Rahul Mazumder’s research while affiliated with Massachusetts Institute of Technology and other places

The matrix-completion problem has attracted a lot of attention, largely as a result of the celebrated Netflix competition. Two popular approaches for solving the problem are nuclear-norm-regularized matrix approximation (Candes and Tao, 2009, Mazumder, Hastie and Tibshirani, 2010), and maximum-margin matrix factorization (Srebro, Rennie and Jaakkola, 2005). These two procedures are in some cases solving equivalent problems, but with quite different algorithms. In this article we bring the two approaches together, leading to an efficient algorithm for large matrix factorization and completion that outperforms both of these. We develop a software package "softImpute" in R for implementing our approaches, and a distributed version for very large matrices using the "Spark" cluster programming environment.

We propose a method to test the correlation of two random fields when they are both spatially autocorrelated. In this scenario, the assumption of independence for the pair of observations in the standard test does not hold, and as a result we reject in many cases where there is no effect (the precision of the null distribution is overestimated). Our method recovers the null distribution taking into account the autocorrelation. It uses Monte-Carlo methods, and focuses on permuting, and then smoothing and scaling one of the variables to destroy the correlation with the other, while maintaining at the same time the initial autocorrelation. With this simulation model, any test based on the independence of two (or more) random fields can be constructed. This research was motivated by a project in biodiversity and conservation in the Biology Department at Stanford University.

Flume experiments were carried out to study the turbulence and its impact on suspension and segregation of grain-sizes under unidirectional flow conditions over the sand-gravel mixture bed. The components of fluid velocity with fluctuations were measured vertically using 3-D Micro-acoustic Doppler velocimeter (ADV). The theoretical models for velocity and sediment suspension have been developed based on the concept of mixing length that includes the damping effect of turbulence due to sediment suspension in the flow over the sand-gravel mixture bed. Statistical analysis of segregation of grain-sizes along downstream of the bed has been performed using the principle of unsupervised learning or clustering problem. Exploratory data analysis suggests that there is a progressive downstream fining of sediment sizes with selective depositions of gravels, sand-gravels and sand materials along the stream, which may be segmented into three regions such as, the upstream, the transitional and the downstream respectively. This contribution is relevant to understand the direction of ancient rivers, the bed material character in the river form, sorting process and its role in controlling the sediment flux through landscape.

The graphical lasso [5] is an algorithm for learning the structure in an undirected Gaussian graphical model, using ℓ[subscript 1] regularization to control the number of zeros in the precision matrix Θ = Σ[superscript −1] [2, 11]. The R package GLASSO [5] is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of GLASSO can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform GLASSO. By studying the “normal equations” we see that, GLASSO is solving the dual of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in [2]. In this dual, the target of estimation is Σ, the covariance matrix, rather than the precision matrix Θ. We propose similar primal algorithms P-GLASSO and DP-GLASSO, that also operate by block-coordinate descent, where Θ is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that DP-GLASSO is superior from several points of view.

We consider the problem of algorithmically recommending items to users on a Yahoo! front page module. Our approach is based on a novel multilevel hierarchical model that we refer to as a User Profile Model with Graphical Lasso (UPG). The UPG provides a personalized recommendation to users by simultaneously incorporating both user covariates and historical user interactions with items in a model based way. In fact, we build a per-item regression model based on a rich set of user covariates and estimate individual user affinity to items by introducing a latent random vector for each user. The vector random effects are assumed to be drawn from a prior with a precision matrix that measures residual partial associations among items. To ensure better estimates of a precision matrix in high-dimensions, the matrix elements are constrained through a Lasso penalty. Our model is fitted through a penalized-quasi likelihood procedure coupled with a scalable EM algorithm. We employ several computational strategies like multi-threading, conjugate gradients and heavily exploit problem structure to scale our computations in the E-step. For the M-step we take recourse to a scalable variant of the Graphical Lasso algorithm for covariance selection. Through extensive experiments on a new data set obtained from Yahoo! front page and a benchmark data set from a movie recommender application, we show that our UPG model significantly improves performance compared to several state-of-the-art methods in the literature, especially those based on a bilinear random effects model (BIRE). In particular, we show that the gains of UPG are significant compared to BIRE when the number of users is large and the number of items to select from is small. For large item sets and relatively small user sets the results of UPG and BIRE are comparable. The UPG leads to faster model building and produces outputs which are interpretable.

We propose a scalable, efficient and statistically motivated computational framework for Graphical Lasso (Friedman et al., 2007b) - a covariance regularization framework that has received significant attention in the statistics community over the past few years. Existing algorithms have trouble in scaling to dimensions larger than a thousand. Our proposal significantly enhances the state-of-the-art for such moderate sized problems and gracefully scales to larger problems where other algorithms become practically infeasible. This requires a few key new ideas. We operate on the primal problem and use a subtle variation of block-coordinate-methods which drastically reduces the computational complexity by orders of magnitude. We provide rigorous theoretical guarantees on the convergence and complexity of our algorithm and demonstrate the effectiveness of our proposal via experiments. We believe that our framework extends the applicability of Graphical Lasso to large-scale modern applications like bioinformatics, collaborative filtering and social networks, among others.

We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $\rho$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $\rho$ is decomposed into connected components. We show that the vertex-partition induced by the thresholded covariance graph is exactly equal to that induced by the estimated concentration graph. This simple rule, when used as a wrapper around existing algorithms, leads to enormous performance gains. For large values of $\rho$, our proposal splits a large graphical lasso problem into smaller tractable problems, making it possible to solve an otherwise infeasible large scale graphical lasso problem.

We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm Soft-Impute iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices: for example it can obtain a rank-80 approximation of a 10(6) × 10(6) incomplete matrix with 10(5) observed entries in 2.5 hours, and can fit a rank 40 approximation to the full Netflix training set in 6.6 hours. Our methods show very good performance both in training and test error when compared to other competitive state-of-the art techniques.

We address the problem of sparse selection in linear models. A number of non-convex penalties have been proposed for this purpose, along with a variety of convex-relaxation algorithms for finding good solutions. In this paper we pursue the coordinate-descent approach for optimization, and study its convergence properties. We characterize the properties of penalties suitable for this approach, study their corresponding threshold functions, and describe a df -standardizing reparametrization that assists our pathwise algorithm. The MC+ penalty (Zhang 2010) is ideally suited to this task, and we use it to demonstrate the performance of our algorithm.

We use convex relaxation techniques to provide a sequence of solutions to the matrix completion problem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm iteratively replaces the missing elements with those obtained from a thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions. Comment: 10 pages, 1 figure