It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\Delta$ is NP-hard if $\Delta\geq 5\cdot 2^{k-1}+1$. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon, Sly and Zhang (2019), the perfect sampler by Qiu, Wang and Zhang (2022), and the deterministic approximation scheme by Feng, Guo, Wang, Wang and Yin (2022).
相關內容
Covariate Selection Based on a Assumpton-free Approach to Linear Regression with Exact Probabilities
In this paper we give a completely new approach to the problem of covariate selection in linear regression. A covariate or a set of covariates is included only if it is better in the sense of least squares than the same number of Gaussian covariates consisting of i.i.d. $N(0,1)$ random variables. The Gaussian P-value is defined as the probability that the Gaussian covariates are better. It is given in terms of the Beta distribution, it is exact and it holds for all data. The covariate selection procedures based on this require only a cut-off value $\alpha$ for the Gaussian P-value: the default value in this paper is $\alpha=0.01$. The resulting procedures are very simple, very fast, do not overfit and require only least squares. In particular there is no regularization parameter, no data splitting, no use of simulations, no shrinkage and no post selection inference is required. The paper includes the results of simulations, applications to real data sets and theorems on the asymptotic behaviour under the standard linear model. Here the stepwise procedure performs overwhelmingly better than any other procedure we are aware of. An R-package {\it gausscov} is available.
A finite cloud of unlabeled points is the simplest representation of many real objects such as rigid shapes considered modulo rigid motion or isometry preserving inter-point distances. The distance matrix uniquely determines any finite cloud of labeled (ordered) points under Euclidean isometry but is intractable for comparing clouds of unlabeled points due to a huge number of permutations. The past work for unlabeled points studied the binary problem of isometry detection, incomplete invariants, or approximations to Hausdorff-style distances, which require minimizations over infinitely many general isometries. This paper introduces the first continuous and complete isometry invariants for finite clouds of unlabeled points considered under isometry in any Euclidean space. The continuity under perturbations of points in the bottleneck distance is proved in terms of new metrics that are exactly computable in polynomial time in the number of points for a fixed dimension.
Understanding the computational complexity of fragments of the Constraint Satisfaction Problem (CSP) has been instrumental in the formulation of Feder-Vardi's Dichotomy Conjecture and its positive resolution by Bulatov and Zhuk. An approximation version of the CSP - known as the promise CSP - has recently gained prominence as an exciting generalisation of the CSP that captures many fundamental computational problems. In this work, we establish a computational complexity dichotomy for a natural fragment of the promise CSP consisting of homomorphism problems involving a class of 3-uniform hypergraphs.
The pandemic COVID-19 disease has had a dramatic impact on almost all countries around the world so that many hospitals have been overwhelmed with Covid-19 cases. As medical resources are limited, deciding on the proper allocation of these resources is a very crucial issue. Besides, uncertainty is a major factor that can affect decisions, especially in medical fields. To cope with this issue, we use fuzzy logic (FL) as one of the most suitable methods in modeling systems with high uncertainty and complexity. We intend to make use of the advantages of FL in decisions on cases that need to treat in ICU. In this study, an interval type-2 fuzzy expert system is proposed for prediction of ICU admission in COVID-19 patients. For this prediction task, we also developed an adaptive neuro-fuzzy inference system (ANFIS). Finally, the results of these fuzzy systems are compared to some well-known classification methods such as Naive Bayes (NB), Case-Based Reasoning (CBR), Decision Tree (DT), and K Nearest Neighbor (KNN). The results show that the type-2 fuzzy expert system and ANFIS models perform competitively in terms of accuracy and F-measure compared to the other system modeling techniques.
Parameter inference, i.e. inferring the posterior distribution of the parameters of a statistical model given some data, is a central problem to many scientific disciplines. Posterior inference with generative models is an alternative to methods such as Markov Chain Monte Carlo, both for likelihood-based and simulation-based inference. However, assessing the accuracy of posteriors encoded in generative models is not straightforward. In this paper, we introduce `distance to random point' (DRP) coverage testing as a method to estimate coverage probabilities of generative posterior estimators. Our method differs from previously-existing coverage-based methods, which require posterior evaluations. We prove that our approach is necessary and sufficient to show that a posterior estimator is optimal. We demonstrate the method on a variety of synthetic examples, and show that DRP can be used to test the results of posterior inference analyses in high-dimensional spaces. We also show that our method can detect non-optimal inferences in cases where existing methods fail.
This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf-sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential-algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.
We present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.
We derive unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection. The discrete orthogonal convolution kernels of the variable-step BDF2 method is commonly utilized recently for solving the phase field models. In this paper, we further prove some new inequalities, concerning the vector forms, for the kernels especially dealing with the nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.
We construct estimators for the parameters of a parabolic SPDE with one spatial dimension based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime. The asymptotic variances are shown to be substantially smaller compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate efficiency gains compared to previous estimation methods numerically and in Monte Carlo simulations.
Causal inference on populations embedded in social networks poses technical challenges, since the typical no interference assumption may no longer hold. For instance, in the context of social research, the outcome of a study unit will likely be affected by an intervention or treatment received by close neighbors. While inverse probability-of-treatment weighted (IPW) estimators have been developed for this setting, they are often highly inefficient. In this work, we assume that the network is a union of disjoint components and propose doubly robust (DR) estimators combining models for treatment and outcome that are consistent and asymptotically normal if either model is correctly specified. We present empirical results that illustrate the DR property and the efficiency gain of DR over IPW estimators when both the outcome and treatment models are correctly specified. Simulations are conducted for networks with equal and unequal component sizes and outcome data with and without a multilevel structure. We apply these methods in an illustrative analysis using the Add Health network, examining the impact of maternal college education on adolescent school performance, both direct and indirect.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191