We derive a new Bayesian Information Criterion (BIC) from first principles by formulating the problem of estimating the number of clusters in an observed data set as maximization of the posterior probability of the candidate models. Given that some mild assumptions are satisfied, we provide a general BIC expression for a broad class of data distributions. This serves as an important milestone when deriving the BIC for specific data distributions. Along this line, we provide a closed-form BIC expression for multivariate Gaussian distributed observations. We show that incorporating data structure of the clustering problem into the derivation of the BIC results in an expression whose penalty term is different from that of the original BIC. We propose a two-step cluster enumeration algorithm. First, a model-based unsupervised learning algorithm partitions the data according to a given set of candidate models. Subsequently, the optimal cluster number is determined as the one associated to the model for which the proposed BIC is maximal. The performance of the proposed criterion is tested using synthetic and real data sets. Despite the fact that the original BIC is a generic criterion which does not include information about the specific model selection problem at hand, it has been widely used in the literature to estimate the number of clusters in an observed data set. We, therefore, consider it as a benchmark comparison. Simulation results show that our proposed criterion outperforms the existing cluster enumeration methods that are based on the original BIC.

Analysis of an organization's computer network activity is a key component of early detection and mitigation of insider threat, a growing concern for many organizations. Raw system logs are a prototypical example of streaming data that can quickly scale beyond the cognitive power of a human analyst. As a prospective filter for the human analyst, we present an online unsupervised deep learning approach to detect anomalous network activity from system logs in real time. Our models decompose anomaly scores into the contributions of individual user behavior features for increased interpretability to aid analysts reviewing potential cases of insider threat. Using the CERT Insider Threat Dataset v6.2 and threat detection recall as our performance metric, our novel deep and recurrent neural network models outperform Principal Component Analysis, Support Vector Machine and Isolation Forest based anomaly detection baselines. For our best model, the events labeled as insider threat activity in our dataset had an average anomaly score in the 95.53 percentile, demonstrating our approach's potential to greatly reduce analyst workloads.

Reza Bosagh Zadeh is Founder CEO at Matroid and Adjunct Professor at Stanford University. His work focuses on Machine Learning, Distributed Computing, and Discrete Applied Mathematics. Reza received his PhD in Computational Mathematics from Stanford University under the supervision of Gunnar Carlsson. His awards include a KDD Best Paper Award and the Gene Golub Outstanding Thesis Award. He has served on the Technical Advisory Boards of Microsoft and Databricks.

Clustering is widely studied in statistics and machine learning, with applications in a variety of fields. As opposed to popular algorithms such as agglomerative hierarchical clustering or k-means which return a single clustering solution, Bayesian nonparametric models provide a posterior over the entire space of partitions, allowing one to assess statistical properties, such as uncertainty on the number of clusters. However, an important problem is how to summarize the posterior; the huge dimension of partition space and difficulties in visualizing it add to this problem. In a Bayesian analysis, the posterior of a real-valued parameter of interest is often summarized by reporting a point estimate such as the posterior mean along with 95% credible intervals to characterize uncertainty. In this paper, we extend these ideas to develop appropriate point estimates and credible sets to summarize the posterior of the clustering structure based on decision and information theoretic techniques.

Like the first edition, the current one is a welcome edition to researchers and academicians equally…. Almost all of the chapters are revised.… The Material is nicely reorganized and repackaged, with the general layout being the same as that of the first edition.… If you bought the first edition, I suggest that you buy the second editon for maximum effect, and if you haven't, then I still strongly recommend you have this book at your desk. Is it a good investment, statistically speaking!

We introduce Graph-Sparse Logistic Regression, a new algorithm for classification for the case in which the support should be sparse but connected on a graph. We val- idate this algorithm against synthetic data and benchmark it against L1-regularized Logistic Regression. We then explore our technique in the bioinformatics context of proteomics data on the interactome graph. We make all our experimental code public and provide GSLR as an open source package.

The superior performance of ensemble methods with infinite models are well known. Most of these methods are based on optimization problems in infinite-dimensional spaces with some regularization, for instance, boosting methods and convex neural networks use $L^1$-regularization with the non-negative constraint. However, due to the difficulty of handling $L^1$-regularization, these problems require early stopping or a rough approximation to solve it inexactly. In this paper, we propose a new ensemble learning method that performs in a space of probability measures, that is, our method can handle the $L^1$-constraint and the non-negative constraint in a rigorous way. Such an optimization is realized by proposing a general purpose stochastic optimization method for learning probability measures via parameterization using transport maps on base models. As a result of running the method, a transport map to output an infinite ensemble is obtained, which forms a residual-type network. From the perspective of functional gradient methods, we give a convergence rate as fast as that of a stochastic optimization method for finite dimensional nonconvex problems. Moreover, we show an interior optimality property of a local optimality condition used in our analysis.

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. We first show that both notions coincide, so that insights from one part of the literature can be used in the other. We then show that the metric induced by a characteristic kernel cannot reliably distinguish between distributions that are far apart in the total variation norm as soon as the underlying space of measures is infinite dimensional. In addition, we provide a characterization of characteristic kernels in terms of eigenvalues and -functions and apply this characterization to the case of continuous kernels on (locally) compact spaces. In the compact case we further show that characteristic kernels exist if and only if the space is metrizable. As special cases of our general theory we investigate translation-invariant kernels on compact Abelian groups and isotropic kernels on spheres. The latter are of particular interest for forecast evaluation of probabilistic predictions on spherical domains as frequently encountered in meteorology and climatology.

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This "implicit regularization" feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e. phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control --- measured entrywise and by the spectral norm --- which might be of independent interest.

In this paper we propose the use of quantum genetic algorithm to optimize the support vector machine (SVM) for human action recognition. The Microsoft Kinect sensor can be used for skeleton tracking, which provides the joints' position data. However, how to extract the motion features for representing the dynamics of a human skeleton is still a challenge due to the complexity of human motion. We present a highly efficient features extraction method for action classification, that is, using the joint angles to represent a human skeleton and calculating the variance of each angle during an action time window. Using the proposed representation, we compared the human action classification accuracy of two approaches, including the optimized SVM based on quantum genetic algorithm and the conventional SVM with grid search. Experimental results on the MSR-12 dataset show that the conventional SVM achieved an accuracy of $ 93.85\% $. The proposed approach outperforms the conventional method with an accuracy of $ 96.15\% $.