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Non-parametric, additive models are able to capture complex data dependencies in a flexible, yet interpretable way. However, choosing the format of the additive components often requires non-trivial data exploration. Here, as an alternative, we propose PrAda-net, a one-hidden-layer neural network, trained with proximal gradient descent and adaptive lasso. PrAda-net automatically adjusts the size and architecture of the neural network to reflect the complexity and structure of the data. The compact network obtained by PrAda-net can be translated to additive model components, making it suitable for non-parametric statistical modelling with automatic model selection. We demonstrate PrAda-net on simulated data, where wecompare the test error performance, variable importance and variable subset identification properties of PrAda-net to other lasso-based regularization approaches for neural networks. We also apply PrAda-net to the massive U.K. black smoke data set, to demonstrate how PrAda-net can be used to model complex and heterogeneous data with spatial and temporal components. In contrast to classical, statistical non-parametric approaches, PrAda-net requires no preliminary modeling to select the functional forms of the additive components, yet still results in an interpretable model representation.

A functional dynamic factor model for time-dependent functional data is proposed. We decompose a functional time series into a predictive low-dimensional common component consisting of a finite number of factors and an infinite-dimensional idiosyncratic component that has no predictive power. The conditions under which all model parameters, including the number of factors, become identifiable are discussed. Our identification results lead to a simple-to-use two-stage estimation procedure based on functional principal components. As part of our estimation procedure, we solve the separation problem between the common and idiosyncratic functional components. In particular, we obtain a consistent information criterion that provides joint estimates of the number of factors and dynamic lags of the common component. Finally, we illustrate the applicability of our method in a simulation study and to the problem of modeling and predicting yield curves. In an out-of-sample experiment, we demonstrate that our model performs well compared to the widely used term structure Nelson-Siegel model for yield curves.

Choosing a shrinkage method can be done by selecting a penalty from a list of pre-specified penalties or by constructing a penalty based on the data. If a list of penalties for a class of linear models is given, we provide comparisons based on sample size and number of non-zero parameters under a predictive stability criterion based on data perturbation. These comparisons provide recommendations for penalty selection in a variety of settings. If the preference is to construct a penalty customized for a given problem, then we propose a technique based on genetic algorithms, again using a predictive criterion. We find that, in general, a custom penalty never performs worse than any commonly used penalties but that there are cases the custom penalty reduces to a recognizable penalty. Since penalty selection is mathematically equivalent to prior selection, our method also constructs priors. The techniques and recommendations we offer are intended for finite sample cases. In this context, we argue that predictive stability under perturbation is one of the few relevant properties that can be invoked when the true model is not known. Nevertheless, we study variable inclusion in simulations and, as part of our shrinkage selection strategy, we include oracle property considerations. In particular, we see that the oracle property typically holds for penalties that satisfy basic regularity conditions and therefore is not restrictive enough to play a direct role in penalty selection. In addition, our real data example also includes considerations merging from model mis-specification.

Inference about a scalar parameter of interest typically relies on the asymptotic normality of common likelihood pivots, such as the signed likelihood root, the score and Wald statistics. Nevertheless, the resulting inferential procedures are known to perform poorly when the dimension of the nuisance parameter is large relative to the sample size and when the information about the parameters is limited. In many such cases, the use of asymptotic normality of analytical modifications of the signed likelihood root is known to recover inferential performance. It is proved here that parametric bootstrap of standard likelihood pivots results in as accurate inferences as analytical modifications of the signed likelihood root do in stratified models with stratum specific nuisance parameters. We focus on the challenging case where the number of strata increases as fast or faster than the stratum samples size. It is also shown that this equivalence holds regardless of whether constrained or unconstrained bootstrap is used. This is in contrast to when the number of strata is fixed or increases slower than the stratum sample size, where we show that constrained bootstrap corrects inference to a higher order than unconstrained bootstrap. Simulation experiments support the theoretical findings and demonstrate the excellent performance of bootstrap in extreme scenarios.

Dimension reduction is an important tool for analyzing high-dimensional data. The predictor envelope is a method of dimension reduction for regression that assumes certain linear combinations of the predictors are immaterial to the regression. The method can result in substantial gains in estimation efficiency and prediction accuracy over traditional maximum likelihood and least squares estimates. While predictor envelopes have been developed and studied for independent data, no work has been done adapting predictor envelopes to spatial data. In this work, the predictor envelope is adapted to a popular spatial model to form the spatial predictor envelope (SPE). Maximum likelihood estimates for the SPE are derived, along with asymptotic distributions for the estimates given certain assumptions, showing the SPE estimates to be asymptotically more efficient than estimates of the original spatial model. The effectiveness of the proposed model is illustrated through simulation studies and the analysis of a geo-chemical data set.

We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of Deshpande, Hellerstein and Kletenik. Our second algorithm, which is simple, is based on the algorithm solving the SBFE problem for $k$-of-$n$ functions, due to Salloum, Breuer, and Ben-Dov. It achieves a $(B-1)$ approximation factor, where $B$ is the number of blocks of 0's and 1's in the standard vector representation of the symmetric Boolean function. As part of the design of the first algorithm, we prove that the goal value of any symmetric Boolean function is less than $n(n+1)/2$. Finally, we give an example showing that for symmetric Boolean functions, minimum expected verification cost and minimum expected evaluation cost are not necessarily equal. This contrasts with a previous result, given by Das, Jafarpour, Orlitsky, Pan and Suresh, which showed that equality holds in the unit-cost case.

Conventional supervised learning methods, especially deep ones, are found to be sensitive to out-of-distribution (OOD) examples, largely because the learned representation mixes the semantic factor with the variation factor due to their domain-specific correlation, while only the semantic factor causes the output. To address the problem, we propose a Causal Semantic Generative model (CSG) based on a causal reasoning so that the two factors are modeled separately, and develop methods for OOD prediction from a single training domain, which is common and challenging. The methods are based on the causal invariance principle, with a novel design for both efficient learning and easy prediction. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor by fitting training data, and this semantic-identification guarantees the boundedness of OOD generalization error and the success of adaptation. Empirical study shows improved OOD performance over prevailing baselines.

This work focuses on combining nonparametric topic models with Auto-Encoding Variational Bayes (AEVB). Specifically, we first propose iTM-VAE, where the topics are treated as trainable parameters and the document-specific topic proportions are obtained by a stick-breaking construction. The inference of iTM-VAE is modeled by neural networks such that it can be computed in a simple feed-forward manner. We also describe how to introduce a hyper-prior into iTM-VAE so as to model the uncertainty of the prior parameter. Actually, the hyper-prior technique is quite general and we show that it can be applied to other AEVB based models to alleviate the {\it collapse-to-prior} problem elegantly. Moreover, we also propose HiTM-VAE, where the document-specific topic distributions are generated in a hierarchical manner. HiTM-VAE is even more flexible and can generate topic distributions with better variability. Experimental results on 20News and Reuters RCV1-V2 datasets show that the proposed models outperform the state-of-the-art baselines significantly. The advantages of the hyper-prior technique and the hierarchical model construction are also confirmed by experiments.

We present a semi-parametric approach to photographic image synthesis from semantic layouts. The approach combines the complementary strengths of parametric and nonparametric techniques. The nonparametric component is a memory bank of image segments constructed from a training set of images. Given a novel semantic layout at test time, the memory bank is used to retrieve photographic references that are provided as source material to a deep network. The synthesis is performed by a deep network that draws on the provided photographic material. Experiments on multiple semantic segmentation datasets show that the presented approach yields considerably more realistic images than recent purely parametric techniques. The results are shown in the supplementary video at //youtu.be/U4Q98lenGLQ

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.