We study matrix completion problem with side information. Side information has been considered in several matrix completion applications, and is generally shown to be useful empirically. Recently, Xu et al. studied the effect of side information for matrix completion under a theoretical viewpoint, showing that sample complexity can be significantly reduced given completely clean features. However, since in reality most given features are noisy or even weakly informative, how to develop a general model to handle general feature set, and how much the noisy features can help matrix recovery in theory, is still an important issue to investigate. In this paper, we propose a novel model that balances between features and observations simultaneously, enabling us to leverage feature information yet to be robust to feature noise. Moreover, we study the effectof general features in theory, and show that by using our model, the sample complexity can still be lower than matrix completion as long as features are sufficiently informative. This result provides a theoretical insight of usefulness for general side information. Finally, we consider synthetic data and two real applications - relationship prediction and semi-supervised clustering, showing that our model outperforms other methods for matrix completion with features both in theory and practice.

Normalized random measures (NRMs) provide a broad class of discrete random measures that are often used as priors for Bayesian nonparametric models. Dirichlet process is a well-known example of NRMs. Most of posterior inference methods for NRM mixture models rely on MCMC methods since they are easy to implement and their convergence is well studied. However, MCMC often suffers from slow convergence when the acceptance rate is low. Tree-based inference is an alternative deterministic posterior inference method, where Bayesian hierarchical clustering (BHC) or incremental Bayesian hierarchical clustering (IBHC) have been developed for DP or NRM mixture (NRMM) models, respectively. Although IBHC is a promising method for posterior inference for NRMM models due to its efficiency and applicability to online inference, its convergence is not guaranteed since it uses heuristics that simply selects the best solution after multiple trials are made. In this paper, we present a hybrid inference algorithm for NRMM models, which combines the merits of both MCMC and IBHC. Trees built by IBHC outlinespartitions of data, which guides Metropolis-Hastings procedure to employ appropriate proposals. Inheriting the nature of MCMC, our tree-guided MCMC (tgMCMC) is guaranteed to converge, and enjoys the fast convergence thanks to the effective proposals guided by trees. Experiments on both synthetic and real world datasets demonstrate the benefit of our method.

We introduce a unifying generalization of the Lovász theta function, and the associated geometric embedding, for graphs with weights on both nodes and edges. We show how it can be computed exactly by semidefinite programming, and how to approximate it using SVM computations. We show how the theta function can be interpreted as a measure of diversity in graphs and use this idea, and the graph embedding in algorithms for Max-Cut, correlation clustering and document summarization, all of which are well represented as problems on weighted graphs.

We show that the objective function of conventional k-means clustering can be expressed as the Frobenius norm of the difference of a data matrix and a low rank approximation of that data matrix. In short, we show that k-means clustering is a matrix factorization problem. These notes are meant as a reference and intended to provide a guided tour towards a result that is often mentioned but seldom made explicit in the literature.

Mode clustering is a nonparametric method for clustering that defines clusters using the basins of attraction of a density estimator's modes. We provide several enhancements to mode clustering: (i) a soft variant of cluster assignment, (ii) a measure of connectivity between clusters, (iii) a technique for choosing the bandwidth, (iv) a method for denoising small clusters, and (v) an approach to visualizing the clusters. Combining all these enhancements gives us a complete procedure for clustering in multivariate problems. We also compare mode clustering to other clustering methods in several examples

In this paper we present a new model and an algorithm for unsupervised clustering of 2-D data such as images. We assume that the data comes from a union of multilinear subspaces (UOMS) model, which is a specific structured case of the much studied union of subspaces (UOS) model. For segmentation under this model, we develop Multilinear Subspace Clustering (MSC) algorithm and evaluate its performance on the YaleB and Olivietti image data sets. We show that MSC is highly competitive with existing algorithms employing the UOS model in terms of clustering performance while enjoying improvement in computational complexity.

Given a set of pairwise comparisons, the classical ranking problem computes a single ranking that best represents the preferences of all users. In this paper, we study the problem of inferring individual preferences, arising in the context of making personalized recommendations. In particular, we assume that there are $n$ users of $r$ types; users of the same type provide similar pairwise comparisons for $m$ items according to the Bradley-Terry model. We propose an efficient algorithm that accurately estimates the individual preferences for almost all users, if there are $r \max \{m, n\}\log m \log^2 n$ pairwise comparisons per type, which is near optimal in sample complexity when $r$ only grows logarithmically with $m$ or $n$. Our algorithm has three steps: first, for each user, compute the \emph{net-win} vector which is a projection of its $\binom{m}{2}$-dimensional vector of pairwise comparisons onto an $m$-dimensional linear subspace; second, cluster the users based on the net-win vectors; third, estimate a single preference for each cluster separately. The net-win vectors are much less noisy than the high dimensional vectors of pairwise comparisons and clustering is more accurate after the projection as confirmed by numerical experiments. Moreover, we show that, when a cluster is only approximately correct, the maximum likelihood estimation for the Bradley-Terry model is still close to the true preference.

Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, whose number, orientations, and dimensions are all unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from undersampling due to complexity and speed constraints on the acquisition device or mechanism. More pertinently, even if the high-dimensional data set is available it is often desirable to first project the data points into a lower-dimensional space and to perform clustering there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality reduction through random projection on the performance of three subspace clustering algorithms, all of which are based on principles from sparse signal recovery. Specifically, we analyze the thresholding based subspace clustering (TSC) algorithm, the sparse subspace clustering (SSC) algorithm, and an orthogonal matching pursuit variant thereof (SSC-OMP). We find, for all three algorithms, that dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation. Moreover, these results are order-wise optimal in the sense that reducing the dimensionality further leads to a fundamentally ill-posed clustering problem. Our findings carry over to the noisy case as illustrated through analytical results for TSC and simulations for SSC and SSC-OMP. Extensive experiments on synthetic and real data complement our theoretical findings.

Blind Source Separation (BSS) has proven to be a powerful tool for the analysis of composite patterns in engineering and science. We introduce Convex Analysis of Mixtures (CAM) for separating non-negative well-grounded sources, which learns the mixing matrix by identifying the lateral edges of the convex data scatter plot. We prove a sufficient and necessary condition for identifying the mixing matrix through edge detection, which also serves as the foundation for CAM to be applied not only to the exact-determined and over-determined cases, but also to the under-determined case. We show the optimality of the edge detection strategy, even for cases where source well-groundedness is not strictly satisfied. The CAM algorithm integrates plug-in noise filtering using sector-based clustering, an efficient geometric convex analysis scheme, and stability-based model order selection. We demonstrate the principle of CAM on simulated data and numerically mixed natural images. The superior performance of CAM against a panel of benchmark BSS techniques is demonstrated on numerically mixed gene expression data. We then apply CAM to dissect dynamic contrast-enhanced magnetic resonance imaging data taken from breast tumors and time-course microarray gene expression data derived from in-vivo muscle regeneration in mice, both producing biologically plausible decomposition results.

Despite its popularity, it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, whereas for other statistical problems the theoretical population goal is clearly defined (as in regression or classification), for some of the clustering methodologies it is difficult to specify the population goal to which the data-based clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of clustering by focusing on two main objectives: to provide an explicit formulation for the ideal population goal of the modal clustering methodology, which understands clusters as regions of high density; and to present two new loss functions, applicable in fact to any clustering methodology, to evaluate the performance of a data-based clustering algorithm with respect to the ideal population goal. In particular, it is shown that only mild conditions on a sequence of density estimators are needed to ensure that the sequence of modal clusterings that they induce is consistent.