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Register a new userHeterogeneous Tensor Mixture Models in High Dimensions
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Added by
Emma Jingfei Zhang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We consider the problem of jointly modeling and clustering populations of tensors by introducing a flexible high-dimensional tensor mixture model with heterogeneous covariances. The proposed mixture model exploits the intrinsic structures of tensor data, and is assumed to have means that are low-rank and internally sparse as well as heterogeneous covariances that are separable and conditionally sparse. We develop an efficient high-dimensional expectation-conditional-maximization (HECM) algorithm that breaks the challenging optimization in the M-step into several simpler conditional optimization problems, each of which is convex, admits regularization and has closed-form updating formulas. We show that the proposed HECM algorithm, with an appropriate initialization, converges geometrically to a neighborhood that is within statistical precision of the true parameter. Such a theoretical analysis is highly nontrivial due to the dual non-convexity arising from both the EM-type estimation and the non-convex objective function in the M-step. The efficacy of our proposed method is demonstrated through simulation studies and an application to an autism spectrum disorder study, where our analysis identifies important brain regions for diagnosis.
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Heterogeneity is an important feature of modern data sets and a central task is to extract information from large-scale and heterogeneous data. In this paper, we consider multiple high-dimensional linear models and adopt the definition of maximin effect (Meinshausen, B{u}hlmann, AoS, 43(4), 1801--1830) to summarize the information contained in this heterogeneous model. We define the maximin effect for a targeted population whose covariate distribution is possibly different from that of the observed data. We further introduce a ridge-type maximin effect to simultaneously account for reward optimality and statistical stability. To identify the high-dimensional maximin effect, we estimate the regression covariance matrix by a debiased estimator and use it to construct the aggregation weights for the maximin effect. A main challenge for statistical inference is that the estimated weights might have a mixture distribution and the resulted maximin effect estimator is not necessarily asymptotic normal. To address this, we devise a novel sampling approach to construct the confidence interval for any linear contrast of high-dimensional maximin effects. The coverage and precision properties of the proposed confidence interval are studied. The proposed method is demonstrated over simulations and a genetic data set on yeast colony growth under different environments.
In high-dimensions, the prior tails can have a significant effect on both posterior computation and asymptotic concentration rates. To achieve optimal rates while keeping the posterior computations relatively simple, an empirical Bayes approach has recently been proposed, featuring thin-tailed conjugate priors with data-driven centers. While conjugate priors ease some of the computational burden, Markov chain Monte Carlo methods are still needed, which can be expensive when dimension is high. In this paper, we develop a variational approximation to the empirical Bayes posterior that is fast to compute and retains the optimal concentration rate properties of the original. In simulations, our method is shown to have superior performance compared to existing variational approximations in the literature across a wide range of high-dimensional settings.
We propose a new method for changepoint estimation in partially-observed, high-dimensional time series that undergo a simultaneous change in mean in a sparse subset of coordinates. Our first methodological contribution is to introduce a MissCUSUM transformation (a generalisation of the popular Cumulative Sum statistics), that captures the interaction between the signal strength and the level of missingness in each coordinate. In order to borrow strength across the coordinates, we propose to project these MissCUSUM statistics along a direction found as the solution to a penalised optimisation problem tailored to the specific sparsity structure. The changepoint can then be estimated as the location of the peak of the absolute value of the projected univariate series. In a model that allows different missingness probabilities in different component series, we identify that the key interaction between the missingness and the signal is a weighted sum of squares of the signal change in each coordinate, with weights given by the observation probabilities. More specifically, we prove that the angle between the estimated and oracle projection directions, as well as the changepoint location error, are controlled with high probability by the sum of two terms, both involving this weighted sum of squares, and representing the error incurred due to noise and the error due to missingness respectively. A lower bound confirms that our changepoint estimator, which we call MissInspect, is optimal up to a logarithmic factor. The striking effectiveness of the MissInspect methodology is further demonstrated both on simulated data, and on an oceanographic data set covering the Neogene period.
Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges. This paper gives a selective overview on recent advance on high-dimensional factor models and their applications to statistics including Factor-Adjusted Robust Model selection (FarmSelect) and Factor-Adjusted Robust Multiple testing (FarmTest). We show that classical methods, especially principal component analysis (PCA), can be tailored to many new problems and provide powerful tools for statistical estimation and inference. We highlight PCA and its connections to matrix perturbation theory, robust statistics, random projection, false discovery rate, etc., and illustrate through several applications how insights from these fields yield solutions to modern challenges. We also present far-reaching connections between factor models and popular statistical learning problems, including network analysis and low-rank matrix recovery.
We propose a Bayesian methodology for estimating spiked covariance matrices with jointly sparse structure in high dimensions. The spiked covariance matrix is reparametrized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the operator norm as well as that for the principal subspace with respect to the projection operator norm loss. We also study the posterior contraction rate of the principal subspace with respect to the two-to-infinity norm loss, a novel loss function measuring the distance between subspaces that is able to capture element-wise eigenvector perturbations. We show that the posterior contraction rate with respect to the two-to-infinity norm loss is tighter than that with respect to the routinely used projection operator norm loss under certain low-rank and bounded coherence conditions. In addition, a point estimator for the principal subspace is proposed with the rate-optimal risk bound with respect to the projection operator norm loss. These results are based on a collection of concentration and large deviation inequalities for the matrix spike-and-slab LASSO prior. The numerical performance of the proposed methodology is assessed through synthetic examples and the analysis of a real-world face data example.
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