The link between Gaussian random fields and Markov random fields is well established based on a stochastic partial differential equation in Euclidean spaces, where the Mat\'ern covariance functions are essential. However, the Mat\'ern covariance functions are not always positive definite on circles and spheres. In this manuscript, we focus on the extension of this link to circles, and show that the link between Gaussian random fields and Markov random fields on circles is valid based on the circular Mat\'ern covariance function instead. First, we show that this circular Mat\'ern function is the covariance of the stationary solution to the stochastic differential equation on the circle with a formally defined white noise space measure. Then, for the corresponding conditional autoregressive model, we derive a closed form formula for its covariance function. Together with a closed form formula for the circular Mat\'ern covariance function, the link between these two random fields can be established explicitly. Additionally, it is known that the estimator of the mean is not consistent on circles, we provide an equivalent Gaussian measure explanation for this non-ergodicity issue.
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We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying Markov random processes parameterized by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" (using bigger step sizes) than the latter (using smaller step sizes). Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, convexity, the Polyak-Lojasiewicz condition, and general non-convexity. We apply our framework to two problems in control and reinforcement learning. First, we look at the standard online actor-critic algorithm over finite state and action spaces and derive a convergence rate of O(k^(-2/5)), which recovers the best known rate derived specifically for this problem. Second, we study an online actor-critic algorithm for the linear-quadratic regulator and show that a convergence rate of O(k^(-2/3)) is achieved. This is the first time such a result is known in the literature. Finally, we support our theoretical analysis with numerical simulations where the convergence rates are visualized.
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.
The problem of continuous inverse optimal control (over finite time horizon) is to learn the unknown cost function over the sequence of continuous control variables from expert demonstrations. In this article, we study this fundamental problem in the framework of energy-based model, where the observed expert trajectories are assumed to be random samples from a probability density function defined as the exponential of the negative cost function up to a normalizing constant. The parameters of the cost function are learned by maximum likelihood via an "analysis by synthesis" scheme, which iterates (1) synthesis step: sample the synthesized trajectories from the current probability density using the Langevin dynamics via back-propagation through time, and (2) analysis step: update the model parameters based on the statistical difference between the synthesized trajectories and the observed trajectories. Given the fact that an efficient optimization algorithm is usually available for an optimal control problem, we also consider a convenient approximation of the above learning method, where we replace the sampling in the synthesis step by optimization. Moreover, to make the sampling or optimization more efficient, we propose to train the energy-based model simultaneously with a top-down trajectory generator via cooperative learning, where the trajectory generator is used to fast initialize the synthesis step of the energy-based model. We demonstrate the proposed methods on autonomous driving tasks, and show that they can learn suitable cost functions for optimal control.
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.
Molecular mechanics (MM) potentials have long been a workhorse of computational chemistry. Leveraging accuracy and speed, these functional forms find use in a wide variety of applications in biomolecular modeling and drug discovery, from rapid virtual screening to detailed free energy calculations. Traditionally, MM potentials have relied on human-curated, inflexible, and poorly extensible discrete chemical perception rules or applying parameters to small molecules or biopolymers, making it difficult to optimize both types and parameters to fit quantum chemical or physical property data. Here, we propose an alternative approach that uses graph neural networks to perceive chemical environments, producing continuous atom embeddings from which valence and nonbonded parameters can be predicted using invariance-preserving layers. Since all stages are built from smooth neural functions, the entire process is modular and end-to-end differentiable with respect to model parameters, allowing new force fields to be easily constructed, extended, and applied to arbitrary molecules. We show that this approach is not only sufficiently expressive to reproduce legacy atom types, but that it can learn to accurately reproduce and extend existing molecular mechanics force fields. Trained with arbitrary loss functions, it can construct entirely new force fields self-consistently applicable to both biopolymers and small molecules directly from quantum chemical calculations, with superior fidelity than traditional atom or parameter typing schemes. When trained on the same quantum chemical small molecule dataset used to parameterize the openff-1.2.0 small molecule force field augmented with a peptide dataset, the resulting espaloma model shows superior accuracy vis-\`a-vis experiments in computing relative alchemical free energy calculations for a popular benchmark set.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results establish the convergence rates of GANs under a collection of integral probability metrics defined through H\"older classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structures or have H\"older densities, when the network architectures are chosen properly. In particular, for distributions concentrated around a low-dimensional set, we show that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into the generator and discriminator approximation error and the statistical error, which may be of independent interest.
One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
Traditional methods for link prediction can be categorized into three main types: graph structure feature-based, latent feature-based, and explicit feature-based. Graph structure feature methods leverage some handcrafted node proximity scores, e.g., common neighbors, to estimate the likelihood of links. Latent feature methods rely on factorizing networks' matrix representations to learn an embedding for each node. Explicit feature methods train a machine learning model on two nodes' explicit attributes. Each of the three types of methods has its unique merits. In this paper, we propose SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction), a new framework for link prediction which combines the power of all the three types into a single graph neural network (GNN). GNN is a new type of neural network which directly accepts graphs as input and outputs their labels. In SEAL, the input to the GNN is a local subgraph around each target link. We prove theoretically that our local subgraphs also reserve a great deal of high-order graph structure features related to link existence. Another key feature is that our GNN can naturally incorporate latent features and explicit features. It is achieved by concatenating node embeddings (latent features) and node attributes (explicit features) in the node information matrix for each subgraph, thus combining the three types of features to enhance GNN learning. Through extensive experiments, SEAL shows unprecedentedly strong performance against a wide range of baseline methods, including various link prediction heuristics and network embedding methods.
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