We consider a voting setting where candidates have preferences about the outcome of the election and are free to join or leave the election. The corresponding candidacy game, where candidates choose strategically to participate or not, has been studied by Dutta et al. [6], who showed that no non-dictatorial voting procedure satisfying unanimity is candidacy-strategyproof, that is, is such that the joint action where all candidates enter the election is always a pure strategy Nash equilibrium. In [7] Dutta et al. also showed that for some voting tree procedures, there are candidacy games with no pure Nash equilibria, and that for the rule that outputs the sophisticated winner of voting by successive elimination, all games have a pure Nash equilibrium. No results were known about other voting rules. Here we prove several such results. For four candidates, the message is, roughly, that most scoring rules (with the exception of Borda) do not guarantee the existence of a pure Nash equilibrium but that Condorcet-consistent rules, for an odd number of voters, do. For five candidates, most rules we study no longer have this guarantee. Finally, we identify one prominent rule that guarantees the existence of a pure Nash equilibrium for any number of candidates (and for an odd number of voters): the Copeland rule. We also show that under mild assumptions on the voting rule, the existence of strong equilibria cannot be guaranteed.

In this paper we introduce three methods for re-scaling data sets aiming at improving the likelihood of clustering validity indexes to return the true number of spherical Gaussian clusters with additional noise features. Our method obtains feature re-scaling factors taking into account the structure of a given data set and the intuitive idea that different features may have different degrees of relevance at different clusters. We experiment with the Silhouette (using squared Euclidean, Manhattan, and the p$^{th}$ power of the Minkowski distance), Dunn's, Calinski-Harabasz and Hartigan indexes on data sets with spherical Gaussian clusters with and without noise features. We conclude that our methods indeed increase the chances of estimating the true number of clusters in a data set.

We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvement over the naive method of combining PCA with regression. To achieve our results, we first observe that ridge regression can be used to obtain a "smooth projection" onto the top principal components. We then sharpen this approximation to true projection using a low-degree polynomial approximation to the matrix step function. Step function approximation is a topic of long-term interest in scientific computing. We extend prior theory by constructing polynomials with simple iterative structure and rigorously analyzing their behavior under limited precision.

Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, the problem is resistant to standard optimization techniques. We construct MIPs with the same solutions whose objective functions are convex. Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. Computing values of this tightest convex extension is NP-hard. However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. For some decompositions, common loss and regularization functions, we derive a closed form.

Clustering is an essential data mining tool that aims to discover inherent cluster structure in data. As such, the study of clusterability, which evaluates whether data possesses such structure, is an integral part of cluster analysis. Yet, despite their central role in the theory and application of clustering, current notions of clusterability fall short in two crucial aspects that render them impractical; most are computationally infeasible and others fail to classify the structure of real datasets. In this paper, we propose a novel approach to clusterability evaluation that is both computationally efficient and successfully captures the structure of real data. Our method applies multimodality tests to the (one-dimensional) set of pairwise distances based on the original, potentially high-dimensional data. We present extensive analyses of our approach for both the Dip and Silverman multimodality tests on real data as well as 17,000 simulations, demonstrating the success of our approach as the first practical notion of clusterability.

Numbers and numerical vectors account for a large portion of data. However, recently the amount of string data generated has increased dramatically. Consequently, classifying string data is a common problem in many fields. The most widely used approach to this problem is to convert strings into numerical vectors using string kernels and subsequently apply a support vector machine that works in a numerical vector space. However, this non-one-to-one conversion involves a loss of information and makes it impossible to evaluate, using probability theory, the generalization error of a learning machine, considering that the given data to train and test the machine are strings generated according to probability laws. In this study, we approach this classification problem by constructing a classifier that works in a set of strings. To evaluate the generalization error of such a classifier theoretically, probability theory for strings is required. Therefore, we first extend a limit theorem on the asymptotic behavior of a consensus sequence of strings, which is the counterpart of the mean of numerical vectors, as demonstrated in the probability theory on a metric space of strings developed by one of the authors and his colleague in a previous study [18]. Using the obtained result, we then demonstrate that our learning machine classifies strings in an asymptotically optimal manner. Furthermore, we demonstrate the usefulness of our machine in practical data analysis by applying it to predicting protein--protein interactions using amino acid sequences.

Despite the importance of sparsity signal models and the increasing prevalence of high-dimensional streaming data, there are relatively few algorithms for dynamic filtering of time-varying sparse signals. Of the existing algorithms, fewer still provide strong performance guarantees. This paper examines two algorithms for dynamic filtering of sparse signals that are based on efficient l1 optimization methods. We first present an analysis for one simple algorithm (BPDN-DF) that works well when the system dynamics are known exactly. We then introduce a novel second algorithm (RWL1-DF) that is more computationally complex than BPDN-DF but performs better in practice, especially in the case where the system dynamics model is inaccurate. Robustness to model inaccuracy is achieved by using a hierarchical probabilistic data model and propagating higher-order statistics from the previous estimate (akin to Kalman filtering) in the sparse inference process. We demonstrate the properties of these algorithms on both simulated data as well as natural video sequences. Taken together, the algorithms presented in this paper represent the first strong performance analysis of dynamic filtering algorithms for time-varying sparse signals as well as state-of-the-art performance in this emerging application.

Large margin classifiers, such as the support vector machine (SVM) [1] and the maximum entropy discrimination (MED) classifier [2], have enjoyed great popularity in the signal processing and machine learning communities due to their broad applicability, robust performance, and the availability of fast software implementations. When the training data is representative of the test data, the performance of MED/SVM has theoretical guarantees that have been validated in practice [1], [3], [4]. Moreover, since the decision boundary of the MED/SVM is solely defined by a few support vectors, the algorithm can tolerate random feature distortions and perturbations. However, in many real applications, anomalous measurements are inherent to the data set due to strong environmental noise or possible sensor failures. Such anomalies arise in industrial process monitoring, video surveillance, tactical multi-modal sensing, and, more generally, any application that involves unattended sensors in difficult environments (Figure 1).

Storytelling algorithms aim to 'connect the dots' between disparate documents by linking starting and ending documents through a series of intermediate documents. Existing storytelling algorithms are based on notions of coherence and connectivity, and thus the primary way by which users can steer the story construction is via design of suitable similarity functions. We present an alternative approach to storytelling wherein the user can interactively and iteratively provide 'must use' constraints to preferentially support the construction of some stories over others. The three innovations in our approach are distance measures based on (inferred) topic distributions, the use of constraints to define sets of linear inequalities over paths, and the introduction of slack and surplus variables to condition the topic distribution to preferentially emphasize desired terms over others. We describe experimental results to illustrate the effectiveness of our interactive storytelling approach over multiple text datasets.

Text documents are structured on multiple levels of detail: individual words are related by syntax, but larger units of text are related by discourse structure. Existing language models generally fail to account for discourse structure, but it is crucial if we are to have language models that reward coherence and generate coherent texts. We present and empirically evaluate a set of multi-level recurrent neural network language models, called Document-Context Language Models (DCLM), which incorporate contextual information both within and beyond the sentence. In comparison with word-level recurrent neural network language models, the DCLM models obtain slightly better predictive likelihoods, and considerably better assessments of document coherence.