We propose a robust, scalable, integrated methodology for community detection and community comparison in graphs. In our procedure, we first embed a graph into an appropriate Euclidean space to obtain a low-dimensional representation, and then cluster the vertices into communities. We next employ nonparametric graph inference techniques to identify structural similarity among these communities. These two steps are then applied recursively on the communities, allowing us to detect more fine-grained structure. We describe a hierarchical stochastic blockmodel---namely, a stochastic blockmodel with a natural hierarchical structure---and establish conditions under which our algorithm yields consistent estimates of model parameters and motifs, which we define to be stochastically similar groups of subgraphs. Finally, we demonstrate the effectiveness of our algorithm in both simulated and real data. Specifically, we address the problem of locating similar subcommunities in a partially reconstructed Drosophila connectome and in the social network Friendster.

We derive fundamental sample complexity bounds for recovering sparse and structured signals for linear and nonlinear observation models including sparse regression, group testing, multivariate regression and problems with missing features. In general, sparse signal processing problems can be characterized in terms of the following Markovian property. We are given a set of $N$ variables $X_1,X_2,\ldots,X_N$, and there is an unknown subset of variables $S \subset \{1,\ldots,N\}$ that are relevant for predicting outcomes $Y$. More specifically, when $Y$ is conditioned on $\{X_n\}_{n\in S}$ it is conditionally independent of the other variables, $\{X_n\}_{n \not \in S}$. Our goal is to identify the set $S$ from samples of the variables $X$ and the associated outcomes $Y$. We characterize this problem as a version of the noisy channel coding problem. Using asymptotic information theoretic analyses, we establish mutual information formulas that provide sufficient and necessary conditions on the number of samples required to successfully recover the salient variables. These mutual information expressions unify conditions for both linear and nonlinear observations. We then compute sample complexity bounds for the aforementioned models, based on the mutual information expressions in order to demonstrate the applicability and flexibility of our results in general sparse signal processing models.

When getting started with a new tool like XGBoost, it can be helpful to review a few talks on the topic before diving into the code. Tianqi Chen, the creator of the library gave a talk to the LA Data Science group in June 2016 titled "XGBoost: A Scalable Tree Boosting System". There is more information on the DataScience LA blog. Tong He, a contributor to XGBoost for the R interface gave a talk at the NYC Data Science Academy in December 2015 titled "XGBoost: eXtreme Gradient Boosting". There is more information about this talk on the NYC Data Science Academy blog.

Multi-view data clustering refers to categorizing a data set by making good use of related information from multiple representations of the data. It becomes important nowadays because more and more data can be collected in a variety of ways, in different settings and from different sources, so each data set can be represented by different sets of features to form different views of it. Many approaches have been proposed to improve clustering performance by exploring and integrating heterogeneous information underlying different views. In this paper, we propose a new multi-view fuzzy clustering approach called MinimaxFCM by using minimax optimization based on well-known Fuzzy c means. In MinimaxFCM the consensus clustering results are generated based on minimax optimization in which the maximum disagreements of different weighted views are minimized. Moreover, the weight of each view can be learned automatically in the clustering process. In addition, there is only one parameter to be set besides the fuzzifier. The detailed problem formulation, updating rules derivation, and the in-depth analysis of the proposed MinimaxFCM are provided here. Experimental studies on nine multi-view data sets including real world image and document data sets have been conducted. We observed that MinimaxFCM outperforms related multi-view clustering approaches in terms of clustering accuracy, demonstrating the great potential of MinimaxFCM for multi-view data analysis.

A brain computer interface (BCI) is a system which provides direct communication between the mind of a person and the outside world by using only brain activity (EEG). The event-related potential (ERP)-based BCI problem consists of a binary pattern recognition. Linear discriminant analysis (LDA) is widely used to solve this type of classification problems, but it fails when the number of features is large relative to the number of observations. In this work we propose a penalized version of the sparse discriminant analysis (SDA), called Kullback-Leibler penalized sparse discriminant analysis (KLSDA). This method inherits both the discriminative feature selection and classification properties of SDA and it also improves SDA performance through the addition of Kullback-Leibler class discrepancy information. The KLSDA method is design to automatically select the optimal regularization parameters. Numerical experiments with two real ERP-EEG datasets show that this new method outperforms standard SDA.

The aim of this paper is to provide some theoretical understanding of Bayesian non-negative matrix factorization methods. We derive an oracle inequality for a quasi-Bayesian estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence. We illustrate our theoretical results with a short numerical study along with a discussion on existing implementations.

We introduce a new criterion to determine the order of an autoregressive model fitted to time series data. It has the benefits of the two well-known model selection techniques, the Akaike information criterion and the Bayesian information criterion. When the data is generated from a finite order autoregression, the Bayesian information criterion is known to be consistent, and so is the new criterion. When the true order is infinity or suitably high with respect to the sample size, the Akaike information criterion is known to be efficient in the sense that its prediction performance is asymptotically equivalent to the best offered by the candidate models; in this case, the new criterion behaves in a similar manner. Different from the two classical criteria, the proposed criterion adaptively achieves either consistency or efficiency depending on the underlying true model. In practice where the observed time series is given without any prior information about the model specification, the proposed order selection criterion is more flexible and robust compared with classical approaches. Numerical results are presented demonstrating the adaptivity of the proposed technique when applied to various datasets.

We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in $O(1/\varepsilon)$ iterations for smooth objectives, and stochastic gradient descent that converges in $O(1/\varepsilon^2)$ iterations for objectives that are sum of smooth functions. We provide the first improvement in this line of research. Our result is based on the variance reduction trick recently introduced to convex optimization, as well as a brand new analysis of variance reduction that is suitable for non-convex optimization. For objectives that are sum of smooth functions, our first-order minibatch stochastic method converges with an $O(1/\varepsilon)$ rate, and is faster than full gradient descent by $\Omega(n^{1/3})$. We demonstrate the effectiveness of our methods on empirical risk minimizations with non-convex loss functions and training neural nets.

In this paper, we present two new communication-efficient methods for distributed minimization of an average of functions. The first algorithm is an inexact variant of the DANE algorithm that allows any local algorithm to return an approximate solution to a local subproblem. We show that such a strategy does not affect the theoretical guarantees of DANE significantly. In fact, our approach can be viewed as a robustification strategy since the method is substantially better behaved than DANE on data partition arising in practice. It is well known that DANE algorithm does not match the communication complexity lower bounds. To bridge this gap, we propose an accelerated variant of the first method, called AIDE, that not only matches the communication lower bounds but can also be implemented using a purely first-order oracle. Our empirical results show that AIDE is superior to other communication efficient algorithms in settings that naturally arise in machine learning applications.

Incremental clustering approaches have been proposed for handling large data when given data set is too large to be stored. The key idea of these approaches is to find representatives to represent each cluster in each data chunk and final data analysis is carried out based on those identified representatives from all the chunks. However, most of the incremental approaches are used for single view data. As large multi-view data generated from multiple sources becomes prevalent nowadays, there is a need for incremental clustering approaches to handle both large and multi-view data. In this paper we propose a new incremental clustering approach called incremental minimax optimization based fuzzy clustering (IminimaxFCM) to handle large multi-view data. In IminimaxFCM, representatives with multiple views are identified to represent each cluster by integrating multiple complementary views using minimax optimization. The detailed problem formulation, updating rules derivation, and the in-depth analysis of the proposed IminimaxFCM are provided. Experimental studies on several real world multi-view data sets have been conducted. We observed that IminimaxFCM outperforms related incremental fuzzy clustering in terms of clustering accuracy, demonstrating the great potential of IminimaxFCM for large multi-view data analysis.