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Multifidelity methods are widely used for estimation of quantities of interest (QoIs) in uncertainty quantification using simulation codes of differing costs and accuracies. Many methods approximate numerical-valued statistics that represent only limited information of the QoIs. In this paper, we generalize the ideas in \cite{xu2021bandit} to develop a multifidelity method that approximates the distribution of scalar-valued QoI. Under a linear model hypothesis, we propose an exploration-exploitation strategy to reconstruct the full distribution, not just statistics, of a scalar-valued QoI using samples from a subset of low-fidelity regressors. We derive an informative asymptotic bound for the mean 1-Wasserstein distance between the estimator and the true distribution, and use it to adaptively allocate computational budget for parametric estimation and non-parametric approximation of the probability distribution. Assuming the linear model is correct, we prove that such a procedure is consistent and converges to the optimal policy (and hence optimal computational budget allocation) under an upper bound criterion as the budget goes to infinity. As a corollary, we obtain convergence of the approximated distribution in the mean 1-Wasserstein metric. The major advantages of our approach are that convergence to the full distribution of the output is attained under appropriate assumptions, and that the procedure and implementation require neither a hierarchical model setup, knowledge of cross-model information or correlation, nor \textit{a priori} known model statistics. Numerical experiments are provided in the end to support our theoretical analysis.

The Longest Common Subsequence (LCS) of two strings is a fundamental string similarity measure with a classical dynamic programming solution taking quadratic time. Despite significant efforts, little progress was made in improving the runtime. Even in the realm of approximation, not much was known for linear time algorithms beyond the trivial $\sqrt{n}$-approximation. Recent breakthrough result provided a $n^{0.497}$-factor approximation algorithm [HSSS19], which was more recently improved to a $n^{0.4}$-factor one [BCD21]. The latter paper also showed a $n^{2-2.5\alpha}$ time algorithm which outputs a $n^{\alpha}$ approximation to the LCS, but so far no sub-polynomial approximation is known in truly subquadratic time. In this work, we show an algorithm which runs in $O(n)$ time, and outputs a $n^{o(1)}$-factor approximation to LCS$(x,y)$, with high probability, for any pair of length $n$ input strings. Our entire algorithm is merely an efficient black-box reduction to the Block-LIS problem, introduced very recently in [ANSS21], and solving the Block-LIS problem directly.

We propose a novel methodology for drawing iid realizations from any target distribution on the Euclidean space with arbitrary dimension. No assumption of compact support is necessary for the validity of our theory and method. Our idea is to construct an appropriate infinite sequence of concentric closed ellipsoids, represent the target distribution as an infinite mixture on the central ellipsoid and the ellipsoidal annuli, and to construct efficient perfect samplers for the mixture components. In contrast with most of the existing works on perfect sampling, ours is not only a theoretically valid method, it is practically applicable to all target distributions on any dimensional Euclidean space and very much amenable to parallel computation. We validate the practicality and usefulness of our methodology by generating 10000 iid realizations from the standard distributions such as normal, Student's t with 5 degrees of freedom and Cauchy, for dimensions d = 1, 5, 10, 50, 100, as well as from a 50-dimensional mixture normal distribution. The implementation time in all the cases are very reasonable, and often less than a minute in our parallel implementation. The results turned out to be highly accurate. We also apply our method to draw 10000 iid realizations from the posterior distributions associated with the well-known Challenger data, a Salmonella data and the 160-dimensional challenging spatial example of the radionuclide count data on Rongelap Island. Again, we are able to obtain quite encouraging results with very reasonable computing time.

In this paper we present an adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck (F-P) equations. F-P equations are usually high-dimensional and defined on an unbounded domain, which limits the application of traditional grid based numerical methods. With the Knothe-Rosenblatt rearrangement, our newly proposed flow-based generative model, called KRnet, provides a family of probability density functions to serve as effective solution candidates for the Fokker-Planck equations, which has a weaker dependence on dimensionality than traditional computational approaches and can efficiently estimate general high-dimensional density functions. To obtain effective stochastic collocation points for the approximation of the F-P equation, we develop an adaptive sampling procedure, where samples are generated iteratively using the approximate density function at each iteration. We present a general framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve strictly a wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms to theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios and that no GNN can perform better than with these ratios. This paper is the first to elucidate approximation ratios of GNNs for combinatorial problems. Furthermore, we prove that adding coloring or weak-coloring to each node feature improves these approximation ratios. This indicates that preprocessing and feature engineering theoretically strengthen model capabilities.

The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, maximizing variance and preservation of pairwise relative distances. The derivation of their asymptotic correlation and numerical experiments tell that a projection usually cannot satisfy both objectives. In a standard classification problem we determine projections on the input data that balance them and compare subsequent results. Next, we extend our application of orthogonal projections to deep learning frameworks. We introduce new variational loss functions that enable integration of additional information via transformations and projections of the target data. In two supervised learning problems, clinical image segmentation and music information classification, the application of the proposed loss functions increase the accuracy.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

Network embedding has attracted considerable research attention recently. However, the existing methods are incapable of handling billion-scale networks, because they are computationally expensive and, at the same time, difficult to be accelerated by distributed computing schemes. To address these problems, we propose RandNE, a novel and simple billion-scale network embedding method. Specifically, we propose a Gaussian random projection approach to map the network into a low-dimensional embedding space while preserving the high-order proximities between nodes. To reduce the time complexity, we design an iterative projection procedure to avoid the explicit calculation of the high-order proximities. Theoretical analysis shows that our method is extremely efficient, and friendly to distributed computing schemes without any communication cost in the calculation. We demonstrate the efficacy of RandNE over state-of-the-art methods in network reconstruction and link prediction tasks on multiple datasets with different scales, ranging from thousands to billions of nodes and edges.

In this paper, we propose an improved quantitative evaluation framework for Generative Adversarial Networks (GANs) on generating domain-specific images, where we improve conventional evaluation methods on two levels: the feature representation and the evaluation metric. Unlike most existing evaluation frameworks which transfer the representation of ImageNet inception model to map images onto the feature space, our framework uses a specialized encoder to acquire fine-grained domain-specific representation. Moreover, for datasets with multiple classes, we propose Class-Aware Frechet Distance (CAFD), which employs a Gaussian mixture model on the feature space to better fit the multi-manifold feature distribution. Experiments and analysis on both the feature level and the image level were conducted to demonstrate improvements of our proposed framework over the recently proposed state-of-the-art FID method. To our best knowledge, we are the first to provide counter examples where FID gives inconsistent results with human judgments. It is shown in the experiments that our framework is able to overcome the shortness of FID and improves robustness. Code will be made available.