Recent studies utilize multiple kernel learning to deal with incomplete-data problem. In this study, we introduce new methods that do not only complete multiple incomplete kernel matrices simultaneously, but also allow control of the flexibility of the model by parameterizing the model matrix. By imposing restrictions on the model covariance, overfitting of the data is avoided. A limitation of kernel matrix estimations done via optimization of an objective function is that the positive definiteness of the result is not guaranteed. In view of this limitation, our proposed methods employ the LogDet divergence, which ensures the positive definiteness of the resulting inferred kernel matrix. We empirically show that our proposed restricted covariance models, employed with LogDet divergence, yield significant improvements in the generalization performance of previous completion methods.
This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We provide a formal mathematical justification for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model.
We consider the problem of matrix completion with side information (\textit{inductive matrix completion}). In real-world applications many side-channel features are typically non-informative making feature selection an important part of the problem. We incorporate feature selection into inductive matrix completion by proposing a matrix factorization framework with group-lasso regularization on side feature parameter matrices. We demonstrate, that the theoretical sample complexity for the proposed method is much lower compared to its competitors in sparse problems, and propose an efficient optimization algorithm for the resulting low-rank matrix completion problem with sparsifying regularizers. Experiments on synthetic and real-world datasets show that the proposed approach outperforms other methods.
The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel matrix completion. The success of ALOHA is due to the concise signal representation in the k-space domain thanks to the duality between structured low-rankness in the k-space domain and the image domain sparsity. Inspired by the recent mathematical discovery that links convolutional neural networks to Hankel matrix decomposition using data-driven framelet basis, here we propose a fully data-driven deep learning algorithm for k-space interpolation. Our network can be also easily applied to non-Cartesian k-space trajectories by simply adding an additional regridding layer. Extensive numerical experiments show that the proposed deep learning method consistently outperforms the existing image-domain deep learning approaches.
Matrix completion is one of the key problems in signal processing and machine learning, with applications ranging from image pro- cessing and data gathering to classification and recommender sys- tems. Recently, deep neural networks have been proposed as la- tent factor models for matrix completion and have achieved state- of-the-art performance. Nevertheless, a major problem with existing neural-network-based models is their limited capabilities to extend to samples unavailable at the training stage. In this paper, we propose a deep two-branch neural network model for matrix completion. The proposed model not only inherits the predictive power of neural net- works, but is also capable of extending to partially observed samples outside the training set, without the need of retraining or fine-tuning. Experimental studies on popular movie rating datasets prove the ef- fectiveness of our model compared to the state of the art, in terms of both accuracy and extendability.
In this paper we consider a general matrix factorization model which covers a large class of existing models with many applications in areas such as machine learning and imaging sciences. To solve this possibly nonconvex, nonsmooth and non-Lipschitz problem, we develop a non-monotone alternating updating method based on a potential function. Our method essentially updates two blocks of variables in turn by inexactly minimizing this potential function, and updates another auxiliary block of variables using an explicit formula. The special structure of our potential function allows us to take advantage of efficient computational strategies for non-negative matrix factorization to perform the alternating minimization over the two blocks of variables. A suitable line search criterion is also incorporated to improve the numerical performance. Under some mild conditions, we show that the line search criterion is well defined, and establish that the sequence generated is bounded and any cluster point of the sequence is a stationary point. Finally, we conduct some numerical experiments using real datasets to compare our method with some existing efficient methods for non-negative matrix factorization and matrix completion. The numerical results show that our method can outperform these methods for these specific applications.
We investigate task clustering for deep-learning based multi-task and few-shot learning in a many-task setting. We propose a new method to measure task similarities with cross-task transfer performance matrix for the deep learning scenario. Although this matrix provides us critical information regarding similarity between tasks, its asymmetric property and unreliable performance scores can affect conventional clustering methods adversely. Additionally, the uncertain task-pairs, i.e., the ones with extremely asymmetric transfer scores, may collectively mislead clustering algorithms to output an inaccurate task-partition. To overcome these limitations, we propose a novel task-clustering algorithm by using the matrix completion technique. The proposed algorithm constructs a partially-observed similarity matrix based on the certainty of cluster membership of the task-pairs. We then use a matrix completion algorithm to complete the similarity matrix. Our theoretical analysis shows that under mild constraints, the proposed algorithm will perfectly recover the underlying "true" similarity matrix with a high probability. Our results show that the new task clustering method can discover task clusters for training flexible and superior neural network models in a multi-task learning setup for sentiment classification and dialog intent classification tasks. Our task clustering approach also extends metric-based few-shot learning methods to adapt multiple metrics, which demonstrates empirical advantages when the tasks are diverse.
In this paper, we propose two new algorithms for transduction with Matrix Completion (MC) problem. The joint MC and prediction tasks are addressed simultaneously to enhance the accuracy, i.e., the label matrix is concatenated to the data matrix forming a stacked matrix. Assuming the data matrix is of low rank, we propose new recommendation methods by posing the problem as a constrained minimization of the Smoothed Rank Function (SRF). We provide convergence analysis for the proposed algorithms. The simulations are conducted on real datasets in two different scenarios of randomly missing pattern with and without block loss. The results confirm that the accuracy of our proposed methods outperforms those of state-of-the-art methods even up to 10% in low observation rates for the scenario without block loss. Our accuracy in the latter scenario, is comparable to state-of-the-art methods while the complexity of the proposed algorithms are reduced up to 4 times.
In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.
Distributed optimization has gained a surge of interest in recent years. In this paper we propose a distributed projection free algorithm named Distributed Conditional Gradient Sliding(DCGS). Compared to the state-of-the-art distributed Frank-Wolfe algorithm, our algorithm attains the same communication complexity under much more realistic assumptions. In contrast to the consensus based algorithm, DCGS is based on the primal-dual algorithm, yielding a modular analysis that can be exploited to improve linear oracle complexity whenever centralized Frank-Wolfe can be improved. We demonstrate this advantage and show that the linear oracle complexity can be reduced to almost the same order of magnitude as the communication complexity, when the feasible set is polyhedral. Finally we present experimental results on Lasso and matrix completion, demonstrating significant performance improvement compared to the existing distributed Frank-Wolfe algorithm.
We consider a challenging multi-label classification problem where both feature matrix $\X$ and label matrix $\Y$ have missing entries. An existing method concatenated $\X$ and $\Y$ as $[\X; \Y]$ and applied a matrix completion (MC) method to fill the missing entries, under the assumption that $[\X; \Y]$ is of low-rank. However, since entries of $\Y$ take binary values in the multi-label setting, it is unlikely that $\Y$ is of low-rank. Moreover, such assumption implies a linear relationship between $\X$ and $\Y$ which may not hold in practice. In this paper, we consider a latent matrix $\Z$ that produces the probability $\sigma(Z_{ij})$ of generating label $Y_{ij}$, where $\sigma(\cdot)$ is nonlinear. Considering label correlation, we assume $[\X; \Z]$ is of low-rank, and propose an MC algorithm based on subgradient descent named co-completion (COCO) motivated by elastic net and one-bit MC. We give a theoretical bound on the recovery effect of COCO and demonstrate its practical usefulness through experiments.
This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clustering of the full graph when the cluster sizes are balanced. A new random degree-based node sampling algorithm is presented which significantly improves upon the performance of the clustering algorithm even when clusters are unbalanced. This framework improves the phase transitions for matrix-decomposition-based clustering with regard to computational complexity and minimum cluster size, which are shown to be nearly dimension-free in the low inter-cluster connectivity regime. A third sampling technique is shown to improve balance by randomly sampling nodes based on spatial distribution. We provide analysis and numerical results using a convex clustering algorithm based on matrix completion.
Principal Components Analysis (PCA) is one of the most widely used dimension reduction techniques. Robust PCA (RPCA) refers to the problem of PCA when the data may be corrupted by outliers. Recent work by Cand{\`e}s, Wright, Li, and Ma defined RPCA as a problem of decomposing a given data matrix into the sum of a low-rank matrix (true data) and a sparse matrix (outliers). The column space of the low-rank matrix then gives the PCA solution. This simple definition has lead to a large amount of interesting new work on provably correct, fast, and practical solutions to RPCA. More recently, the dynamic (time-varying) version of the RPCA problem has been studied and a series of provably correct, fast, and memory efficient tracking solutions have been proposed. Dynamic RPCA (or robust subspace tracking) is the problem of tracking data lying in a (slowly) changing subspace while being robust to sparse outliers. This article provides an exhaustive review of the last decade of literature on RPCA and its dynamic counterpart (robust subspace tracking), along with describing their theoretical guarantees, discussing the pros and cons of various approaches, and providing empirical comparisons of performance and speed. A brief overview of the (low-rank) matrix completion literature is also provided (the focus is on works not discussed in other recent reviews). This refers to the problem of completing a low-rank matrix when only a subset of its entries are observed. It can be interpreted as a simpler special case of RPCA in which the indices of the outlier corrupted entries are known.
The goal of a recommendation system is to predict the interest of a user in a given item by exploiting the existing set of ratings as well as certain user/item features. A standard approach to modeling this problem is Inductive Matrix Completion where the predicted rating is modeled as an inner product of the user and the item features projected onto a latent space. In order to learn the parameters effectively from a small number of observed ratings, the latent space is constrained to be low-dimensional which implies that the parameter matrix is constrained to be low-rank. However, such bilinear modeling of the ratings can be limiting in practice and non-linear prediction functions can lead to significant improvements. A natural approach to introducing non-linearity in the prediction function is to apply a non-linear activation function on top of the projected user/item features. Imposition of non-linearities further complicates an already challenging problem that has two sources of non-convexity: a) low-rank structure of the parameter matrix, and b) non-linear activation function. We show that one can still solve the non-linear Inductive Matrix Completion problem using gradient descent type methods as long as the solution is initialized well. That is, close to the optima, the optimization function is strongly convex and hence admits standard optimization techniques, at least for certain activation functions, such as Sigmoid and tanh. We also highlight the importance of the activation function and show how ReLU can behave significantly differently than say a sigmoid function. Finally, we apply our proposed technique to recommendation systems and semi-supervised clustering, and show that our method can lead to much better performance than standard linear Inductive Matrix Completion methods.
Valid causal inference in observational studies often requires controlling for confounders. However, in practice measurements of confounders may be noisy, and can lead to biased estimates of causal effects. We show that we can reduce the bias caused by measurement noise using a large number of noisy measurements of the underlying confounders. We propose the use of matrix factorization to infer the confounders from noisy covariates, a flexible and principled framework that adapts to missing values, accommodates a wide variety of data types, and can augment many causal inference methods. We bound the error for the induced average treatment effect estimator and show it is consistent in a linear regression setting, using Exponential Family Matrix Completion preprocessing. We demonstrate the effectiveness of the proposed procedure in numerical experiments with both synthetic data and real clinical data.
Feature missing is a serious problem in many applications, which may lead to low quality of training data and further significantly degrade the learning performance. While feature acquisition usually involves special devices or complex process, it is expensive to acquire all feature values for the whole dataset. On the other hand, features may be correlated with each other, and some values may be recovered from the others. It is thus important to decide which features are most informative for recovering the other features as well as improving the learning performance. In this paper, we try to train an effective classification model with least acquisition cost by jointly performing active feature querying and supervised matrix completion. When completing the feature matrix, a novel target function is proposed to simultaneously minimize the reconstruction error on observed entries and the supervised loss on training data. When querying the feature value, the most uncertain entry is actively selected based on the variance of previous iterations. In addition, a bi-objective optimization method is presented for cost-aware active selection when features bear different acquisition costs. The effectiveness of the proposed approach is well validated by both theoretical analysis and experimental study.
The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as "graphical lasso", but its computational cost becomes prohibitive for large data sets. A recent line of results showed--under mild assumptions--that the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix and solving a maximum determinant matrix completion (MDMC) problem. This paper proves an extension of this result, and describes a Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $\epsilon$-accurate solution in $O(n\log(1/\epsilon))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve the associated MDMC problems with as many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.
While Bayesian methods are extremely popular in statistics and machine learning, their application to massive datasets is often challenging, when possible at all. Indeed, the classical MCMC algorithms are prohibitively slow when both the model dimension and the sample size are large. Variational Bayesian methods aim at approximating the posterior by a distribution in a tractable family. Thus, MCMC are replaced by an optimization algorithm which is orders of magnitude faster. VB methods have been applied in such computationally demanding applications as including collaborative filtering, image and video processing, NLP and text processing... However, despite very nice results in practice, the theoretical properties of these approximations are usually not known. In this paper, we propose a general approach to prove the concentration of variational approximations of fractional posteriors. We apply our theory to two examples: matrix completion, and Gaussian VB.
We use deep learning to model interactions across two or more sets of objects, such as user-movie ratings, protein-drug bindings, or ternary user-item-tag interactions. The canonical representation of such interactions is a matrix (or a higher-dimensional tensor) with an exchangeability property: the encoding's meaning is not changed by permuting rows or columns. We argue that models should hence be Permutation Equivariant (PE): constrained to make the same predictions across such permutations. We present a parameter-sharing scheme and prove that it could not be made any more expressive without violating PE. This scheme yields three benefits. First, we demonstrate state-of-the-art performance on multiple matrix completion benchmarks. Second, our models require a number of parameters independent of the numbers of objects, and thus scale well to large datasets. Third, models can be queried about new objects that were not available at training time, but for which interactions have since been observed. In experiments, our models achieved surprisingly good generalization performance on this matrix extrapolation task, both within domains (e.g., new users and new movies drawn from the same distribution used for training) and even across domains (e.g., predicting music ratings after training on movies).