Gaussian data is pervasive and many learning algorithms (e.g., k-means) model their inputs as a single sample drawn from a multivariate Gaussian. However, in many real-life settings, each input object is best described by multiple samples drawn from a multivariate Gaussian. Such data can arise, for example, in a movie review database where each movie is rated by several users, or in time-series domains such as sensor networks. Here, each input can be naturally described by both a mean vector and covariance matrix which parameterize the Gaussian distribution. In this paper, we consider the problem of clustering such input objects, each represented as a multivariate Gaussian. We formulate the problem using an information theoretic approach and draw several interesting theoretical connections to Bregman divergences and also Bregman matrix divergences. We evaluate our method across several domains, including synthetic data, sensor network data, and a statistical debugging application. 1

Prior work has shown that features which appear to be biologically plausible as well as empirically useful can be found by sparse coding with a prior such as a laplacian (L1) that promotes sparsity. We show how smoother priors can preserve the benefits of these sparse priors while adding stability to the Maximum A-Posteriori (MAP) estimate that makes it more useful for prediction problems. Additionally, we show how to calculate the derivative of the MAP estimate efficiently with implicit differentiation. One prior that can be differentiated this way is KL-regularization. We demonstrate its effectiveness on a wide variety of applications, and find that online optimization of the parameters of the KL-regularized model can significantly improve prediction performance. 1

Classification problems abundantly arise in many computer vision tasks – being of supervised, semi-supervised or unsupervised nature. Even when class labels are not available, a user still might favor certain grouping solutions over others. This bias can be expressed either by providing a clustering criterion or cost function and, in addition to that, by specifying pairwise constraints on the assignment of objects to classes. In this work, we discuss a unifying formulation for labelled and unlabelled data that can incorporate constrained data for model fitting. Our approach models the constraint information by the maximum entropy principle. This modeling strategy allows us (i) to handle constraint violations and soft constraints, and, at the same time, (ii) to speed up the optimization process. Experimental results on face classification and image segmentation indicates that the proposed algorithm is computationally efficient and generates superior groupings when compared with alternative techniques. 1.

A number of real-world domains such as social networks and e-commerce involve heterogeneous data that describes relations between multiple classes of entities. Understanding the natural structure of this type of heterogeneous relational data is essential both for exploratory analysis and for performing various predictive modeling tasks. In this paper, we propose a principled multi-way clustering framework for relational data, wherein different types of entities are simultaneously clustered based not only on their intrinsic attribute values, but also on the multiple relations between the entities. To achieve this, we introduce a relation graph model that describes all the known relations between the different entity classes, in which each relation between a given set of entity classes is represented in the form of multi-modal tensor over an appropriate domain. Our multi-way clustering formulation is driven by the objective of capturing the maximal “information ” in the original relation graph, i.e., accurately approximating the set of tensors corresponding to the various relations. This formulation is applicable to all Bregman divergences (a broad family of loss functions that includes squared Euclidean distance, KL-divergence), and also permits analysis of mixed data types using convex combinations of appropriate Bregman loss functions. Furthermore, we present a large family of structurally different multi-way clustering schemes that preserve various linear summary statistics of the original data. We accomplish the above generalizations by extending a recently proposed key theoretical result, namely the minimum Bregman information principle [1], to the relation graph setting. We also describe an efficient multi-way clustering algorithm based on alternate minimization that generalizes a number of other recently proposed clustering methods. Empirical results on datasets obtained from real-world domains (e.g., movie recommendations, newsgroup articles) demonstrate the generality and efficacy of our framework. 1

We present a method that unifies tracking and video content recognition with applications to Mobile Augmented Reality (MAR). We introduce the Radial Gradient Transform (RGT) and an approximate RGT, yielding the Rotation-Invariant, Fast Feature (RIFF) descriptor. We demonstrate that RIFF is fast enough for real-time tracking, while robust enough for large scale retrieval tasks. At 26 × the speed, our trackingscheme obtains a more accurate global affine motionmodel than the Kanade Lucas Tomasi (KLT) tracker. The same descriptors can achieve 94^% retrieval accuracy from a database of 10 4 images.

Relational clustering has attracted more and more attention due to its phenomenal impact in various important applications which involve multi-type interrelated data objects, such as Web mining, search marketing, bioinformatics, citation analysis, and epidemiology. In this paper, we propose a probabilistic model for relational clustering, which also provides a principal framework to unify various important clustering tasks including traditional attributes-based clustering, semi-supervised clustering, co-clustering and graph clustering. The proposed model seeks to identify cluster structures for each type of data objects and interaction patterns between different types of objects. Under this model, we propose parametric hard and soft relational clustering algorithms under a large number of exponential family distributions. The algorithms are applicable to relational data of various structures and at the same time unifies a number of stat-of-the-art clustering algorithms: co-clustering algorithms, the k-partite graph clustering, and semi-supervised clustering based on hidden Markov random fields.

This paper proposes a technique for jointly quantizing continuous features and the posterior distributions of their class labels based on minimizing empirical information loss, such that the index K of the quantizer region to which a given feature X is assigned approximates a sufficient statistic for its class label Y. We derive an alternating minimization procedure for simultaneously learning codebooks in the Euclidean feature space and in the simplex of posterior class distributions. The resulting quantizer can be used to encode unlabeled points outside the training set and to predict their posterior class distributions, and has an elegant interpretation in terms of lossless source coding. The proposed method is extensively validated on synthetic and real datasets, and is applied to two diverse problems: learning discriminative visual vocabularies for bag-of-features image classification, and image segmentation.

Clustering is often formulated as the maximum likelihood estimation of a mixture model that explains the data. The EM algorithm widely used to solve the resulting optimization problem is inherently a gradient-descent method and is sensitive to initialization. The resulting solution is a local optimum in the neighborhood of the initial guess. This sensitivity to initialization presents a significant challenge in clustering large data sets into many clusters. In this paper, we present a different approach to approximate mixture fitting for clustering. We introduce an exemplar-based likelihood function that approximates the exact likelihood. This formulation leads to a convex minimization problem and an efficient algorithm with guaranteed convergence to the globally optimal solution. The resulting clustering can be thought of as a probabilistic mapping of the data points to the set of exemplars that minimizes the average distance and the information-theoretic cost of mapping. We present experimental results illustrating the performance of our algorithm and its comparison with the conventional approach to mixture model clustering. 1

Low dimensional data representations are crucial to numerous applications in machine learning, statistics, and signal processing. Nonnegative matrix approximation (NNMA) is a method for dimensionality reduction that respects the nonnegativity of the input data while constructing a low-dimensional approximation. NNMA has been used in a multitude of applications, though without commensurate theoretical development. In this report we describe generic methods for minimizing generalized divergences between the input and its low rank approximant. Some of our general methods are even extensible to arbitrary convex penalties. Our methods yield efficient multiplicative iterative schemes for solving the proposed problems. We also consider interesting extensions such as the use of penalty functions, non-linear relationships via “link ” functions, weighted errors, and multi-factor approximations. We present some experiments as an illustration of our algorithms. For completeness, the report also includes a brief literature survey of the various algorithms and the applications of NNMA. Keywords: Nonnegative matrix factorization, weighted approximation, Bregman divergence, multiplicative

Abstract. Finding a point which minimizes the maximal distortion with respect to a dataset is an important estimation problem that has recently received growing attentions in machine learning, with the advent of one class classification. In this paper, we study the problem from a general standpoint, and suppose that the distortion is a Bregman divergence, without restriction. Applications of this formulation can be found in machine learning, statistics, signal processing and computational geometry. We propose two theoretically founded generalizations of a popular smallest enclosing ball approximation algorithm for Euclidean spaces coined by Bădoiu and Clarkson in 2002. Experiments clearly display the advantages of being able to tune the divergence depending on the data’s domain. As an additional result, we unveil an useful bijection between Bregman divergences and a family of popular averages that includes the arithmetic, geometric, harmonic and power means. 1