenge lies in leveraging the similarity structure of the graph to enhance matrix recovery. Existing approaches, primarily based on graph Laplacian regularization, suffer from several limitations: (1) they focus only on the similarity between neighboring variables, while overlooking long-range correlations; (2) they are highly sensitive to false edges in the graphs and (3) they lack theoretical guarantees regarding statistical and computational complexities. To address these issues, we propose in this paper a novel graph regularized matrix completion algorithm called GSGD, based on preconditioned projected gradient descent approach. We demonstrate that GSGD effectively captures the higher-order correlation information behind the graphs, and achieves superior robustness and stability against the false edges. Theoretically, we prove that GSGD achieves linear convergence to the global optimum with near-optimal sample complexity, providing the first theoretical guarantees for both recovery accuracy and efficacy in the perspective of nonconvex optimization. Our numerical experiments on both synthetic and real-world data further validate that GSGD achieves superior recovery accuracy and scalability compared with several popular alternatives.
This paper addresses the problem of mixed-membership estimation in networks, where the goal is to efficiently estimate the latent mixed-membership structure from the observed network. Recognizing the widespread availability and valuable information carried by node covariates, we propose a novel network model that incorporates both community information, as represented by the Degree-Corrected Mixed Membership (DCMM) model, and node covariate similarities to determine connections. We investigate the regularized maximum likelihood estimation (MLE) for this model and demonstrate that our approach achieves optimal estimation accuracy for both the similarity matrix and the mixed-membership, in terms of both the Frobenius norm and the entrywise loss. Since directly analyzing the original convex optimization problem is intractable, we employ nonconvex optimization to facilitate the analysis. A key contribution of our work is identifying a crucial assumption that bridges the gap between convex and nonconvex solutions, enabling the transfer of statistical guarantees from the nonconvex approach to its convex counterpart. Importantly, our analysis extends beyond the MLE loss and the mean squared error (MSE) used in matrix completion problems, generalizing to all the convex loss functions. Consequently, our analysis techniques extend to a broader set of applications, including ranking problems based on pairwise comparisons. Finally, simulation experiments validate our theoretical findings, and real-world data analyses confirm the practical relevance of our model.
We consider the noisy matrix sensing problem in the over-parameterization setting, where the estimated rank $r$ is larger than the true rank $r_\star$ of the target matrix $X_\star$. Specifically, our main objective is to recover a matrix $ X_\star \in \mathbb{R}^{n_1 \times n_2} $ with rank $ r_\star $ from noisy measurements using an over-parameterized factorization $ LR^\top $, where $ L \in \mathbb{R}^{n_1 \times r}, \, R \in \mathbb{R}^{n_2 \times r} $ and $ \min\{n_1, n_2\} \ge r > r_\star $, with $ r_\star $ being unknown. Recently, preconditioning methods have been proposed to accelerate the convergence of matrix sensing problem compared to vanilla gradient descent, incorporating preconditioning terms $ (L^\top L + \lambda I)^{-1} $ and $ (R^\top R + \lambda I)^{-1} $ into the original gradient. However, these methods require careful tuning of the damping parameter $\lambda$ and are sensitive to step size. To address these limitations, we propose the alternating preconditioned gradient descent (APGD) algorithm, which alternately updates the two factor matrices, eliminating the need for the damping parameter $\lambda$ and enabling faster convergence with larger step sizes. We theoretically prove that APGD convergences to a near-optimal error at a linear rate. We further show that APGD can be extended to deal with other low-rank matrix estimation tasks, also with a theoretical guarantee of linear convergence. To validate the effectiveness and scalability of the proposed APGD, we conduct simulated and real-world experiments on a wide range of low-rank estimation problems, including noisy matrix sensing, weighted PCA, 1-bit matrix completion, and matrix completion. The extensive results demonstrate that APGD consistently achieves the fastest convergence and the lowest computation time compared to the existing alternatives.
In this work, we propose norm-bounded low-rank adaptation (NB-LoRA) for parameter-efficient fine tuning. We introduce two parameterizations that allow explicit bounds on each singular value of the weight adaptation matrix, which can therefore satisfy any prescribed unitarily invariant norm bound, including the Schatten norms (e.g., nuclear, Frobenius, spectral norm). The proposed parameterizations are unconstrained and complete, i.e. they cover all matrices satisfying the prescribed rank and norm constraints. Experiments on vision fine-tuning benchmarks show that the proposed approach can achieve good adaptation performance while avoiding model catastrophic forgetting and also substantially improve robustness to a wide range of hyper-parameters, including adaptation rank, learning rate and number of training epochs. We also explore applications in privacy-preserving model merging and low-rank matrix completion.
Many models used in machine learning have become so large that even computer computation of the full gradient of the loss function is impractical. This has made it necessary to efficiently train models using limited available information, such as batch size and learning rate. We have theoretically analyzed the use of Riemannian stochastic gradient descent (RSGD) and found that using an increasing batch size leads to faster RSGD convergence than using a constant batch size not only with a constant learning rate but also with a decaying learning rate, such as cosine annealing decay and polynomial decay. In particular, RSGD has a better convergence rate $O(\frac{1}{\sqrt{T}})$ than the existing rate $O(\frac{\sqrt{\log T}}{\sqrt[4]{T}})$ with a diminishing learning rate, where $T$ is the number of iterations. The results of experiments on principal component analysis and low-rank matrix completion problems confirmed that, except for the MovieLens dataset and a constant learning rate, using a polynomial growth batch size or an exponential growth batch size results in better performance than using a constant batch size.
The Netflix problem (from machine learning) asks the following. Given a ratings matrix in which each entry $(i,j)$ represents the rating of movie $j$ by customer $i$, if customer $i$ has watched movie $j$, and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. The remaining entries are predicted so as to minimize the {\it rank} of the completed matrix. In this survey we study a more general problem, in which instead of knowing specific matrix elements, we know linear relations on such elements. We describe applications of these results to embeddings of graphs in surfaces (more precisely, embeddings with rotation systems, and embeddings modulo 2).
The main goal of group testing is to identify a small number of defective items in a large population of items. A test on a subset of items is positive if the subset contains at least one defective item and negative otherwise. In non-adaptive design, all tests can be tested simultaneously and represented by a measurement matrix in which a row and a column represent a test and an item, respectively. An entry in row $i$ and column $j$ is 1 if item $j$ belongs to the test $i$ and is 0 otherwise. Given an unknown set of defective items, the objective is to design a measurement matrix such that, by observing its corresponding outcome vector, the defective items can be recovered efficiently. The basic trait of this approach is that the measurement matrix has remained unchanged throughout the course of generating the outcome vector and recovering defective items. In this paper, we study the case in which some entries in the measurement matrix are erased, called \emph{the missing measurement matrix}, before the recovery phase of the defective items, and our objective is to fully recover the measurement matrix from the missing measurement matrix. In particular, we show that some specific rows with erased entries provide information aiding the recovery while others do not. Given measurement matrices and erased entries follow the Bernoulli distribution, we show that before the erasing event happens, sampling sufficient sets of defective items and their corresponding outcome vectors can help us recover the measurement matrix from the missing measurement matrix.
We consider a robust and self-reliant (or "egoistic") variation of the rigid body localization (RBL) problem, in which a primary rigid body seeks to estimate the pose (i.e., location and orientation) of another rigid body (or "target"), relative to its own, without the assistance of external infrastructure, without prior knowledge of the shape of the target, and taking into account the possibility that the available observations are incomplete. Three complementary contributions are then offered for such a scenario. The first is a method to estimate the translation vector between the center point of both rigid bodies, which unlike existing techniques does not require that both objects have the same shape or even the same number of landmark points. This technique is shown to significantly outperform the state-of-the-art (SotA) under complete information, but to be sensitive to data erasures, even when enhanced by matrix completion methods. The second contribution, designed to offer improved performance in the presence of incomplete information, offers a robust alternative to the latter, at the expense of a slight relative loss under complete information. Finally, the third contribution is a scheme for the estimation of the rotation matrix describing the relative orientation of the target rigid body with respect to the primary. Comparisons of the proposed schemes and SotA techniques demonstrate the advantage of the contributed methods in terms of root mean square error (RMSE) performance under fully complete information and incomplete conditions.
Inspired by the impact of the Goemans-Williamson algorithm on combinatorial optimization, we construct an analogous relax-then-sample strategy for low-rank optimization problems. First, for orthogonally constrained quadratic optimization problems, we derive a semidefinite relaxation and a randomized rounding scheme, which obtains provably near-optimal solutions, mimicking the blueprint from Goemans and Williamson for the Max-Cut problem. We then extend our approach to generic low-rank optimization problems by developing new semidefinite relaxations that are both tighter and more broadly applicable than those in prior works. Although our original proposal introduces large semidefinite matrices as decision variables, we show that most of the blocks in these matrices can be safely omitted without altering the optimal value, hence improving the scalability of our approach. Using several examples (including matrix completion, basis pursuit, and reduced-rank regression), we show how to reduce the size of our relaxation even further. Finally, we numerically illustrate the effectiveness and scalability of our relaxation and our sampling scheme on orthogonally constrained quadratic optimization and matrix completion problems.
Robust matrix completion (RMC) is a widely used machine learning tool that simultaneously tackles two critical issues in low-rank data analysis: missing data entries and extreme outliers. This paper proposes a novel scalable and learnable non-convex approach, coined Learned Robust Matrix Completion (LRMC), for large-scale RMC problems. LRMC enjoys low computational complexity with linear convergence. Motivated by the proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. Furthermore, this paper proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from fix-number iterations to infinite iterations. The superior empirical performance of LRMC is verified with extensive experiments against state-of-the-art on synthetic datasets and real applications, including video background subtraction, ultrasound imaging, face modeling, and cloud removal from satellite imagery.
Matrix completion has gained considerable interest in recent years. The goal of matrix completion is to predict the unknown entries of a partially observed matrix using its known entries. Although common applications feature discrete rating-scale data, such as user-product rating matrices in recommender systems or surveys in the social and behavioral sciences, methods for matrix completion are almost always designed for and studied in the context of continuous data. Furthermore, only a small subset of the literature considers matrix completion in the presence of corrupted observations despite their common occurrence in practice. Examples include attacks on recommender systems (i.e., malicious users deliberately manipulating ratings to influence the recommender system to their advantage), or careless respondents in surveys (i.e., respondents providing answers irrespective of what the survey asks of them due to a lack of attention). We introduce a matrix completion algorithm that is tailored towards the discrete nature of rating-scale data and robust to the presence of corrupted observations. In addition, we investigate the performance of the proposed method and its competitors with discrete rating-scale (rather than continuous) data as well as under various missing data mechanisms and types of corrupted observations.
Noisy matrix completion has attracted significant attention due to its applications in recommendation systems, signal processing and image restoration. Most existing works rely on (weighted) least squares methods under various low-rank constraints. However, minimizing the sum of squared residuals is not always efficient, as it may ignore the potential structural information in the residuals. In this study, we propose a novel residual spectral matching criterion that incorporates not only the numerical but also locational information of residuals. This criterion is the first in noisy matrix completion to adopt the perspective of low-rank perturbation of random matrices and exploit the spectral properties of sparse random matrices. We derive optimal statistical properties by analyzing the spectral properties of sparse random matrices and bounding the effects of low-rank perturbations and partial observations. Additionally, we propose algorithms that efficiently approximate solutions by constructing easily computable pseudo-gradients. The iterative process of the proposed algorithms ensures convergence at a rate consistent with the optimal statistical error bound. Our method and algorithms demonstrate improved numerical performance in both simulated and real data examples, particularly in environments with high noise levels.
We propose to transfer representational knowledge from multiple sources to a target noisy matrix completion task by aggregating singular subspaces information. Under our representational similarity framework, we first integrate linear representation information by solving a two-way principal component analysis problem based on a properly debiased matrix-valued dataset. After acquiring better column and row representation estimators from the sources, the original high-dimensional target matrix completion problem is then transformed into a low-dimensional linear regression, of which the statistical efficiency is guaranteed. A variety of extensional arguments, including post-transfer statistical inference and robustness against negative transfer, are also discussed alongside. Finally, extensive simulation results and a number of real data cases are reported to support our claims.
The theory of evidence reasoning has been applied to collective decision-making in recent years. However, existing distributed evidence fusion methods lead to participants' preference leakage and fusion failures as they directly exchange raw evidence and do not assess evidence credibility like centralized credible evidence fusion (CCEF) does. To do so, a privacy-preserving distributed credible evidence fusion method with three-level consensus (PCEF) is proposed in this paper. In evidence difference measure (EDM) neighbor consensus, an evidence-free equivalent expression of EDM among neighbored agents is derived with the shared dot product protocol for pignistic probability and the identical judgment of two events with maximal subjective probabilities, so that evidence privacy is guaranteed due to such irreversible evidence transformation. In EDM network consensus, the non-neighbored EDMs are inferred and neighbored EDMs reach uniformity via interaction between linear average consensus (LAC) and low-rank matrix completion with rank adaptation to guarantee EDM consensus convergence and no solution of inferring raw evidence in numerical iteration style. In fusion network consensus, a privacy-preserving LAC with a self-cancelling differential privacy term is proposed, where each agent adds its randomness to the sharing content and step-by-step cancels such randomness in consensus iterations. Besides, the sufficient condition of the convergence to the CCEF is explored, and it is proven that raw evidence is impossibly inferred in such an iterative consensus. The simulations show that PCEF is close to CCEF both in credibility and fusion results and obtains higher decision accuracy with less time-comsuming than existing methods.
Background subtraction refers to extracting the foreground from an observed video, and is the fundamental problem of various applications. There are two kinds of popular methods to deal with background separation, namely, robust principal component analysis (RPCA) and low-rank matrix factorization (LRMF). Nevertheless, the drawback of RPCA requires tuning penalty parameter to attain an ideal result. Compared with RPCA, the ℓ1 -norm based LRMF does not involve extra parameters tuning, but it is challenging to optimize the ℓ1 -norm based minimization because of the nonsmooth ℓ1 -norm. In addition, it becomes time-consuming to find the optimal solution. In this work, we propose to employ smooth ℓ1,ϵ -norm, an approximation of ℓ1 -norm, to tackle background subtraction. Thus, the proposed model inherits the superiority of LRMF and even becomes tractable. Then the resultant optimization problem is solved by alternating minimization and gradient descent where the step-size of the gradient descent is adaptively updated via backtracking line searching approach. The proposed method is proved to be locally convergent. Experimental results on synthetic and real-world data demonstrate that our method outperforms the state-of-the-art algorithms in terms of reconstruction loss, computational speed and hardware performance.
With the wide deployment of digital image capturing equipment, the need of denoising to produce a crystal clear image from noisy capture environment has become indispensable. In this article, a novel type-2 fuzzy-based filter is proposed for denoising images corrupted by impulse noise, especially for the high density of salt-and-pepper noise. It operates two stages, namely, type-2 fuzzy identifier and matrix completion denoiser. In the proposed method, the type-2 fuzzy identifier is first employed to identify and trim the entries contaminated by impulse noise in the data matrix from fuzzy system. Then, the trimmed data matrix is utilized to retrieve the noiseless data matrix with the matrix completion technology. Herein, a novel matrix completion technique is developed without a priori rank information compared to its counterparts. Simulation results are presented, which vividly show the denoised images obtained by the proposed method can achieve crystal clear image with strong structural integrity, and are showing good performance in terms of peak signal-to-noise ratio.
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.
Accurately estimating spatiotemporal traffic states on freeways is a significant challenge due to limited sensor deployment and potential data corruption. In this study, we propose an efficient and robust low-rank model for precise spatiotemporal traffic speed state estimation (TSE) using lowpenetration vehicle trajectory data. Leveraging traffic wave priors, an oblique grid-based matrix is first designed to transform the inherent dependencies of spatiotemporal traffic states into the algebraic low-rankness of a matrix. Then, with the enhanced traffic state low-rankness in the oblique matrix, a low-rank matrix completion method is tailored to explicitly capture spatiotemporal traffic propagation characteristics and precisely reconstruct traffic states. In addition, an anomaly-tolerant module based on a sparse matrix is developed to accommodate corrupted data input and thereby improve the TSE model robustness. Notably, driven by the understanding of traffic waves, the computational complexity of the proposed efficient method is only correlated with the problem size itself, not with dataset size and hyperparameter selection prevalent in existing studies. Extensive experiments demonstrate the effectiveness, robustness, and efficiency of the proposed model. The performance of the proposed method achieves up to a 12% improvement in Root Mean Squared Error (RMSE) in the TSE scenarios and an 18% improvement in RMSE in the robust TSE scenarios, and it runs more than 20 times faster than the state-of-the-art (SOTA) methods.
Rating is a typical user explicit feedback that visually reflects how much a user likes a related item. The (rating) matrix completion is essentially a rating prediction process, which is also a significant problem in recommender systems. Recently, graph neural networks (GNNs) have been widely used in matrix completion, which captures users' preferences over items by formulating a rating matrix as a bipartite graph. However, existing methods are susceptible due to data sparsity and long-tail distribution in real-world scenarios. Moreover, the messaging mechanism of GNNs makes it difficult to capture high-order correlations and constraints between nodes, which are essentially useful in recommendation tasks. To tackle these challenges, we propose a Multi-Channel Hypergraph Contrastive Learning framework for matrix completion, named MHCL. Specifically, MHCL adaptively learns hypergraph structures to capture high-order correlations between nodes and jointly captures local and global collaborative relationships through attention-based cross-view aggregation. Additionally, to consider the magnitude and order information of ratings, we treat different rating subgraphs as different channels, encourage alignment between adjacent ratings, and further achieve the mutual enhancement between different ratings through multi-channel cross-rating contrastive learning. Extensive experiments on five public datasets demonstrate that the proposed method significantly outperforms the current state-of-the-art approaches.
Recent analysis on the training dynamics of Transformers has unveiled an interesting characteristic: the training loss plateaus for a significant number of training steps, and then suddenly (and sharply) drops to near--optimal values. To understand this phenomenon in depth, we formulate the low-rank matrix completion problem as a masked language modeling (MLM) task, and show that it is possible to train a BERT model to solve this task to low error. Furthermore, the loss curve shows a plateau early in training followed by a sudden drop to near-optimal values, despite no changes in the training procedure or hyper-parameters. To gain interpretability insights into this sudden drop, we examine the model's predictions, attention heads, and hidden states before and after this transition. Concretely, we observe that (a) the model transitions from simply copying the masked input to accurately predicting the masked entries; (b) the attention heads transition to interpretable patterns relevant to the task; and (c) the embeddings and hidden states encode information relevant to the problem. We also analyze the training dynamics of individual model components to understand the sudden drop in loss.