Classifying samples into categories becomes intractable when a single sample can have millions to billions of features, such as in genetics or imaging data. Principal Components Analysis (PCA) is widely used to identify a low-dimensional representation of such features for further analysis. However, PCA ignores class labels, such as whether or not a subject has cancer, thereby discarding information that could substantially improve downstream classification performance. We describe an approach, "Linear Optimal Low-rank" projection (LOL), which extends PCA by incorporating the class labels in a fashion that is advantageous over existing supervised dimensionality reduction techniques. We prove, and substantiate with synthetic experiments, that LOL leads to a better representation of the data for subsequent classification than other linear approaches, while adding negligible computational cost. We then demonstrate that LOL substantially outperforms PCA in differentiating cancer patients from healthy controls using genetic data, and in differentiating gender using magnetic resonance imaging data with $>$500 million features and 400 gigabytes of data. LOL therefore allows the solution of previous intractable problems, yet requires only a few minutes to run on a desktop computer.

This paper introduces a latent position network model, called the generalised random dot product graph, comprising as special cases the stochastic blockmodel, mixed membership stochastic blockmodel, and random dot product graph. In this model, nodes are represented as random vectors on $\mathbb{R}^d$, and the probability of an edge between nodes $i$ and $j$ is given by the bilinear form $X_i^T I_{p,q} X_j$, where $I_{p,q} = \mathrm{diag}(1,\ldots, 1, -1, \ldots, -1)$ with $p$ ones and $q$ minus ones, where $p+q=d$. As we show, this provides the only possible representation of nodes in $\mathbb{R}^d$ such that mixed membership is encoded as the corresponding convex combination of latent positions. The positions are identifiable only up to transformation in the indefinite orthogonal group $O(p,q)$, and we discuss some consequences for typical follow-on inference tasks, such as clustering and prediction.

The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describe a comprehensive paradigm for statistical inference on random dot product graphs, a paradigm centered on spectral embeddings of adjacency and Laplacian matrices. We examine the analogues, in graph inference, of several canonical tenets of classical Euclidean inference: in particular, we summarize a body of existing results on the consistency and asymptotic normality of the adjacency and Laplacian spectral embeddings, and the role these spectral embeddings can play in the construction of single- and multi-sample hypothesis tests for graph data. We investigate several real-world applications, including community detection and classification in large social networks and the determination of functional and biologically relevant network properties from an exploratory data analysis of the Drosophila connectome. We outline requisite background and current open problems in spectral graph inference.

We present semiparametric spectral modeling of the complete larval Drosophila mushroom body connectome. Motivated by a thorough exploratory data analysis of the network via Gaussian mixture modeling (GMM) in the adjacency spectral embedding (ASE) representation space, we introduce the latent structure model (LSM) for network modeling and inference. LSM is a generalization of the stochastic block model (SBM) and a special case of the random dot product graph (RDPG) latent position model, and is amenable to semiparametric GMM in the ASE representation space. The resulting connectome code derived via semiparametric GMM composed with ASE captures latent connectome structure and elucidates biologically relevant neuronal properties.

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 prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

Inference for the stochastic blockmodel is currently of burgeoning interest in the statistical community, as well as in various application domains as diverse as social networks, citation networks, brain connectivity networks (connectomics), etc. Recent theoretical developments have shown that spectral embedding of graphs yields tractable distributional results; in particular, a random dot product latent position graph formulation of the stochastic blockmodel informs a mixture of normal distributions for the adjacency spectral embedding. We employ this new theory to provide an empirical Bayes methodology for estimation of block memberships of vertices in a random graph drawn from the stochastic blockmodel, and demonstrate its practical utility. The posterior inference is conducted using a Metropolis-within-Gibbs algorithm. The theory and methods are illustrated through Monte Carlo simulation studies, both within the stochastic blockmodel and beyond, and experimental results on a Wikipedia data set are presented.

Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In thispaper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering for the stochastic blockmodel and the degree-corrected stochastic blockmodel. We also show an analogous result for the more general random dot product graph model.

It is common to find collections/measurements of related objects, such as the same article in different languages, similar talks given by different presenters, similar weather patterns in different years, etc. It remains to determine how much the available big data helps us in statistical analysis; simply throwing every collected dataset into the mix may not yield an optimal output. Thus it is natural and important to understand theoretically when and how additional datasets improve the performance of various statistical analysis tasks such as regression, clustering, classification, etc. This is our motivation to explore the following classification problem.

We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in \citet{sussman2012universally} to construct consistent estimates for the latent positions, and we show that the appropriately scaled differences between the estimated and true latent positions converge to a mixture of Gaussian random variables. As a corollary, we obtain a central limit theorem for the first eigenvector of the adjacency matrix of an Erd\"os-Renyi random graph.