We focus on spectral clustering of unlabeled graphs and review some results on clustering methods which achieve weak or strong consistent identification in data generated by such models. We also present a new algorithm which appears to perform optimally both theoretically using asymptotic theory and empirically.

We introduce a dimension reduction method for visualizing the clustering structure obtained from a finite mixture of Gaussian densities. Information on the dimension reduction subspace is obtained from the variation on group means and, depending on the estimated mixture model, on the variation on group covariances. The proposed method aims at reducing the dimensionality by identifying a set of linear combinations, ordered by importance as quantified by the associated eigenvalues, of the original features which capture most of the cluster structure contained in the data. Observations may then be projected onto such a reduced subspace, thus providing summary plots which help to visualize the clustering structure. These plots can be particularly appealing in the case of high-dimensional data and noisy structure. The new constructed variables capture most of the clustering information available in the data, and they can be further reduced to improve clustering performance. We illustrate the approach on both simulated and real data sets.

In this paper, we address the problem of identifying linear structural equation models. We first extend the edge set half-trek criterion to cover a broader class of models. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification algorithm of non-parametric models.

In this paper, we present a novel way to summarize the structure of large graphs, based on non-parametric estimation of edge density in directed multigraphs. Following coclustering approach, we use a clustering of the vertices, with a piecewise constant estimation of the density of the edges across the clusters, and address the problem of automatically and reliably inferring the number of clusters, which is the granularity of the coclustering. We use a model selection technique with data-dependent prior and obtain an exact evaluation criterion for the posterior probability of edge density estimation models. We demonstrate, both theoretically and empirically, that our data-dependent modeling technique is consistent, resilient to noise, valid non asymptotically and asymptotically behaves as an universal approximator of the true edge density in directed multigraphs. We evaluate our method using artificial graphs and present its practical interest on real world graphs. The method is both robust and scalable. It is able to extract insightful patterns in the unsupervised learning setting and to provide state of the art accuracy when used as a preparation step for supervised learning.

Approximate Newton methods are a standard optimization tool which aim to maintain the benefits of Newton's method, such as a fast rate of convergence, whilst alleviating its drawbacks, such as computationally expensive calculation or estimation of the inverse Hessian. In this work we investigate approximate Newton methods for policy optimization in Markov Decision Processes (MDPs). We first analyse the structure of the Hessian of the objective function for MDPs. We show that, like the gradient, the Hessian exhibits useful structure in the context of MDPs and we use this analysis to motivate two Gauss-Newton Methods for MDPs. Like the Gauss-Newton method for non-linear least squares, these methods involve approximating the Hessian by ignoring certain terms in the Hessian which are difficult to estimate. The approximate Hessians possess desirable properties, such as negative definiteness, and we demonstrate several important performance guarantees including guaranteed ascent directions, invariance to affine transformation of the parameter space, and convergence guarantees. We finally provide a unifying perspective of key policy search algorithms, demonstrating that our second Gauss-Newton algorithm is closely related to both the EM-algorithm and natural gradient ascent applied to MDPs, but performs significantly better in practice on a range of challenging domains.

The output scores of a neural network classifier are converted to probabilities via normalizing over the scores of all competing categories. Computing this partition function, $Z$, is then linear in the number of categories, which is problematic as real-world problem sets continue to grow in categorical types, such as in visual object recognition or discriminative language modeling. We propose three approaches for sublinear estimation of the partition function, based on approximate nearest neighbor search and kernel feature maps and compare the performance of the proposed approaches empirically.

We present a sparse knowledge gradient (SpKG) algorithm for adaptively selecting the targeted regions within a large RNA molecule to identify which regions are most amenable to interactions with other molecules. Experimentally, such regions can be inferred from fluorescence measurements obtained by binding a complementary probe with fluorescence markers to the targeted regions. We use a biophysical model which shows that the fluorescence ratio under the log scale has a sparse linear relationship with the coefficients describing the accessibility of each nucleotide, since not all sites are accessible (due to the folding of the molecule). The SpKG algorithm uniquely combines the Bayesian ranking and selection problem with the frequentist $\ell_1$ regularized regression approach Lasso. We use this algorithm to identify the sparsity pattern of the linear model as well as sequentially decide the best regions to test before experimental budget is exhausted. Besides, we also develop two other new algorithms: batch SpKG algorithm, which generates more suggestions sequentially to run parallel experiments; and batch SpKG with a procedure which we call length mutagenesis. It dynamically adds in new alternatives, in the form of types of probes, are created by inserting, deleting or mutating nucleotides within existing probes. In simulation, we demonstrate these algorithms on the Group I intron (a mid-size RNA molecule), showing that they efficiently learn the correct sparsity pattern, identify the most accessible region, and outperform several other policies.

Gaussian process (GP) regression is a powerful tool for building predictive models for spatial systems. However, it does not scale efficiently for large datasets. Particularly, for high-dimensional spatial datasets, i.e., spatial datasets that contain exogenous variables, the performance of GP regression further deteriorates. This paper presents the Sparse Pseudo-input Local Kriging (SPLK) which approximates the full GP for spatial datasets with exogenous variables. SPLK employs orthogonal cuts which decompose the domain into smaller subdomains and then applies a sparse approximation of the full GP in each subdomain. We obtain the continuity of the global predictor by imposing continuity constraints on the boundaries of the neighboring subdomains. The domain decomposition scheme applies independent covariance structures in each region, and as a result, SPLK captures heterogeneous covariance structures. SPLK achieves computational efficiency by utilizing sparse approximation in each subdomain which enables SPLK to accommodate large subdomains that contain many data points and possess a homogenous covariance structure. We Apply the proposed method to real and simulated datasets. We conclude that the combination of orthogonal cuts and sparse approximation makes the proposed method an efficient algorithm for high-dimensional large spatial datasets.

A typical goal of supervised dimension reduction is to find a low-dimensional subspace of the input space such that the projected input variables preserve maximal information about the output variables. The dependence maximization approach solves the supervised dimension reduction problem through maximizing a statistical dependence between projected input variables and output variables. A well-known statistical dependence measure is mutual information (MI) which is based on the Kullback-Leibler (KL) divergence. However, it is known that the KL divergence is sensitive to outliers. On the other hand, quadratic MI (QMI) is a variant of MI based on the $L_2$ distance which is more robust against outliers than the KL divergence, and a computationally efficient method to estimate QMI from data, called least-squares QMI (LSQMI), has been proposed recently. For these reasons, developing a supervised dimension reduction method based on LSQMI seems promising. However, not QMI itself, but the derivative of QMI is needed for subspace search in supervised dimension reduction, and the derivative of an accurate QMI estimator is not necessarily a good estimator of the derivative of QMI. In this paper, we propose to directly estimate the derivative of QMI without estimating QMI itself. We show that the direct estimation of the derivative of QMI is more accurate than the derivative of the estimated QMI. Finally, we develop a supervised dimension reduction algorithm which efficiently uses the proposed derivative estimator, and demonstrate through experiments that the proposed method is more robust against outliers than existing methods.

Hierarchical latent tree analysis (HLTA) is recently proposed as a new method for topic detection. It differs fundamentally from the LDA-based methods in terms of topic definition, topic-document relationship, and learning method. It has been shown to discover significantly more coherent topics and better topic hierarchies. However, HLTA relies on the Expectation-Maximization (EM) algorithm for parameter estimation and hence is not efficient enough to deal with large datasets. In this paper, we propose a method to drastically speed up HLTA using a technique inspired by recent advances in the moments method. Empirical experiments show that our method greatly improves the efficiency of HLTA. It is as efficient as the state-of-the-art LDA-based method for hierarchical topic detection and finds substantially better topics and topic hierarchies.