Modern high-dimensional point process data, especially those from neuroscience experiments, often involve observations from multiple conditions and/or experiments. Networks of interactions corresponding to these conditions are expected to share many edges, but also exhibit unique, condition-specific ones. However, the degree of similarity among the networks from different conditions is generally unknown. Existing approaches for multivariate point processes do not take these structures into account and do not provide inference for jointly estimated networks. To address these needs, we propose a joint estimation procedure for networks of high-dimensional point processes that incorporates easy-to-compute weights in order to data-adaptively encourage similarity between the estimated networks. We also propose a powerful hierarchical multiple testing procedure for edges of all estimated networks, which takes into account the data-driven similarity structure of the multi-experiment networks. Compared to conventional multiple testing procedures, our proposed procedure greatly reduces the number of tests and results in improved power, while tightly controlling the family-wise error rate. Unlike existing procedures, our method is also free of assumptions on dependency between tests, offers flexibility on p-values calculated along the hierarchy, and is robust to misspecification of the hierarchical structure. We verify our theoretical results via simulation studies and demonstrate the application of the proposed procedure using neuronal spike train data.
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Functional connectivity (FC) for quantifying interactions between regions of the brain is commonly estimated from functional magnetic resonance imaging (fMRI). There has been increasing interest in the potential of multimodal imaging to obtain more robust estimates of FC in high-dimensional settings. Recent work has found uses for graphical algorithms in combining fMRI signals with structural connectivity estimated from diffusion tensor imaging (DTI) for FC estimation. At the same time new algorithms focused on de novo identification of graphical subnetworks with significant levels of connectivity are finding other biological applications with great success. Such algorithms develop notions of graphical influence that aid in revealing subnetworks of interest while maintaining rigorous statistical control on discoveries. We develop a novel algorithm adapting some of these methods to FC estimation with computational efficiency and scalability. Our proposed algorithm leverages a graphical random walk on DTI data to define a new measure of structural influence that highlights connected components of maximal interest. The subnetwork topology is then compared to a suitable null hypothesis using permutation testing. Finally, individual discovered components are tested for significance. Extensive simulations show our method is comparable in power to those currently in use while being fast, robust, and simple to implement. We also analyze task-fMRI data from the Human Connectome Project database and find novel insights into brain interactions during the performance of a motor task. It is anticipated that the transparency and flexibility of our approach will prove valuable as further understanding of the structure-function relationship informs the future of network estimation. Scalability will also only become more important as neurological data become more granular and grow in dimension.
A significant obstacle in the development of robust machine learning models is covariate shift, a form of distribution shift that occurs when the input distributions of the training and test sets differ while the conditional label distributions remain the same. Despite the prevalence of covariate shift in real-world applications, a theoretical understanding in the context of modern machine learning has remained lacking. In this work, we examine the exact high-dimensional asymptotics of random feature regression under covariate shift and present a precise characterization of the limiting test error, bias, and variance in this setting. Our results motivate a natural partial order over covariate shifts that provides a sufficient condition for determining when the shift will harm (or even help) test performance. We find that overparameterized models exhibit enhanced robustness to covariate shift, providing one of the first theoretical explanations for this intriguing phenomenon. Additionally, our analysis reveals an exact linear relationship between in-distribution and out-of-distribution generalization performance, offering an explanation for this surprising recent empirical observation.
The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.
We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This results also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.
In many practices, scientists are particularly interested in detecting which of the predictors are truly associated with a multivariate response. It is more accurate to model multiple responses as one vector rather than separating each component one by one. This is particularly true for complex traits having multiple correlated components. A Bayesian multivariate variable selection (BMVS) approach is proposed to select important predictors influencing the multivariate response from a candidate pool with an ultrahigh dimension. By applying the sample-size-dependent spike and slab priors, the BMVS approach satisfies the strong selection consistency property under certain conditions, which represents the advantages of BMVS over other existing Bayesian multivariate regression-based approaches. The proposed approach considers the covariance structure of multiple responses without assuming independence and integrates the estimation of covariance-related parameters together with all regression parameters into one framework through a fast updating MCMC procedure. It is demonstrated through simulations that the BMVS approach outperforms some other relevant frequentist and Bayesian approaches. The proposed BMVS approach possesses the flexibility of wide applications, including genome-wide association studies with multiple correlated phenotypes and a large scale of genetic variants and/or environmental variables, as demonstrated in the real data analyses section. The computer code and test data of the proposed method are available as an R package.
The bilevel functional data under consideration has two sources of repeated measurements. One is to densely and repeatedly measure a variable from each subject at a series of regular time/spatial points, which is named as functional data. The other is to repeatedly collect one functional data at each of the multiple visits. Compared to the well-established single-level functional data analysis approaches, those that are related to high-dimensional bilevel functional data are limited. In this article, we propose a high-dimensional functional mixed-effect model (HDFMM) to analyze the association between the bilevel functional response and a large scale of scalar predictors. We utilize B-splines to smooth and estimate the infinite-dimensional functional coefficient, a sandwich smoother to estimate the covariance function and integrate the estimation of covariance-related parameters together with all regression parameters into one framework through a fast updating MCMC procedure. We demonstrate that the performance of the HDFMM method is promising under various simulation studies and a real data analysis. As an extension of the well-established linear mixed model, the HDFMM model extends the response from repeatedly measured scalars to repeatedly measured functional data/curves, while maintaining the ability to account for the relatedness among samples and control for confounding factors.
This paper investigates the problem of online statistical inference of model parameters in stochastic optimization problems via the Kiefer-Wolfowitz algorithm with random search directions. We first present the asymptotic distribution for the Polyak-Ruppert-averaging type Kiefer-Wolfowitz (AKW) estimators, whose asymptotic covariance matrices depend on the function-value query complexity and the distribution of search directions. The distributional result reflects the trade-off between statistical efficiency and function query complexity. We further analyze the choices of random search directions to minimize the asymptotic covariance matrix, and conclude that the optimal search direction depends on the optimality criteria with respect to different summary statistics of the Fisher information matrix. Based on the asymptotic distribution result, we conduct online statistical inference by providing two construction procedures of valid confidence intervals. We provide numerical experiments verifying our theoretical results with the practical effectiveness of the procedures.
Target-Based Sentiment Analysis aims to detect the opinion aspects (aspect extraction) and the sentiment polarities (sentiment detection) towards them. Both the previous pipeline and integrated methods fail to precisely model the innate connection between these two objectives. In this paper, we propose a novel dynamic heterogeneous graph to jointly model the two objectives in an explicit way. Both the ordinary words and sentiment labels are treated as nodes in the heterogeneous graph, so that the aspect words can interact with the sentiment information. The graph is initialized with multiple types of dependencies, and dynamically modified during real-time prediction. Experiments on the benchmark datasets show that our model outperforms the state-of-the-art models. Further analysis demonstrates that our model obtains significant performance gain on the challenging instances under multiple-opinion aspects and no-opinion aspect situations.
Inferencing with network data necessitates the mapping of its nodes into a vector space, where the relationships are preserved. However, with multi-layered networks, where multiple types of relationships exist for the same set of nodes, it is crucial to exploit the information shared between layers, in addition to the distinct aspects of each layer. In this paper, we propose a novel approach that first obtains node embeddings in all layers jointly via DeepWalk on a \textit{supra} graph, which allows interactions between layers, and then fine-tunes the embeddings to encourage cohesive structure in the latent space. With empirical studies in node classification, link prediction and multi-layered community detection, we show that the proposed approach outperforms existing single- and multi-layered network embedding algorithms on several benchmarks. In addition to effectively scaling to a large number of layers (tested up to $37$), our approach consistently produces highly modular community structure, even when compared to methods that directly optimize for the modularity function.
Amortized inference has led to efficient approximate inference for large datasets. The quality of posterior inference is largely determined by two factors: a) the ability of the variational distribution to model the true posterior and b) the capacity of the recognition network to generalize inference over all datapoints. We analyze approximate inference in variational autoencoders in terms of these factors. We find that suboptimal inference is often due to amortizing inference rather than the limited complexity of the approximating distribution. We show that this is due partly to the generator learning to accommodate the choice of approximation. Furthermore, we show that the parameters used to increase the expressiveness of the approximation play a role in generalizing inference rather than simply improving the complexity of the approximation.
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