Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.
We consider the problem of matrix completion on an $n \times m$ matrix. We introduce the problem of Interpretable Matrix Completion that aims to provide meaningful insights for the low-rank matrix using side information. We show that the problem can be reformulated as a binary convex optimization problem. We design OptComplete, based on a novel concept of stochastic cutting planes to enable efficient scaling of the algorithm up to matrices of sizes $n=10^6$ and $m=10^6$. We report experiments on both synthetic and real-world datasets that show that OptComplete has favorable scaling behavior and accuracy when compared with state-of-the-art methods for other types of matrix completion, while providing insight on the factors that affect the matrix.
Matrix completion is one of the key problems in signal processing and machine learning. In recent years, deep-learning-based models have achieved state-of-the-art results in matrix completion. Nevertheless, they suffer from two drawbacks: (i) they can not be extended easily to rows or columns unseen during training; and (ii) their results are often degraded in case discrete predictions are required. This paper addresses these two drawbacks by presenting a deep matrix factorization model and a generic method to allow joint training of the factorization model and the discretization operator. Experiments on a real movie rating dataset show the efficacy of the proposed models.
Level-set optimization formulations with data-driven constraints minimize a regularization functional subject to matching observations to a given error level. These formulations are widely used, particularly for matrix completion and sparsity promotion in data interpolation and denoising. The misfit level is typically measured in the l2 norm, or other smooth metrics. In this paper, we present a new flexible algorithmic framework that targets nonsmooth level-set constraints, including L1, Linf, and even L0 norms. These constraints give greater flexibility for modeling deviations in observation and denoising, and have significant impact on the solution. Measuring error in the L1 and L0 norms makes the result more robust to large outliers, while matching many observations exactly. We demonstrate the approach for basis pursuit denoise (BPDN) problems as well as for extensions of BPDN to matrix factorization, with applications to interpolation and denoising of 5D seismic data. The new methods are particularly promising for seismic applications, where the amplitude in the data varies significantly, and measurement noise in low-amplitude regions can wreak havoc for standard Gaussian error models.
Learning by integrating multiple heterogeneous data sources is a common requirement in many tasks. Collective Matrix Factorization (CMF) is a technique to learn shared latent representations from arbitrary collections of matrices. It can be used to simultaneously complete one or more matrices, for predicting the unknown entries. Classical CMF methods assume linearity in the interaction of latent factors which can be restrictive and fails to capture complex non-linear interactions. In this paper, we develop the first deep-learning based method, called dCMF, for unsupervised learning of multiple shared representations, that can model such non-linear interactions, from an arbitrary collection of matrices. We address optimization challenges that arise due to dependencies between shared representations through Multi-Task Bayesian Optimization and design an acquisition function adapted for collective learning of hyperparameters. Our experiments show that dCMF significantly outperforms previous CMF algorithms in integrating heterogeneous data for predictive modeling. Further, on two tasks - recommendation and prediction of gene-disease association - dCMF outperforms state-of-the-art matrix completion algorithms that can utilize auxiliary sources of information.
Recently, techniques for applying convolutional neural networks to graph-structured data have emerged. Graph convolutional neural networks (GCNNs) have been used to address node and graph classification and matrix completion. Although the performance has been impressive, the current implementations have limited capability to incorporate uncertainty in the graph structure. Almost all GCNNs process a graph as though it is a ground-truth depiction of the relationship between nodes, but often the graphs employed in applications are themselves derived from noisy data or modelling assumptions. Spurious edges may be included; other edges may be missing between nodes that have very strong relationships. In this paper we adopt a Bayesian approach, viewing the observed graph as a realization from a parametric family of random graphs. We then target inference of the joint posterior of the random graph parameters and the node (or graph) labels. We present the Bayesian GCNN framework and develop an iterative learning procedure for the case of assortative mixed-membership stochastic block models. We present the results of experiments that demonstrate that the Bayesian formulation can provide better performance when there are very few labels available during the training process.
Inferring air quality from a limited number of observations is an essential task for monitoring and controlling air pollution. Existing inference methods typically use low spatial resolution data collected by fixed monitoring stations and infer the concentration of air pollutants using additional types of data, e.g., meteorological and traffic information. In this work, we focus on street-level air quality inference by utilizing data collected by mobile stations. We formulate air quality inference in this setting as a graph-based matrix completion problem and propose a novel variational model based on graph convolutional autoencoders. Our model captures effectively the spatio-temporal correlation of the measurements and does not depend on the availability of additional information apart from the street-network topology. Experiments on a real air quality dataset, collected with mobile stations, shows that the proposed model outperforms state-of-the-art approaches.
In this paper, we introduce a novel and robust approach to Quantized Matrix Completion (QMC). First, we propose a rank minimization problem with constraints induced by quantization bounds. Next, we form an unconstrained optimization problem by regularizing the rank function with Huber loss. Huber loss is leveraged to control the violation from quantization bounds due to two properties: 1- It is differentiable, 2- It is less sensitive to outliers than the quadratic loss. A Smooth Rank Approximation is utilized to endorse lower rank on the genuine data matrix. Thus, an unconstrained optimization problem with differentiable objective function is obtained allowing us to advantage from Gradient Descent (GD) technique. Novel and firm theoretical analysis on problem model and convergence of our algorithm to the global solution are provided. Another contribution of our work is that our method does not require projections or initial rank estimation unlike the state- of-the-art. In the Numerical Experiments Section, the noticeable outperformance of our proposed method in learning accuracy and computational complexity compared to those of the state-of- the-art literature methods is illustrated as the main contribution.
The last decade has witnessed an explosion in the development of models, theory and computational algorithms for "big data" analysis. In particular, distributed computing has served as a natural and dominating paradigm for statistical inference. However, the existing literature on parallel inference almost exclusively focuses on Euclidean data and parameters. While this assumption is valid for many applications, it is increasingly more common to encounter problems where the data or the parameters lie on a non-Euclidean space, like a manifold for example. Our work aims to fill a critical gap in the literature by generalizing parallel inference algorithms to optimization on manifolds. We show that our proposed algorithm is both communication efficient and carries theoretical convergence guarantees. In addition, we demonstrate the performance of our algorithm to the estimation of Fr\'echet means on simulated spherical data and the low-rank matrix completion problem over Grassmann manifolds applied to the Netflix prize data set.
Matrix completion is a widely used technique for image inpainting and personalized recommender system, etc. In this work, we focus on accelerating the matrix completion using faster randomized singular value decomposition (rSVD). Firstly, two fast randomized algorithms (rSVD-PI and rSVD- BKI) are proposed for handling sparse matrix. They make use of an eigSVD procedure and several accelerating skills. Then, with the rSVD-BKI algorithm and a new subspace recycling technique, we accelerate the singular value thresholding (SVT) method in [1] to realize faster matrix completion. Experiments show that the proposed rSVD algorithms can be 6X faster than the basic rSVD algorithm [2] while keeping same accuracy. For image inpainting and movie-rating estimation problems, the proposed accelerated SVT algorithm consumes 15X and 8X less CPU time than the methods using svds and lansvd respectively, without loss of accuracy.
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `complete' guarantee for ST-miss. This means that we can prove that under assumptions on only the algorithm inputs, the output subspace estimates are close to the true data subspaces at all times. Our guarantees hold under mild and easily interpretable assumptions, and allow the underlying subspace to change with time in a piecewise constant fashion. In contrast, all existing guarantees for ST are partial results and assume a fixed unknown subspace. Extensive numerical experiments are shown to back up our theoretical claims. Finally, our solution can be interpreted as a provably correct mini-batch and memory-efficient solution to low-rank Matrix Completion (MC).
Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. In any problem where one seeks to recover a structure rather than the exact initial matrices, techniques developed thus far relying on certificates of optimality, will fail. We present an important extension of these methods, that allows the treatment of such problems.
Visual complexity identifies the level of intricacy and details in an image or the level of difficulty to describe the image. It is an important concept in a variety of areas such as cognitive psychology, computer vision and visualization, and advertisement. Yet, efforts to create large, downloadable image datasets with diverse content and unbiased groundtruthing are lacking. In this work, we introduce Savoias, a visual complexity dataset that compromises of more than 1,400 images from seven image categories relevant to the above research areas, namely Scenes, Advertisements, Visualization and infographics, Objects, Interior design, Art, and Suprematism. The images in each category portray diverse characteristics including various low-level and high-level features, objects, backgrounds, textures and patterns, text, and graphics. The ground truth for Savoias is obtained by crowdsourcing more than 37,000 pairwise comparisons of images using the forced-choice methodology and with more than 1,600 contributors. The resulting relative scores are then converted to absolute visual complexity scores using the Bradley-Terry method and matrix completion. When applying five state-of-the-art algorithms to analyze the visual complexity of the images in the Savoias dataset, we found that the scores obtained from these baseline tools only correlate well with crowdsourced labels for abstract patterns in the Suprematism category (Pearson correlation r=0.84). For the other categories, in particular, the objects and advertisement categories, low correlation coefficients were revealed (r=0.3 and 0.56, respectively). These findings suggest that (1) state-of-the-art approaches are mostly insufficient and (2) Savoias enables category-specific method development, which is likely to improve the impact of visual complexity analysis on specific application areas, including computer vision.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defective vision, or assessment of gait in patients with neurological disorders. Since the data collection is often costly and inconvenient, estimation of progression from sparse observations is of great interest for practitioners. From the statistical standpoint, such data is often analyzed in the context of a mixed-effect model where time is treated as both a fixed-effect (population progression curve) and a random-effect (individual variability). Alternatively, researchers analyze Gaussian processes or functional data where observations are assumed to be drawn from a certain distribution of processes. These models are flexible but rely on probabilistic assumptions, require very careful implementation, specific to the given problem, and tend to be slow in practice. In this study, we propose an alternative elementary framework for analyzing longitudinal data, relying on matrix completion. Our method yields estimates of progression curves by iterative application of the Singular Value Decomposition. Our framework covers multivariate longitudinal data, regression, and can be easily extended to other settings. As it relies on existing tools for matrix algebra it is efficient and easy to implement. We apply our methods to understand trends of progression of motor impairment in children with Cerebral Palsy. Our model approximates individual progression curves and explains 30% of the variability. Low-rank representation of progression trends enables identification of different progression trends in subtypes of Cerebral Palsy.
We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a low-rank matrix from various information deficits by using the principle of low-rank completion. However, the current theoretical guarantees for low-rank MC do not apply to clipped matrices, as the deficit depends on the underlying values. Therefore, the feasibility of clipped matrix completion (CMC) is not trivial. In this paper, we first provide a theoretical guarantee for the exact recovery of CMC by using a trace-norm minimization algorithm. Furthermore, we propose practical CMC algorithms by extending ordinary MC methods. Our extension is to use the squared hinge loss in place of the squared loss for reducing the penalty of over-estimation on clipped entries. We also propose a novel regularization term tailored for CMC. It is a combination of two trace-norm terms, and we theoretically bound the recovery error under the regularization. We demonstrate the effectiveness of the proposed methods through experiments using both synthetic and benchmark data for recommendation systems.
Multi-target prediction (MTP) is concerned with the simultaneous prediction of multiple target variables of diverse type. Due to its enormous application potential, it has developed into an active and rapidly expanding research field that combines several subfields of machine learning, including multivariate regression, multi-label classification, multi-task learning, dyadic prediction, zero-shot learning, network inference, and matrix completion. In this paper, we present a unifying view on MTP problems and methods. First, we formally discuss commonalities and differences between existing MTP problems. To this end, we introduce a general framework that covers the above subfields as special cases. As a second contribution, we provide a structured overview of MTP methods. This is accomplished by identifying a number of key properties, which distinguish such methods and determine their suitability for different types of problems. Finally, we also discuss a few challenges for future research.
In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based sparse and low-rank recovery is of considerable interest and it in fact is a main driver of the recent progress in nonconvex and nonsmooth optimization. This paper gives an overview of this topic in various fields in signal processing, statistics and machine learning, including compressive sensing (CS), sparse regression and variable selection, sparse signals separation, sparse principal component analysis (PCA), large covariance and inverse covariance matrices estimation, matrix completion, and robust PCA. We present recent developments of nonconvex regularization based sparse and low-rank recovery in these fields, addressing the issues of penalty selection, applications and the convergence of nonconvex algorithms. Code is available at https://github.com/FWen/ncreg.git.
We consider the problem of ranking a set of items from pairwise comparisons in the presence of features associated with the items. Recent works have established that $O(n\log(n))$ samples are needed to rank well when there is no feature information present. However, this might be sub-optimal in the presence of associated features. We introduce a new probabilistic preference model called feature-Bradley-Terry-Luce (f-BTL) model that generalizes the standard BTL model to incorporate feature information. We present a new least squares based algorithm called fBTL-LS which we show requires much lesser than $O(n\log(n))$ pairs to obtain a good ranking -- precisely our new sample complexity bound is of $O(\alpha\log \alpha)$, where $\alpha$ denotes the number of `independent items' of the set, in general $\alpha << n$. Our analysis is novel and makes use of tools from classical graph matching theory to provide tighter bounds that sheds light on the true complexity of the ranking problem, capturing the item dependencies in terms of their feature representations. This was not possible with earlier matrix completion based tools used for this problem. We also prove an information theoretic lower bound on the required sample complexity for recovering the underlying ranking, which essentially shows the tightness of our proposed algorithms. The efficacy of our proposed algorithms are validated through extensive experimental evaluations on a variety of synthetic and real world datasets.