In the past few years, Quora a community-driven social platform for question and answering, has grown exponentially from a small community of users into one of the largest and reliable source of Q&A on the Internet. Quora has a built-in social structure integrated to its backbone; users can follow each other, follow question, topics etc. Apart from the social connections that Quora provides, it has developed a knowledge base nicely organized via hierarchy and relatedness of topics. In this paper, we consider a massive dataset of more than four years and analyze the dynamics of topical growth over time; how various factors affect the popularity of a topic or its acceptance in Q&A community. We also propose a regression model to predict the popularity of the topics and discuss the important discriminating features. We achieve a high prediction accuracy (correlation coefficient ~0.773) with low root mean square error (~1.065). We further categorize thetopics into a few broad classes by implementing a simple Latent Dirichlet Allocation (LDA) model on the question texts associated with the topics. In comparison to the data sample with no categorization, this stratification of the topics enhances the prediction accuracies for several categories. However, for certain categories there seems to a slight decrease in the accuracy values and we present an in-depth discussion analyzing the cause for the same pointing out potential ways for improvement. We believe that this thorough measurement study will have a direct application to a service like recommending trending topics in Quora.

We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years. Its least squares solution can be approximated by either a spectral or a semidefinite programming (SDP) relaxation, followed by a rounding procedure. We perform extensive numerical simulations on both synthetic and real-world data sets, showing that our proposed method compares favorably to other algorithms from the recent literature. Existing theoretical guarantees on the group synchronization problem imply lower bounds on the largest amount of noise permissible in the ranking data while still achieving exact recovery. We propose a similar synchronization-based algorithm for the rank-aggregation problem, which integrates in a globally consistent ranking pairwise comparisons given by different rating systems on the same set of items. We also discuss the problem of semi-supervised ranking when there is available information on the ground truth rank of a subset of players, and propose an algorithm based on SDP which recovers the ranks of the remaining players. Finally, synchronization-based ranking, combined with a spectral technique for the densest subgraph problem, allows one to extract locally-consistent partial rankings, in other words, to identify the rank of a small subset of players whose pairwise comparisons are less noisy than the rest of the data, which other methods are not able to identify.

We establish bounds on the KL divergence between two multivariate Gaussian distributions in terms of the Hamming distance between the edge sets of the corresponding graphical models. We show that the KL divergence is bounded below by a constant when the graphs differ by at least one edge; this is essentially the tightest possible bound, since classes of graphs exist for which the edge discrepancy increases but the KL divergence remains bounded above by a constant. As a natural corollary to our KL lower bound, we also establish a sample size requirement for correct model selection via maximum likelihood estimation. Our results rigorize the notion that it is essential to estimate the edge structure of a Gaussian graphical model accurately in order to approximate the true distribution to close precision.

We define the beta diffusion tree, a random tree structure with a set of leaves that defines a collection of overlapping subsets of objects, known as a feature allocation. A generative process for the tree structure is defined in terms of particles (representing the objects) diffusing in some continuous space, analogously to the Dirichlet diffusion tree (Neal, 2003), which defines a tree structure over partitions (i.e., non-overlapping subsets) of the objects. Unlike in the Dirichlet diffusion tree, multiple copies of a particle may exist and diffuse along multiple branches in the beta diffusion tree, and an object may therefore belong to multiple subsets of particles. We demonstrate how to build a hierarchically-clustered factor analysis model with the beta diffusion tree and how to perform inference over the random tree structures with a Markov chain Monte Carlo algorithm. We conclude with several numerical experiments on missing data problems with data sets of gene expression microarrays, international development statistics, and intranational socioeconomic measurements.

We propose a novel parameterized family of Mixed Membership Mallows Models (M4) to account for variability in pairwise comparisons generated by a heterogeneous population of noisy and inconsistent users. M4 models individual preferences as a user-specific probabilistic mixture of shared latent Mallows components. Our key algorithmic insight for estimation is to establish a statistical connection between M4 and topic models by viewing pairwise comparisons as words, and users as documents. This key insight leads us to explore Mallows components with a separable structure and leverage recent advances in separable topic discovery. While separability appears to be overly restrictive, we nevertheless show that it is an inevitable outcome of a relatively small number of latent Mallows components in a world of large number of items. We then develop an algorithm based on robust extreme-point identification of convex polygons to learn the reference rankings, and is provably consistent with polynomial sample complexity guarantees. We demonstrate that our new model is empirically competitive with the current state-of-the-art approaches in predicting real-world preferences.

Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum.

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

Diffusion magnetic resonance imaging (dMRI) data allow to reconstruct the 3D pathways of axons within the white matter of the brain as a tractography. The analysis of tractographies has drawn attention from the machine learning and pattern recognition communities providing novel challenges such as finding an appropriate representation space for the data. Many of the current learning algorithms require the input to be from a vectorial space. This requirement contrasts with the intrinsic nature of the tractography because its basic elements, called streamlines or tracks, have different lengths and different number of points and for this reason they cannot be directly represented in a common vectorial space. In this work we propose the adoption of the dissimilarity representation which is an Euclidean embedding technique defined by selecting a set of streamlines called prototypes and then mapping any new streamline to the vector of distances from prototypes. We investigate the degree of approximation of this projection under different prototype selection policies and prototype set sizes in order to characterise its use on tractography data. Additionally we propose the use of a scalable approximation of the most effective prototype selection policy that provides fast and accurate dissimilarity approximations of complete tractographies.

This paper presents the first theoretical results showing that stable identification of overcomplete $\mu$-coherent dictionaries $\Phi \in \mathbb{R}^{d\times K}$ is locally possible from training signals with sparsity levels $S$ up to the order $O(\mu^{-2})$ and signal to noise ratios up to $O(\sqrt{d})$. In particular the dictionary is recoverable as the local maximum of a new maximisation criterion that generalises the K-means criterion. For this maximisation criterion results for asymptotic exact recovery for sparsity levels up to $O(\mu^{-1})$ and stable recovery for sparsity levels up to $O(\mu^{-2})$ as well as signal to noise ratios up to $O(\sqrt{d})$ are provided. These asymptotic results translate to finite sample size recovery results with high probability as long as the sample size $N$ scales as $O(K^3dS \tilde \varepsilon^{-2})$, where the recovery precision $\tilde \varepsilon$ can go down to the asymptotically achievable precision. Further, to actually find the local maxima of the new criterion, a very simple Iterative Thresholding and K (signed) Means algorithm (ITKM), which has complexity $O(dKN)$ in each iteration, is presented and its local efficiency is demonstrated in several experiments.

We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint or penalization is considered, and generalization is achieved by (early) stopping an empirical iteration. We consider a nonparametric setting, in the framework of reproducing kernel Hilbert spaces, and prove finite sample bounds on the excess risk under general regularity conditions. Our study provides a new class of efficient regularized learning algorithms and gives insights on the interplay between statistics and optimization in machine learning.