In this paper, we propose a novel solution for non-convex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of sub-problems corresponding to each variable, and then iteratively optimize each sub-problem using a fixed updating rule. However, due to the intrinsic non-convexity of the original optimization problem, the optimization can usually be trapped into spurious local minimum even when each sub-problem can be optimally solved at each iteration. Meanwhile, learning-based approaches, such as deep unfolding algorithms, are highly limited by the lack of labelled data and restricted explainability. To tackle these issues, we propose a meta-learning based alternating minimization (MLAM) method, which aims to minimize a partial of the global losses over iterations instead of carrying minimization on each sub-problem, and it tends to learn an adaptive strategy to replace the handcrafted counterpart resulting in advance on superior performance. Meanwhile, the proposed MLAM still maintains the original algorithmic principle, which contributes to a better interpretability. We evaluate the proposed method on two representative problems, namely, bi-linear inverse problem: matrix completion, and non-linear problem: Gaussian mixture models. The experimental results validate that our proposed approach outperforms AM-based methods in standard settings, and is able to achieve effective optimization in challenging cases while other comparing methods would typically fail.
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as both an iteratively reweighted least squares (IRLS) algorithm and a saddle-escaping smoothing Newton method applied to a non-convex rank surrogate objective. It combines the favorable data efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. Our method attains a local quadratic convergence rate already for a number of samples that is close to the information theoretical limit. We show in numerical experiments that unlike many state-of-the-art approaches, our approach is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples.
A model-based collaborative filtering (CF) approach utilizing fast adaptive randomized singular value decomposition (SVD) is proposed for the matrix completion problem in recommender system. Firstly, a fast adaptive PCA frameworkis presented which combines the fixed-precision randomized matrix factorization algorithm [1] and accelerating skills for handling large sparse data. Then, a novel termination mechanism for the adaptive PCA is proposed to automatically determine a number of latent factors for achieving the near optimal prediction accuracy during the subsequent model-based CF. The resulted CF approach has good accuracy while inheriting high runtime efficiency. Experiments on real data show that, the proposed adaptive PCA is up to 2.7X and 6.7X faster than the original fixed-precision SVD approach [1] and svds in Matlab repsectively, while preserving accuracy. The proposed model-based CF approach is able to efficiently process the MovieLens data with 20M ratings and exhibits more than 10X speedup over the regularized matrix factorization based approach [2] and the fast singular value thresholding approach [3] with comparable or better accuracy. It also owns the advantage of parameter free. Compared with the deep-learning-based CF approach, the proposed approach is much more computationally efficient, with just marginal performance loss.
It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low rank are insufficient to perform image restoration well. In this work, we propose a Regularized Deep Matrix Factorized (RDMF) model for image restoration, which utilizes the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. We demonstrate the effectiveness of our RDMF model with extensive experiments, in which our method surpasses the state of art models in common examples, especially for the restoration from very few observations. Our work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularization.
Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data popularized as Robust PCA (Candes et al., 2011; Chandrasekaran et al., 2009). Compressed sensing, matrix completion, and their variants (Eldar and Kutyniok, 2012; Foucart and Rauhut, 2013) have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that $m\times n$ matrices that can be expressed as the sum of a rank-$r$ matrix and a $s$-sparse matrix can be recovered by computationally tractable methods from $\mathcal{O}(r(m+n-r)+s)\log(mn/s)$ linear measurements. More specifically, we establish that the low-rank plus sparse matrix set is closed provided the incoherence of the low-rank component is upper bounded as $\mu<\sqrt{mn}/(r\sqrt{s})$, and subsequently, the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided $p/mn$, $s/p$, and $r(m+n-r)/p$ remain fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operator's RIC's are sufficiently small. These results also provably solve convex and non-convex formulation of Robust PCA with the asymptotically optimal fraction of corruptions $\alpha=\mathcal{O}\left(1/(\mu r) \right)$, where $s = \alpha^2 mn$, and improve the previously best known guarantees by not requiring that the fraction of corruptions is spread in every column and row by being upper bounded by $\alpha$. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/static-background separation, and multispectral imaging.
Recommendation models can effectively estimate underlying user interests and predict one's future behaviors by factorizing an observed user-item rating matrix into products of two sets of latent factors. However, the user-specific embedding factors can only be learned in a transductive way, making it difficult to handle new users on-the-fly. In this paper, we propose an inductive collaborative filtering framework that contains two representation models. The first model follows conventional matrix factorization which factorizes a group of key users' rating matrix to obtain meta latents. The second model resorts to attention-based structure learning that estimates hidden relations from query to key users and learns to leverage meta latents to inductively compute embeddings for query users via neural message passing. Our model enables inductive representation learning for users and meanwhile guarantees equivalent representation capacity as matrix factorization. Experiments demonstrate that our model achieves promising results for recommendation on few-shot users with limited training ratings and new unseen users which are commonly encountered in open-world recommender systems.
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.
Modern large scale datasets are often plagued with missing entries. For tabular data with missing values, a flurry of imputation algorithms solve for a complete matrix which minimizes some penalized reconstruction error. However, almost none of them can estimate the uncertainty of its imputations. This paper proposes a probabilistic and scalable framework for missing value imputation with quantified uncertainty. Our model, the Low Rank Gaussian Copula, augments a standard probabilistic model, Probabilistic Principal Component Analysis, with marginal transformations for each column that allow the model to better match the distribution of the data. It naturally handles Boolean, ordinal, and real-valued observations and quantifies the uncertainty in each imputation. The time required to fit the model scales linearly with the number of rows and the number of columns in the dataset. Empirical results show the method yields state-of-the-art imputation accuracy across a wide range of data types, including those with high rank. Our uncertainty measure predicts imputation error well: entries with lower uncertainty do have lower imputation error (on average). Moreover, for real-valued data, the resulting confidence intervals are well-calibrated.
Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approaches can still be computationally expensive especially for ill-conditioned matrices: the convergence rate of gradient descent depends linearly on the condition number of the low-rank matrix, while the per-iteration cost of alternating minimization is often prohibitive for large matrices. The goal of this paper is to set forth a competitive algorithmic approach dubbed Scaled Gradient Descent (ScaledGD) which can be viewed as pre-conditioned or diagonally-scaled gradient descent, where the pre-conditioners are adaptive and iteration-varying with a minimal computational overhead. With tailored variants for low-rank matrix sensing, robust principal component analysis and matrix completion, we theoretically show that ScaledGD achieves the best of both worlds: it converges linearly at a rate independent of the condition number of the low-rank matrix similar as alternating minimization, while maintaining the low per-iteration cost of gradient descent. Our analysis is also applicable to general loss functions that are restricted strongly convex and smooth over low-rank matrices. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties over a wide range of low-rank matrix estimation tasks.
Large-scale population-based studies in medicine are a key resource towards better diagnosis, monitoring, and treatment of diseases. They also serve as enablers of clinical decision support systems, in particular Computer Aided Diagnosis (CADx) using machine learning (ML). Numerous ML approaches for CADx have been proposed in literature. However, these approaches assume full data availability, which is not always feasible in clinical data. To account for missing data, incomplete data samples are either removed or imputed, which could lead to data bias and may negatively affect classification performance. As a solution, we propose an end-to-end learning of imputation and disease prediction of incomplete medical datasets via Multigraph Geometric Matrix Completion (MGMC). MGMC uses multiple recurrent graph convolutional networks, where each graph represents an independent population model based on a key clinical meta-feature like age, sex, or cognitive function. Graph signal aggregation from local patient neighborhoods, combined with multigraph signal fusion via self-attention, has a regularizing effect on both matrix reconstruction and classification performance. Our proposed approach is able to impute class relevant features as well as perform accurate classification on two publicly available medical datasets. We empirically show the superiority of our proposed approach in terms of classification and imputation performance when compared with state-of-the-art approaches. MGMC enables disease prediction in multimodal and incomplete medical datasets. These findings could serve as baseline for future CADx approaches which utilize incomplete datasets.
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on which the implicit regularization drives all norms (and quasi-norms) towards infinity. Our results suggest that, rather than perceiving the implicit regularization via norms, a potentially more useful interpretation is minimization of rank. We demonstrate empirically that this interpretation extends to a certain class of non-linear neural networks, and hypothesize that it may be key to explaining generalization in deep learning.
Structured low-rank (SLR) algorithms, which exploit annihilation relations between the Fourier samples of a signal resulting from different properties, is a powerful image reconstruction framework in several applications. This scheme relies on low-rank matrix completion to estimate the annihilation relations from the measurements. The main challenge with this strategy is the high computational complexity of matrix completion. We introduce a deep learning (DL) approach to significantly reduce the computational complexity. Specifically, we use a convolutional neural network (CNN)-based filterbank that is trained to estimate the annihilation relations from imperfect (under-sampled and noisy) k-space measurements of Magnetic Resonance Imaging (MRI). The main reason for the computational efficiency is the pre-learning of the parameters of the non-linear CNN from exemplar data, compared to SLR schemes that learn the linear filterbank parameters from the dataset itself. Experimental comparisons show that the proposed scheme can enable calibration-less parallel MRI; it can offer performance similar to SLR schemes while reducing the runtime by around three orders of magnitude. Unlike pre-calibrated and self-calibrated approaches, the proposed uncalibrated approach is insensitive to motion errors and affords higher acceleration. The proposed scheme also incorporates image domain priors that are complementary, thus significantly improving the performance over that of SLR schemes.
In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank matrix completion formulation. This framework is centered on the fundamental duality between the compactness (e.g., sparsity) of the continuous signal and the rank of a structured matrix, whose entries are functions of the signal. This property enables the reformulation of the signal recovery as a low-rank structured matrix completion, which comes with performance guarantees. We will also review fast algorithms that are comparable in complexity to current compressed sensing methods, which enables the application of the framework to large-scale magnetic resonance (MR) recovery problems. The remarkable flexibility of the formulation can be used to exploit signal properties that are difficult to capture by current sparse and low-rank optimization strategies. We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic MRI.
Matrix completion is often applied to data with entries missing not at random (MNAR). For example, consider a recommendation system where users tend to only reveal ratings for items they like. In this case, a matrix completion method that relies on entries being revealed at uniformly sampled row and column indices can yield overly optimistic predictions of unseen user ratings. Recently, various papers have shown that we can reduce this bias in MNAR matrix completion if we know the probabilities of different matrix entries being missing. These probabilities are typically modeled using logistic regression or naive Bayes, which make strong assumptions and lack guarantees on the accuracy of the estimated probabilities. In this paper, we suggest a simple approach to estimating these probabilities that avoids these shortcomings. Our approach follows from the observation that missingness patterns in real data often exhibit low nuclear norm structure. We can then estimate the missingness probabilities by feeding the (always fully-observed) binary matrix specifying which entries are revealed or missing to an existing nuclear-norm-constrained matrix completion algorithm by Davenport et al. [2014]. Thus, we tackle MNAR matrix completion by solving a different matrix completion problem first that recovers missingness probabilities. We establish finite-sample error bounds for how accurate these probability estimates are and how well these estimates debias standard matrix completion losses for the original matrix to be completed. Our experiments show that the proposed debiasing strategy can improve a variety of existing matrix completion algorithms, and achieves downstream matrix completion accuracy at least as good as logistic regression and naive Bayes debiasing baselines that require additional auxiliary information.
This paper examines how homestead policies, which opened vast frontier lands for settlement, influenced the development of American frontier states. It uses a treatment propensity-weighted matrix completion model to estimate the counterfactual size of these states without homesteading. In simulation studies, the method shows lower bias and variance than other estimators, particularly in higher complexity scenarios. The empirical analysis reveals that homestead policies significantly and persistently reduced state government expenditure and revenue. These findings align with continuous difference-in-differences estimates using 1.46 million land patent records. This study's extension of the matrix completion method to include propensity score weighting for causal effect estimation in panel data, especially in staggered treatment contexts, enhances policy evaluation by improving the precision of long-term policy impact assessments.
In matrix factorization, available graph side-information may not be well suited for the matrix completion problem, having edges that disagree with the latent-feature relations learnt from the incomplete data matrix. We show that removing these $\textit{contested}$ edges improves prediction accuracy and scalability. We identify the contested edges through a highly-efficient graphical lasso approximation. The identification and removal of contested edges adds no computational complexity to state-of-the-art graph-regularized matrix factorization, remaining linear with respect to the number of non-zeros. Computational load even decreases proportional to the number of edges removed. Formulating a probabilistic generative model and using expectation maximization to extend graph-regularised alternating least squares (GRALS) guarantees convergence. Rich simulated experiments illustrate the desired properties of the resulting algorithm. On real data experiments we demonstrate improved prediction accuracy with fewer graph edges (empirical evidence that graph side-information is often inaccurate). A 300 thousand dimensional graph with three million edges (Yahoo music side-information) can be analyzed in under ten minutes on a standard laptop computer demonstrating the efficiency of our graph update.
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, we present theoretical and empirical arguments questioning a nascent view by which implicit regularization in matrix factorization can be captured using simple mathematical norms. Our results point to the possibility that the language of standard regularizers may not be rich enough to fully encompass the implicit regularization brought forth by gradient-based optimization.
Several complex tasks that arise in organizations can be simplified by mapping them into a matrix completion problem. In this paper, we address a key challenge faced by our company: predicting the efficiency of artists in rendering visual effects (VFX) in film shots. We tackle this challenge by using a two-fold approach: first, we transform this task into a constrained matrix completion problem with entries bounded in the unit interval [0, 1]; second, we propose two novel matrix factorization models that leverage our knowledge of the VFX environment. Our first approach, expertise matrix factorization (EMF), is an interpretable method that structures the latent factors as weighted user-item interplay. The second one, survival matrix factorization (SMF), is instead a probabilistic model for the underlying process defining employees' efficiencies. We show the effectiveness of our proposed models by extensive numerical tests on our VFX dataset and two additional datasets with values that are also bounded in the [0, 1] interval.
We propose an inductive matrix completion model without using side information. By factorizing the (rating) matrix into the product of low-dimensional latent embeddings of rows (users) and columns (items), a majority of existing matrix completion methods are transductive, since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, most previous works use content (side information), such as user's age or movie's genre, to make predictions. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side information is available other than the matrix to complete, can we still learn an inductive matrix completion model? In this paper, we propose an Inductive Graph-based Matrix Completion (IGMC) model to address this problem. IGMC trains a graph neural network (GNN) based purely on 1-hop subgraphs around (user, item) pairs generated from the rating matrix and maps these subgraphs to their corresponding ratings. It achieves highly competitive performance with state-of-the-art transductive baselines. In addition, IGMC is inductive -- it can generalize to users/items unseen during the training (given that their interactions exist), and can even transfer to new tasks. Our transfer learning experiments show that a model trained out of the MovieLens dataset can be directly used to predict Douban movie ratings with surprisingly good performance. Our work demonstrates that: 1) it is possible to train inductive matrix completion models without using side information while achieving similar or better performances than state-of-the-art transductive methods; 2) local graph patterns around a (user, item) pair are effective predictors of the rating this user gives to the item; and 3) Long-range dependencies might not be necessary for modeling recommender systems.
Recommender systems are widely used to recommend the most appealing items to users. These recommendations can be generated by applying collaborative filtering methods. The low-rank matrix completion method is the state-of-the-art collaborative filtering method. In this work, we show that the skewed distribution of ratings in the user-item rating matrix of real-world datasets affects the accuracy of matrix-completion-based approaches. Also, we show that the number of ratings that an item or a user has positively correlates with the ability of low-rank matrix-completion-based approaches to predict the ratings for the item or the user accurately. Furthermore, we use these insights to develop four matrix completion-based approaches, i.e., Frequency Adaptive Rating Prediction (FARP), Truncated Matrix Factorization (TMF), Truncated Matrix Factorization with Dropout (TMF + Dropout) and Inverse Frequency Weighted Matrix Factorization (IFWMF), that outperforms traditional matrix-completion-based approaches for the users and the items with few ratings in the user-item rating matrix.