Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an "inverse" of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019). We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
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When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new theory about inference with missing data. It proves that as the sample size increases and the extent of missingness decreases, the mean-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random (or "partially missing at random"), this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency of the posterior mode. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.
We propose nonparametric estimators for the second-order central moments of spherical random fields within a functional data context. We consider a measurement framework where each field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterizing the form of our estimators. We determine their uniform rates of convergence as the number of fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover validate and demonstrate the practical feasibility of our estimation procedure in a simulation setting, assuming a fixed number of samples per field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.
We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. We show that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. The leading error terms can be fully characterized by the difference of the Stieltjes transforms on the contour. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm (CGA). As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limit, allowing for precise estimates on the algorithms as the number of iterations diverges. For this concrete random matrix model, beyond the first order expansion, we derive a mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.
Flexibly modeling how an entire density changes with covariates is an important but challenging generalization of mean and quantile regression. While existing methods for density regression primarily consist of covariate-dependent discrete mixture models, we consider a continuous latent variable model in general covariate spaces, which we call DR-BART. The prior mapping the latent variable to the observed data is constructed via a novel application of Bayesian Additive Regression Trees (BART). We prove that the posterior induced by our model concentrates quickly around true generative functions that are sufficiently smooth. We also analyze the performance of DR-BART on a set of challenging simulated examples, where it outperforms various other methods for Bayesian density regression. Lastly, we apply DR-BART to two real datasets from educational testing and economics, to study student growth and predict returns to education. Our proposed sampler is efficient and allows one to take advantage of BART's flexibility in many applied settings where the entire distribution of the response is of primary interest. Furthermore, our scheme for splitting on latent variables within BART facilitates its future application to other classes of models that can be described via latent variables, such as those involving hierarchical or time series data.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
We present a new clustering method in the form of a single clustering equation that is able to directly discover groupings in the data. The main proposition is that the first neighbor of each sample is all one needs to discover large chains and finding the groups in the data. In contrast to most existing clustering algorithms our method does not require any hyper-parameters, distance thresholds and/or the need to specify the number of clusters. The proposed algorithm belongs to the family of hierarchical agglomerative methods. The technique has a very low computational overhead, is easily scalable and applicable to large practical problems. Evaluation on well known datasets from different domains ranging between 1077 and 8.1 million samples shows substantial performance gains when compared to the existing clustering techniques.
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.
Adding attributes for nodes to network embedding helps to improve the ability of the learned joint representation to depict features from topology and attributes simultaneously. Recent research on the joint embedding has exhibited a promising performance on a variety of tasks by jointly embedding the two spaces. However, due to the indispensable requirement of globality based information, present approaches contain a flaw of in-scalability. Here we propose \emph{SANE}, a scalable attribute-aware network embedding algorithm with locality, to learn the joint representation from topology and attributes. By enforcing the alignment of a local linear relationship between each node and its K-nearest neighbors in topology and attribute space, the joint embedding representations are more informative comparing with a single representation from topology or attributes alone. And we argue that the locality in \emph{SANE} is the key to learning the joint representation at scale. By using several real-world networks from diverse domains, We demonstrate the efficacy of \emph{SANE} in performance and scalability aspect. Overall, for performance on label classification, SANE successfully reaches up to the highest F1-score on most datasets, and even closer to the baseline method that needs label information as extra inputs, compared with other state-of-the-art joint representation algorithms. What's more, \emph{SANE} has an up to 71.4\% performance gain compared with the single topology-based algorithm. For scalability, we have demonstrated the linearly time complexity of \emph{SANE}. In addition, we intuitively observe that when the network size scales to 100,000 nodes, the "learning joint embedding" step of \emph{SANE} only takes $\approx10$ seconds.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.
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