Graph representation learning aims at transforming graph data into meaningful low-dimensional vectors to facilitate the employment of machine learning and data mining algorithms designed for general data. Most current graph representation learning approaches are transductive, which means that they require all the nodes in the graph are known when learning graph representations and these approaches cannot naturally generalize to unseen nodes. In this paper, we present a Fast Inductive Graph Representation Learning framework (FI-GRL) to learn nodes' low-dimensional representations. Our approach can obtain accurate representations for seen nodes with provable theoretical guarantees and can easily generalize to unseen nodes. Specifically, in order to explicitly decouple nodes' relations expressed by the graph, we transform nodes into a randomized subspace spanned by a random projection matrix. This stage is guaranteed to preserve the projection-cost of the normalized random walk matrix which is highly related to the normalized cut of the graph. Then feature extraction is achieved by conducting singular value decomposition on the obtained matrix sketch. By leveraging the property of projection-cost preservation on the matrix sketch, the obtained representation result is nearly optimal. To deal with unseen nodes, we utilize folding-in technique to learn their meaningful representations. Empirically, when the amount of seen nodes are larger than that of unseen nodes, FI-GRL always achieves excellent results. Our algorithm is fast, simple to implement and theoretically guaranteed. Extensive experiments on real datasets demonstrate the superiority of our algorithm on both efficacy and efficiency over both macroscopic level (clustering) and microscopic level (structural hole detection) applications.

Understanding and analyzing graphs is an essential topic that has been widely studied over the past decades. Many real world problems can be formulated as link predictions in graphs (Gehrke, Ginsparg, and Kleinberg 2003; Freeman 2000; Theocharidis et al. 2009; Goyal, Sapienza, and Ferrara 2018). For example, link prediction in an author collaboration network (Gehrke, Ginsparg, and Kleinberg 2003) can be used to predict potential future author collaboration. Similarly, new connections between proteins can be discovered using protein interaction networks (Pavlopoulos, Wegener, and Schneider 2008), and new friendships can be predicted using social networks (Wasserman and Faust 1994). Recent work on obtaining such predictions use graph representation learning. These methods represent each node in the network with a fixed dimensional embedding, and map link prediction in the network space to a nearest neighbor search in the embedding space (Goyal and Ferrara 2018). It has been shown that such techniques can outperform traditional link prediction methods on graphs (Grover and Leskovec 2016; Ou et al. 2016a).

Many datasets are in the form of tables of binned data. Performing regression on these data usually involves either reading off bin heights, ignoring data from neighbouring bins or interpolating between bins thus over or underestimating the true bin integrals. In this paper we propose an elegant method for performing Gaussian Process (GP) regression given such binned data, allowing one to make probabilistic predictions of the latent function which produced the binned data. We look at several applications. First, for differentially private regression; second, to make predictions over other integrals; and third when the input regions are irregularly shaped collections of polytopes. In summary, our method provides an effective way of analysing binned data such that one can use more information from the histogram representation, and thus reconstruct a more useful and precise density for making predictions.

In this paper, we consider the convex and non-convex composition problem with the structure $\frac{1}{n}\sum\nolimits_{i = 1}^n {{F_i}( {G( x )} )}$, where $G( x )=\frac{1}{n}\sum\nolimits_{j = 1}^n {{G_j}( x )} $ is the inner function, and $F_i(\cdot)$ is the outer function. We explore the variance reduction based method to solve the composition optimization. Due to the fact that when the number of inner function and outer function are large, it is not reasonable to estimate them directly, thus we apply the stochastically controlled stochastic gradient (SCSG) method to estimate the gradient of the composition function and the value of the inner function. The query complexity of our proposed method for the convex and non-convex problem is equal to or better than the current method for the composition problem. Furthermore, we also present the mini-batch version of the proposed method, which has the improved the query complexity with related to the size of the mini-batch.

Despite the advances in the representational capacity of approximate distributions for variational inference, the optimization process can still limit the density that is ultimately learned. We demonstrate the drawbacks of biasing the true posterior to be unimodal, and introduce Annealed Variational Objectives (AVO) into the training of hierarchical variational methods. Inspired by Annealed Importance Sampling, the proposed method facilitates learning by incorporating energy tempering into the optimization objective. In our experiments, we demonstrate our method's robustness to deterministic warm up, and the benefits of encouraging exploration in the latent space.

In this paper, we focus on escaping from saddle points in smooth nonconvex optimization problems subject to a convex set $\mathcal{C}$. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set $\mathcal{C}$ is simple for a quadratic objective function. To be more precise, our results hold if one can find a $\rho$-approximate solution of a quadratic program subject to $\mathcal{C}$ in polynomial time, where $\rho<1$ is a positive constant that depends on the structure of the set $\mathcal{C}$. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an $(\epsilon,\gamma)$-second order stationary point (SOSP) in at most $\mathcal{O}(\max\{\epsilon^{-2},\rho^{-3}\gamma^{-3}\})$ iterations. We further characterize the overall arithmetic operations to reach an SOSP when the convex set $\mathcal{C}$ can be written as a set of quadratic constraints. Finally, we extend our results to the stochastic setting and characterize the number of stochastic gradient and Hessian evaluations to reach an $(\epsilon,\gamma)$-SOSP.

When searching for gene pathways leading to specific disease outcomes, additional information on gene characteristics is often available that may facilitate to differentiate genes related to the disease from irrelevant background when connections involving both types of genes are observed and their relationships to the disease are unknown. We propose method to single out irrelevant background genes with the help of auxiliary information through a logistic regression, and cluster relevant genes into cohesive groups using the adjacency matrix. Expectation-maximization algorithm is modified to maximize a joint pseudo-likelihood assuming latent indicators for relevance to the disease and latent group memberships as well as Poisson or multinomial distributed link numbers within and between groups. A robust version allowing arbitrary linkage patterns within the background is further derived. Asymptotic consistency of label assignments under the stochastic blockmodel is proven. Superior performance and robustness in finite samples are observed in simulation studies. The proposed robust method identifies previously missed gene sets underlying autism related neurological diseases using diverse data sources including de novo mutations, gene expressions and protein-protein interactions.

Interpretability and stability are two important features that are desired in many contemporary big data applications arising in economics and finance. While the former is enjoyed to some extent by many existing forecasting approaches, the latter in the sense of controlling the fraction of wrongly discovered features which can enhance greatly the interpretability is still largely underdeveloped in the econometric settings. To this end, in this paper we exploit the general framework of model-X knockoffs introduced recently in Cand\`{e}s, Fan, Janson and Lv (2018), which is nonconventional for reproducible large-scale inference in that the framework is completely free of the use of p-values for significance testing, and suggest a new method of intertwined probabilistic factors decoupling (IPAD) for stable interpretable forecasting with knockoffs inference in high-dimensional models. The recipe of the method is constructing the knockoff variables by assuming a latent factor model that is exploited widely in economics and finance for the association structure of covariates. Our method and work are distinct from the existing literature in that we estimate the covariate distribution from data instead of assuming that it is known when constructing the knockoff variables, our procedure does not require any sample splitting, we provide theoretical justifications on the asymptotic false discovery rate control, and the theory for the power analysis is also established. Several simulation examples and the real data analysis further demonstrate that the newly suggested method has appealing finite-sample performance with desired interpretability and stability compared to some popularly used forecasting methods.

This paper presents an efficient binarized algorithm for both learning and classification of human epileptic seizures from intracranial electroencephalography (iEEG). The algorithm combines local binary patterns with brain-inspired hyperdimensional computing to enable end-to-end learning and inference with binary operations. The algorithm first transforms iEEG time series from each electrode into local binary pattern codes. Then atomic high-dimensional binary vectors are used to construct composite representations of seizures across all electrodes. For the majority of our patients (10 out of 16), the algorithm quickly learns from one or two seizures (i.e., one-/few-shot learning) and perfectly generalizes on 27 further seizures. For other patients, the algorithm requires three to six seizures for learning. Overall, our algorithm surpasses the state-of-the-art methods for detecting 65 novel seizures with higher specificity and sensitivity, and lower memory footprint.

Predicting keywords performance, such as number of impressions, click-through rate (CTR), conversion rate (CVR), revenue per click (RPC), and cost per click (CPC), is critical for sponsored search in the online advertising industry. An interesting phenomenon is that, despite the size of the overall data, the data are very sparse at the individual unit level. To overcome the sparsity and leverage hierarchical information across the data structure, we propose a Dynamic Hierarchical Empirical Bayesian (DHEB) model that dynamically determines the hierarchy through a data-driven process and provides shrinkage-based estimations. Our method is also equipped with an efficient empirical approach to derive inferences through the hierarchy. We evaluate the proposed method in both simulated and real-world datasets and compare to several competitive models. The results favor the proposed method among all comparisons in terms of both accuracy and efficiency. In the end, we design a two-phase system to serve prediction in real time.