Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives of the density functions have received only relatively scant attention. Kernel estimators of density functions are widely used as they exhibit excellent theoretical and practical properties, though their generalization to density derivatives has progressed more slowly due to the mathematical intractabilities encountered in the crucial problem of bandwidth (or smoothing parameter) selection. This paper presents the first fully automatic, data-based bandwidth selectors for multivariate kernel density derivative estimators. This is achieved by synthesizing recent advances in matrix analytic theory which allow mathematically and computationally tractable representations of higher order derivatives of multivariate vector valued functions. The theoretical asymptotic properties as well as the finite sample behaviour of the proposed selectors are studied. {In addition, we explore in detail the applications of the new data-driven methods for two other statistical problems: clustering and bump hunting. The introduced techniques are combined with the mean shift algorithm to develop novel automatic, nonparametric clustering procedures which are shown to outperform mixture-model cluster analysis and other recent nonparametric approaches in practice. Furthermore, the advantage of the use of smoothing parameters designed for density derivative estimation for feature significance analysis for bump hunting is illustrated with a real data example.

In many practical applications of clustering, the objects to be clustered evolve over time, and a clustering result is desired at each time step. In such applications, evolutionary clustering typically outperforms traditional static clustering by producing clustering results that reflect long-term trends while being robust to short-term variations. Several evolutionary clustering algorithms have recently been proposed, often by adding a temporal smoothness penalty to the cost function of a static clustering method. In this paper, we introduce a different approach to evolutionary clustering by accurately tracking the time-varying proximities between objects followed by static clustering. We present an evolutionary clustering framework that adaptively estimates the optimal smoothing parameter using shrinkage estimation, a statistical approach that improves a naive estimate using additional information. The proposed framework can be used to extend a variety of static clustering algorithms, including hierarchical, k-means, and spectral clustering, into evolutionary clustering algorithms. Experiments on synthetic and real data sets indicate that the proposed framework outperforms static clustering and existing evolutionary clustering algorithms in many scenarios.

We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for k-means in two separate directions. First, we provide new results for k-means reconstruction on manifolds and, secondly, we prove reconstruction bounds for higher-order approximation (k-flats), for which no known results were previously available. While the results for k-means are novel, some of the technical tools are well-established in the literature. In the case of k-flats, both the results and the mathematical tools are new.

In this letter, we derive the optimal discriminant functions for modulation classification based on the sampled distribution distance. The proposed method classifies various candidate constellations using a low complexity approach based on the distribution distance at specific testpoints along the cumulative distribution function. This method, based on the Bayesian decision criteria, asymptotically provides the minimum classification error possible given a set of testpoints. Testpoint locations are also optimized to improve classification performance. The method provides significant gains over existing approaches that also use the distribution of the signal features.

The performance of stochastic gradient descent (SGD) depends critically on how learning rates are tuned and decreased over time. We propose a method to automatically adjust multiple learning rates so as to minimize the expected error at any one time. The method relies on local gradient variations across samples. In our approach, learning rates can increase as well as decrease, making it suitable for non-stationary problems. Using a number of convex and non-convex learning tasks, we show that the resulting algorithm matches the performance of SGD or other adaptive approaches with their best settings obtained through systematic search, and effectively removes the need for learning rate tuning.

When using deep, multi-layered architectures to build generative models of data, it is difficult to train all layers at once. We propose a layer-wise training procedure admitting a performance guarantee compared to the global optimum. It is based on an optimistic proxy of future performance, the best latent marginal. We interpret auto-encoders in this setting as generative models, by showing that they train a lower bound of this criterion. We test the new learning procedure against a state of the art method (stacked RBMs), and find it to improve performance. Both theory and experiments highlight the importance, when training deep architectures, of using an inference model (from data to hidden variables) richer than the generative model (from hidden variables to data).

One basic requirement of many studies is the necessity of classifying data. Clustering is a proposed method for summarizing networks. Clustering methods can be divided into two categories named model-based approaches and algorithmic approaches. Since the most of clustering methods depend on their input parameters, it is important to evaluate the result of a clustering algorithm with its different input parameters, to choose the most appropriate one. There are several clustering validity techniques based on inner density and outer density of clusters that represent different metrics to choose the most appropriate clustering independent of the input parameters. According to dependency of previous methods on the input parameters, one challenge in facing with large systems, is to complete data incrementally that effects on the final choice of the most appropriate clustering. Those methods define the existence of high intensity in a cluster, and low intensity among different clusters as the measure of choosing the optimal clustering. This measure has a tremendous problem, not availing all data at the first stage. In this paper, we introduce an efficient measure in which maximum number of repetitions for various initial values occurs.

The classical mixture of Gaussians model is related to K-means via small-variance asymptotics: as the covariances of the Gaussians tend to zero, the negative log-likelihood of the mixture of Gaussians model approaches the K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis & Jordan (2012) used this observation to obtain a novel K-means-like algorithm from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead consider applying small-variance asymptotics directly to the posterior in Bayesian nonparametric models. This framework is independent of any specific Bayesian inference algorithm, and it has the major advantage that it generalizes immediately to a range of models beyond the DP mixture. To illustrate, we apply our framework to the feature learning setting, where the beta process and Indian buffet process provide an appropriate Bayesian nonparametric prior. We obtain a novel objective function that goes beyond clustering to learn (and penalize new) groupings for which we relax the mutual exclusivity and exhaustivity assumptions of clustering. We demonstrate several other algorithms, all of which are scalable and simple to implement. Empirical results demonstrate the benefits of the new framework.

The problem of change-point estimation is considered under a general framework where the data are generated by unknown stationary ergodic process distributions. In this context, the consistent estimation of the number of change-points is provably impossible. However, it is shown that a consistent clustering method may be used to estimate the number of change points, under the additional constraint that the correct number of process distributions that generate the data is provided. This additional parameter has a natural interpretation in many real-world applications. An algorithm is proposed that estimates the number of change-points and locates the changes. The proposed algorithm is shown to be asymptotically consistent; its empirical evaluations are provided.

We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor selection and rank reduction are the most popular strategies for obtaining lower-dimensional approximations of the parameter matrix in such models. We show in this article that important gains in prediction accuracy can be obtained by considering them jointly. We motivate a new class of sparse multivariate regression models, in which the coefficient matrix has low rank and zero rows or can be well approximated by such a matrix. Next, we introduce estimators that are based on penalized least squares, with novel penalties that impose simultaneous row and rank restrictions on the coefficient matrix. We prove that these estimators indeed adapt to the unknown matrix sparsity and have fast rates of convergence. We support our theoretical results with an extensive simulation study and two data analyses.