A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and depends on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier-Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Demographic inference from text has received a surge of attention in the field of natural language processing in the last decade. In this paper, we use personal names to infer religion in South Asia - where religion is a salient social division, and yet, disaggregated data on it remains scarce. Existing work predicts religion using dictionary based method, and therefore, can not classify unseen names. We use character based models which learn character patterns and, therefore, can classify unseen names as well with high accuracy. These models are also much faster and can easily be scaled to large data sets. We improve our classifier by combining the name of an individual with that of their parent/spouse and achieve remarkably high accuracy. Finally, we trace the classification decisions of a convolutional neural network model using layer-wise relevance propagation which can explain the predictions of complex non-linear classifiers and circumvent their purported black box nature. We show how character patterns learned by the classifier are rooted in the linguistic origins of names.

This paper introduces a generative model for planar point processes in a square window, built upon a single realization of a stationary, ergodic point process observed in this window. Inspired by recent advances in gradient descent methods for maximum entropy models, we propose a method to generate similar point patterns by jointly moving particles of an initial Poisson configuration towards a target counting measure. The target measure is generated via a deterministic gradient descent algorithm, so as to match a set of statistics of the given, observed realization. Our statistics are estimators of the multi-scale wavelet phase harmonic covariance, recently proposed in image modeling. They allow one to capture geometric structures through multi-scale interactions between wavelet coefficients. Both our statistics and the gradient descent algorithm scale better with the number of observed points than the classical k-nearest neighbour distances previously used in generative models for point processes, based on the rejection sampling or simulated-annealing. The overall quality of our model is evaluated on point processes with various geometric structures through spectral and topological data analysis.

The idea behind the \emph{unsupervised} learning of \emph{disentangled} representations is that real-world data is generated by a few explanatory factors of variation which can be recovered by unsupervised learning algorithms. In this paper, we provide a sober look at recent progress in the field and challenge some common assumptions. We first theoretically show that the unsupervised learning of disentangled representations is fundamentally impossible without inductive biases on both the models and the data. Then, we train over $14000$ models covering most prominent methods and evaluation metrics in a reproducible large-scale experimental study on eight data sets. We observe that while the different methods successfully enforce properties "encouraged" by the corresponding losses, well-disentangled models seemingly cannot be identified without supervision. Furthermore, different evaluation metrics do not always agree on what should be considered "disentangled" and exhibit systematic differences in the estimation. Finally, increased disentanglement does not seem to necessarily lead to a decreased sample complexity of learning for downstream tasks. Our results suggest that future work on disentanglement learning should be explicit about the role of inductive biases and (implicit) supervision, investigate concrete benefits of enforcing disentanglement of the learned representations, and consider a reproducible experimental setup covering several data sets.

Tensor regression models, such as CP regression and Tucker regression, have many successful applications in neuroimaging analysis where the covariates are of ultrahigh dimensionality and possess complex spatial structures. The high-dimensional covariate arrays, also known as tensors, can be approximated by low-rank structures and fit into the generalized linear models. The resulting tensor regression achieves a significant reduction in dimensionality while remaining efficient in estimation and prediction. Brain functional connectivity is an essential measure of brain activity and has shown significant association with neurological disorders such as Alzheimer's disease. The symmetry nature of functional connectivity is a property that has not been explored in previous tensor regression models. In this work, we propose a sparse symmetric tensor regression that further reduces the number of free parameters and achieves superior performance over symmetrized and ordinary CP regression, under a variety of simulation settings. We apply the proposed method to a study of Alzheimer's disease (AD) and normal ageing from the Berkeley Aging Cohort Study (BACS) and detect two regions of interest that have been identified important to AD.

Model-Agnostic Meta-Learning (MAML) has demonstrated widespread success in training models that can quickly adapt to new tasks via one or few stochastic gradient descent steps. However, the MAML objective is significantly more difficult to optimize compared to standard Empirical Risk Minimization (ERM), and little is understood about how much MAML improves over ERM in terms of the fast adaptability of their solutions in various scenarios. We analytically address this issue in a linear regression setting consisting of a mixture of easy and hard tasks, where hardness is determined by the number of gradient steps required to solve the task. Specifically, we prove that for $\Omega(d_{\text{eff}})$ labelled test samples (for gradient-based fine-tuning) where $d_{\text{eff}}$ is the effective dimension of the problem, in order for MAML to achieve substantial gain over ERM, the optimal solutions of the hard tasks must be closely packed together with the center far from the center of the easy task optimal solutions. We show that these insights also apply in a low-dimensional feature space when both MAML and ERM learn a representation of the tasks, which reduces the effective problem dimension. Further, our few-shot image classification experiments suggest that our results generalize beyond linear regression.

We identify an implicit under-parameterization phenomenon in value-based deep RL methods that use bootstrapping: when value functions, approximated using deep neural networks, are trained with gradient descent using iterated regression onto target values generated by previous instances of the value network, more gradient updates decrease the expressivity of the current value network. We characterize this loss of expressivity in terms of a drop in the rank of the learned value network features, and show that this corresponds to a drop in performance. We demonstrate this phenomenon on widely studies domains, including Atari and Gym benchmarks, in both offline and online RL settings. We formally analyze this phenomenon and show that it results from a pathological interaction between bootstrapping and gradient-based optimization. We further show that mitigating implicit under-parameterization by controlling rank collapse improves performance.

We analyze the classical method of Principal Component Regression (PCR) in the high-dimensional error-in-variables setting. Here, the observed covariates are not only noisy and contain missing data, but the number of covariates can also exceed the sample size. Under suitable conditions, we establish that PCR identifies the unique model parameter with minimum $\ell_2$-norm, and derive non-asymptotic $\ell_2$-rates of convergence that show its consistency. We further provide non-asymptotic out-of-sample prediction performance guarantees that again prove consistency, even in the presence of corrupted unseen data. Notably, our results do not require the out-of-samples covariates to follow the same distribution as that of the in-sample covariates, but rather that they obey a simple linear algebraic constraint. We finish by presenting simulations that illustrate our theoretical results.

Abstract--We present a simple generative model in which spectral graph embedding for subsequent inference succeeds whereas unsupervised graph convolutional networks (GCN) fail. The geometrical insight is that the GCN is unable to look beyond the first non-informative spectral dimension. 's, we observe a Euclidean space and Email: shenc@udel.edu - Ningyuan (Teresa) Huang and Tianyi Chen are with the Department of's are corrupted through the Bernoulli noise The authors thank Wade Shen for providing the motivation for this investigation. Geometry for the canonical case where ASE succeeds but GCN fails. Figure 1 illustrates the failure mode for GCN.

Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model class. We derive an upper bound on the expected mean squared error of a ReLU network used to estimate any quantile conditional on a set of covariates. This upper bound only depends on the best possible approximation error, the number of layers in the network, and the number of nodes per layer. We further show upper bounds that are tight for two large classes of functions: compositions of H\"older functions and members of a Besov space. These tight bounds imply ReLU networks with quantile regression achieve minimax rates for broad collections of function types. Unlike existing work, the theoretical results hold under minimal assumptions and apply to general error distributions, including heavy-tailed distributions. Empirical simulations on a suite of synthetic response functions demonstrate the theoretical results translate to practical implementations of ReLU networks. Overall, the theoretical and empirical results provide insight into the strong performance of ReLU neural networks for quantile regression across a broad range of function classes and error distributions. All code for this paper is publicly available at https://github.com/tansey/quantile-regression.