Optimal experimental design (OED) is the general formalism of sensor placement and decisions about the data collection strategy for engineered or natural experiments. This approach is prevalent in many critical fields such as battery design, numerical weather prediction, geosciences, and environmental and urban studies. State-of-the-art computational methods for experimental design, however, do not accommodate correlation structure in observational errors produced by many expensive-to-operate devices such as X-ray machines, radars, and satellites. Discarding evident data correlations leads to biased results, higher expenses, and waste of valuable resources. We present a general formulation of the OED formalism for model-constrained large-scale Bayesian linear inverse problems, where measurement errors are generally correlated. The proposed approach utilizes the Hadamard product of matrices to formulate the weighted likelihood and is valid for both finite- and infinite-dimensional Bayesian inverse problems. Extensive numerical experiments are carried out for empirical verification of the proposed approach using an advection-diffusion model, where the objective is to optimally place a small set of sensors, under a limited budget, to predict the concentration of a contaminant in a closed and bounded domain.
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We study the two inference problems of detecting and recovering an isolated community of \emph{general} structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph $\mathcal{G}(n,q)$ with edge density $q\in(0,1)$; under the alternative, there is an unknown structure $\Gamma_k$ on $k$ nodes, planted in $\mathcal{G}(n,q)$, such that it appears as an \emph{induced subgraph}. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure $\Gamma_k$ in terms of the parameters $(n,k,q)$, as well as certain properties of $\Gamma_k$, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an "easy-hard-impossible" phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.
Minimum-time navigation within constrained and dynamic environments is of special relevance in robotics. Seeking time-optimality, while guaranteeing the integrity of time-varying spatial bounds, is an appealing trade-off for agile vehicles, such as quadrotors. State of the art approaches, either assume bounds to be static and generate time-optimal trajectories offline, or compromise time-optimality for constraint satisfaction. Leveraging nonlinear model predictive control and a path parametric reformulation of the quadrotor model, we present a real-time control that approximates time-optimal behavior and remains within dynamic corridors. The efficacy of the approach is evaluated according to simulated results, showing itself capable of performing extremely aggressive maneuvers as well as stop-and-go and backward motions.
This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Pecl\'et numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.
One of the possible objectives when designing experiments is to build or formulate a model for predicting future observations. When the primary objective is prediction, some typical approaches in the planning phase are to use well-established small-sample experimental designs in the design phase (e.g., Definitive Screening Designs) and to construct predictive models using widely used model selection algorithms such as LASSO. These design and analytic strategies, however, do not guarantee high prediction performance, partly due to the small sample sizes that prevent partitioning the data into training and validation sets, a strategy that is commonly used in machine learning models to improve out-of-sample prediction. In this work, we propose a novel framework for building high-performance predictive models from experimental data that capitalizes on the advantage of having both training and validation sets. However, instead of partitioning the data into two mutually exclusive subsets, we propose a weighting scheme based on the fractional random weight bootstrap that emulates data partitioning by assigning anti-correlated training and validation weights to each observation. The proposed methodology, called Self-Validated Ensemble Modeling (SVEM), proceeds in the spirit of bagging so that it iterates through bootstraps of anti-correlated weights and fitted models, with the final SVEM model being the average of the bootstrapped models. We investigate the performance of the SVEM algorithm with several model-building approaches such as stepwise regression, Lasso, and the Dantzig selector. Finally, through simulation and case studies, we show that SVEM generally generates models with better prediction performance in comparison to one-shot model selection approaches.
We propose a Jacobi-style distributed algorithm to solve convex, quadratically constrained quadratic programs (QCQPs), which arise from a broad range of applications. While small to medium-sized convex QCQPs can be solved efficiently by interior-point algorithms, large-scale problems pose significant challenges to traditional algorithms that are mainly designed to be implemented on a single computing unit. The exploding volume of data (and hence, the problem size), however, may overwhelm any such units. In this paper, we propose a distributed algorithm for general, non-separable, large-scale convex QCQPs, using a novel idea of predictor-corrector primal-dual update with an adaptive step size. The algorithm enables distributed storage of data as well as parallel distributed computing. We establish the conditions for the proposed algorithm to converge to a global optimum, and implement our algorithm on a computer cluster with multiple nodes using Message Passing Interface (MPI). The numerical experiments are conducted on data sets of various scales from different applications, and the results show that our algorithm exhibits favorable scalability for solving large-scale problems.
Bayesian regression games are a special class of two-player general-sum Bayesian games in which the learner is partially informed about the adversary's objective through a Bayesian prior. This formulation captures the uncertainty in regard to the adversary, and is useful in problems where the learner and adversary may have conflicting, but not necessarily perfectly antagonistic objectives. Although the Bayesian approach is a more general alternative to the standard minimax formulation, the applications of Bayesian regression games have been limited due to computational difficulties, and the existence and uniqueness of a Bayesian equilibrium are only known for quadratic cost functions. First, we prove the existence and uniqueness of a Bayesian equilibrium for a class of convex and smooth Bayesian games by regarding it as a solution of an infinite-dimensional variational inequality (VI) in Hilbert space. We consider two special cases in which the infinite-dimensional VI reduces to a high-dimensional VI or a nonconvex stochastic optimization, and provide two simple algorithms of solving them with strong convergence guarantees. Numerical results on real datasets demonstrate the promise of this approach.
We propose a novel combinatorial inference framework to conduct general uncertainty quantification in ranking problems. We consider the widely adopted Bradley-Terry-Luce (BTL) model, where each item is assigned a positive preference score that determines the Bernoulli distributions of pairwise comparisons' outcomes. Our proposed method aims to infer general ranking properties of the BTL model. The general ranking properties include the "local" properties such as if an item is preferred over another and the "global" properties such as if an item is among the top $K$-ranked items. We further generalize our inferential framework to multiple testing problems where we control the false discovery rate (FDR), and apply the method to infer the top-$K$ ranked items. We also derive the information-theoretic lower bound to justify the minimax optimality of the proposed method. We conduct extensive numerical studies using both synthetic and real datasets to back up our theory.
Relevance search is to find top-ranked entities in a knowledge graph (KG) that are relevant to a query entity. Relevance is ambiguous, particularly over a schema-rich KG like DBpedia which supports a wide range of different semantics of relevance based on numerous types of relations and attributes. As users may lack the expertise to formalize the desired semantics, supervised methods have emerged to learn the hidden user-defined relevance from user-provided examples. Along this line, in this paper we propose a novel generative model over KGs for relevance search, named GREASE. The model applies to meta-path based relevance where a meta-path characterizes a particular type of semantics of relating the query entity to answer entities. It is also extended to support properties that constrain answer entities. Extensive experiments on two large-scale KGs demonstrate that GREASE has advanced the state of the art in effectiveness, expressiveness, and efficiency.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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.
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