We study the design of embeddings into Euclidean space with outliers. Given a metric space $(X,d)$ and an integer $k$, the goal is to embed all but $k$ points in $X$ (called the "outliers") into $\ell_2$ with the smallest possible distortion $c$. Finding the optimal distortion $c$ for a given outlier set size $k$, or alternately the smallest $k$ for a given target distortion $c$ are both NP-hard problems. In fact, it is UGC-hard to approximate $k$ to within a factor smaller than $2$ even when the metric sans outliers is isometrically embeddable into $\ell_2$. We consider bi-criteria approximations. Our main result is a polynomial time algorithm that approximates the outlier set size to within an $O(\log^4 k)$ factor and the distortion to within a constant factor. The main technical component in our result is an approach for constructing a composition of two given embeddings from subsets of $X$ into $\ell_2$ which inherits the distortions of each to within small multiplicative factors. Specifically, given a low $c_S$ distortion embedding from $S\subset X$ into $\ell_2$ and a high(er) $c_X$ distortion embedding from the entire set $X$ into $\ell_2$, we construct a single embedding that achieves the same distortion $c_S$ over pairs of points in $S$ and an expansion of at most $O(\log k)\cdot c_X$ over the remaining pairs of points, where $k=|X\setminus S|$. Our composition theorem extends to embeddings into arbitrary $\ell_p$ metrics for $p\ge 1$, and may be of independent interest. While unions of embeddings over disjoint sets have been studied previously, to our knowledge, this is the first work to consider compositions of nested embeddings.
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We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such "migrating" eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.
By combining a logarithm transformation with a corrected Milstein-type method, the present article proposes an explicit, unconditional boundary and dynamics preserving scheme for the stochastic susceptible-infected-susceptible (SIS) epidemic model that takes value in (0,N). The scheme applied to the model is first proved to have a strong convergence rate of order one. Further, the dynamic behaviors are analyzed for the numerical approximations and it is shown that the scheme can unconditionally preserve both the domain and the dynamics of the model. More precisely, the proposed scheme gives numerical approximations living in the domain (0,N) and reproducing the extinction and persistence properties of the original model for any time discretization step-size h > 0, without any additional requirements on the model parameters. Numerical experiments are presented to verify our theoretical results.
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schr\"odinger equation.
We propose a deep learning based method for simulating the large bending deformation of bilayer plates. Inspired by the greedy algorithm, we propose a pre-training method on a series of nested domains, which can accelerate the convergence of training and find the absolute minimizer more effectively. The proposed method exhibits the capability to converge to an absolute minimizer, overcoming the limitation of gradient flow methods getting trapped in local minimizer basins. We showcase better performance for the relative energy errors and relative $L^2$-errors of the minimizer through several numerical experiments. Furthermore, our method successfully maintains the $L^2$-norm of the isometric constraint, leading to improved solution accuracy.
Linear statistics of point processes yield Monte Carlo estimators of integrals. While the simplest approach relies on a homogeneous Poisson point process, more regularly spread point processes, such as scrambled low-discrepancy sequences or determinantal point processes, can yield Monte Carlo estimators with fast-decaying mean square error. Following the intuition that more regular configurations result in lower integration error, we introduce the repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our main theoretical result is that applying the repulsion operator to a homogeneous Poisson point process yields an unbiased Monte Carlo estimator with lower variance than under the original point process. On the computational side, the evaluation of our estimator is only quadratic in the number of integrand evaluations and can be easily parallelized without any communication across tasks. We illustrate our variance reduction result with numerical experiments and compare it to popular Monte Carlo methods. Finally, we numerically investigate a few open questions on the repulsion operator. In particular, the experiments suggest that the variance reduction also holds when the operator is applied to other motion-invariant point processes.
Compared to widely used likelihood-based approaches, the minimum contrast (MC) method is a computationally efficient method for estimation and inference of parametric stationary point processes. This advantage becomes more pronounced when analyzing complex point process models, such as multivariate log-Gaussian Cox processes (LGCP). Despite its practical advantages, there is very little work on the MC method for multivariate point processes. The aim of this article is to introduce a new MC method for parametric multivariate stationary spatial point processes. A contrast function is calculated based on the trace of the power of the difference between the conjectured $K$-function matrix and its nonparametric unbiased edge-corrected estimator. Under standard assumptions, the asymptotic normality of the MC estimator of the model parameters is derived. The performance of the proposed method is illustrated with bivariate LGCP simulations and a real data analysis of a bivariate point pattern of the 2014 terrorist attacks in Nigeria.
We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem, we propose a Bayesian strategy based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.
This article presents the openCFS submodule scattered data reader for coupling multi-physical simulations performed in different simulation programs. For instance, by considering a forward-coupling of a surface vibration simulation (mechanical system) to an acoustic propagation simulation using time-dependent acoustic absorbing material as a noise mitigation measure. The nearest-neighbor search of the target and source points from the interpolation is performed using the FLANN or the CGAL library. In doing so, the coupled field (e.g., surface velocity) is interpolated from a source representation consisting of field values physically stored and organized in a file directory to a target representation being the quadrature points in the case of the finite element method. A test case of the functionality is presented in the "testsuite" module of the openCFS software called "Abc2dcsvt". This scattered data reader module was successfully applied in numerous studies on flow-induced sound generation. Within this short article, the functionality, and usability of this module are described.
Stochastic inversion problems are typically encountered when it is wanted to quantify the uncertainty affecting the inputs of computer models. They consist in estimating input distributions from noisy, observable outputs, and such problems are increasingly examined in Bayesian contexts where the targeted inputs are affected by stochastic uncertainties. In this regard, a stochastic input can be qualified as meaningful if it explains most of the output uncertainty. While such inverse problems are characterized by identifiability conditions, constraints of "signal to noise", that can formalize this meaningfulness, should be accounted for within the definition of the model, prior to inference. This article investigates the possibility of forcing a solution to be meaningful in the context of parametric uncertainty quantification, through the tools of global sensitivity analysis and information theory (variance, entropy, Fisher information). Such forcings have mainly the nature of constraints placed on the input covariance, and can be made explicit by considering linear or linearizable models. Simulated experiments indicate that, when injected into the modeling process, these constraints can limit the influence of measurement or process noise on the estimation of the input distribution, and let hope for future extensions in a full non-linear framework, for example through the use of linear Gaussian mixtures.
Positive linear programs (LPs) model many graph and operations research problems. One can solve for a $(1+\epsilon)$-approximation for positive LPs, for any selected $\epsilon$, in polylogarithmic depth and near-linear work via variations of the multiplicative weight update (MWU) method. Despite extensive theoretical work on these algorithms through the decades, their empirical performance is not well understood. In this work, we implement and test an efficient parallel algorithm for solving positive LP relaxations, and apply it to graph problems such as densest subgraph, bipartite matching, vertex cover and dominating set. We accelerate the algorithm via a new step size search heuristic. Our implementation uses sparse linear algebra optimization techniques such as fusion of vector operations and use of sparse format. Furthermore, we devise an implicit representation for graph incidence constraints. We demonstrate the parallel scalability with the use of threading OpenMP and MPI on the Stampede2 supercomputer. We compare this implementation with exact libraries and specialized libraries for the above problems in order to evaluate MWU's practical standing for both accuracy and performance among other methods. Our results show this implementation is faster than general purpose LP solvers (IBM CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms state-of-the-art specialized parallel graph algorithms.
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