Hierarchical clustering based on pairwise similarities is a common tool used in a broad range of scientific applications. However, in many problems it may be expensive to obtain or compute similarities between the items to be clustered. This paper investigates the hierarchical clustering of N items based on a small subset of pairwise similarities, significantly less than the complete set of N(N-1)/2 similarities. First, we show that if the intracluster similarities exceed intercluster similarities, then it is possible to correctly determine the hierarchical clustering from as few as 3N log N similarities. We demonstrate this order of magnitude savings in the number of pairwise similarities necessitates sequentially selecting which similarities to obtain in an adaptive fashion, rather than picking them at random. We then propose an active clustering method that is robust to a limited fraction of anomalous similarities, and show how even in the presence of these noisy similarity values we can resolve the hierarchical clustering using only O(N log^2 N) pairwise similarities.

Game-theoretic solution concepts prescribe how rational parties should act, but to become operational the concepts need to be accompanied by algorithms. I will review the state of solving incomplete-information games. They encompass many practical problems such as auctions, negotiations, and security applications. I will discuss them in the context of how they have transformed computer poker. In short, game-theoretic reasoning now scales to many large problems, outperforms the alternatives on those problems, and in some games beats the best humans.

In this paper, we regard clustering as ensembles of k-ary affinity relations and clusters correspond to subsets of objects with maximal average affinity relations. The average affinity relation of a cluster is relaxed and well approximated by a constrained homogenous function. We present an efficient procedure to solve this optimization problem, and show that the underlying clusters can be robustly revealed by using priors systematically constructed from the data. Our method can automatically select some points to form clusters, leaving other points un-grouped; thus it is inherently robust to large numbers of outliers, which has seriously limited the applicability of classical methods. Our method also provides a unified solution to clustering from k-ary affinity relations with k ≥ 2, that is, it applies to both graph-based and hypergraph-based clustering problems. Both theoretical analysis and experimental results show the superiority of our method over classical solutions to the clustering problem, especially when there exists a large number of outliers.

Motivated by an application to unsupervised part-of-speech tagging, we present an algorithm for the Euclidean embedding of large sets of categorical data based on co-occurrence statistics. We use the CODE model of Globerson et al. but constrain the embedding to lie on a high-dimensional unit sphere. This constraint allows for efficient optimization, even in the case of large datasets and high embedding dimensionality. Using k-means clustering of the embedded data, our approach efficiently produces state-of-the-art results. We analyze the reasons why the sphere constraint is beneficial in this application, and conjecture that these reasons might apply quite generally to other large-scale tasks.

The max-norm was proposed as a convex matrix regularizer by Srebro et al (2004) and was shown to be empirically superior to the trace-norm for collaborative filtering problems. Although the max-norm can be computed in polynomial time, there are currently no practical algorithms for solving large-scale optimization problems that incorporate the max-norm. The present work uses a factorization technique of Burer and Monteiro (2003) to devise scalable first-order algorithms for convex programs involving the max-norm. These algorithms are applied to solve huge collaborative filtering, graph cut, and clustering problems. Empirically, the new methods outperform mature techniques from all three areas.

While clinicians can accurately identify different types of heartbeats in electrocardiograms (ECGs) from different patients, researchers have had limited success in applying supervised machine learning to the same task. The problem is made challenging by the variety of tasks, inter- and intra-patient differences, an often severe class imbalance, and the high cost of getting cardiologists to label data for individual patients. We address these difficulties using active learning to perform patient-adaptive and task-adaptive heartbeat classification. When tested on a benchmark database of cardiologist annotated ECG recordings, our method had considerably better performance than other recently proposed methods on the two primary classification tasks recommended by the Association for the Advancement of Medical Instrumentation. Additionally, our method required over 90% less patient-specific training data than the methods to which we compared it.

Cardiovascular disease is the leading cause of death globally, resulting in 17 million deaths each year. Despite the availability of various treatment options, existing techniques based upon conventional medical knowledge often fail to identify patients who might have benefited from more aggressive therapy. In this paper, we describe and evaluate a novel unsupervised machine learning approach for cardiac risk stratification. The key idea of our approach is to avoid specialized medical knowledge, and assess patient risk using symbolic mismatch, a new metric to assess similarity in long-term time-series activity. We hypothesize that high risk patients can be identified using symbolic mismatch, as individuals in a population with unusual long-term physiological activity. We describe related approaches that build on these ideas to provide improved medical decision making for patients who have recently suffered coronary attacks. We first describe how to compute the symbolic mismatch between pairs of long term electrocardiographic (ECG) signals. This algorithm maps the original signals into a symbolic domain, and provides a quantitative assessment of the difference between these symbolic representations of the original signals. We then show how this measure can be used with each of a one-class SVM, a nearest neighbor classifier, and hierarchical clustering to improve risk stratification. We evaluated our methods on a population of 686 cardiac patients with available long-term electrocardiographic data. In a univariate analysis, all of the methods provided a statistically significant association with the occurrence of a major adverse cardiac event in the next 90 days. In a multivariate analysis that incorporated the most widely used clinical risk variables, the nearest neighbor and hierarchical clustering approaches were able to statistically significantly distinguish patients with a roughly two-fold risk of suffering a major adverse cardiac event in the next 90 days.

A number of objective functions in clustering problems can be described with submodular functions. In this paper, we introduce the minimum average cost criterion, and show that the theory of intersecting submodular functions can be used for clustering with submodular objective functions. The proposed algorithm does not require the number of clusters in advance, and it will be determined by the property of a given set of data points. The minimum average cost clustering problem is parameterized with a real variable, and surprisingly, we show that all information about optimal clusterings for all parameters can be computed in polynomial time in total. Additionally, we evaluate the performance of the proposed algorithm through computational experiments.

We present a model that describes the structure in the responses of different brain areas to a set of stimuli in terms of stimulus categories" (clusters of stimuli) and "functional units" (clusters of voxels). We assume that voxels within a unit respond similarly to all stimuli from the same category, and design a nonparametric hierarchical model to capture inter-subject variability among the units. The model explicitly captures the relationship between brain activations and fMRI time courses. A variational inference algorithm derived based on the model can learn categories, units, and a set of unit-category activation probabilities from data. When applied to data from an fMRI study of object recognition, the method finds meaningful and consistent clusterings of stimuli into categories and voxels into units."

Dimensionality reduction is commonly used in the setting of multi-label supervised classification to control the learning capacity and to provide a meaningful representation of the data. We introduce a simple forward probabilistic model which is a multinomial extension of reduced rank regression; we show that this model provides a probabilistic interpretation of discriminative clustering methods with added benefits in terms of number of hyperparameters and optimization. While expectation-maximization (EM) algorithm is commonly used to learn these models, its optimization usually leads to local minimum because it relies on a non-convex cost function with many such local minima. To avoid this problem, we introduce a local approximation of this cost function, which leads to a quadratic non-convex optimization problem over a product of simplices. In order to minimize such functions, we propose an efficient algorithm based on convex relaxation and low-rank representation of our data, which allows to deal with large instances. Experiments on text document classification show that the new model outperforms other supervised dimensionality reduction methods, while simulations on unsupervised clustering show that our probabilistic formulation has better properties than existing discriminative clustering methods.