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序贯蒙特卡洛估计量的有限样本复杂度AI 解读

We present bounds for the finite-sample error of sequential Monte Carlo samplers on static spaces. Our approach explicitly relates the performance of the algorithm to properties of the chosen sequence of distributions and mixing properties of the associated Markov kernels. This allows us to give the first finite-sample comparison to other Monte Carlo schemes. We obtain bounds for the complexity of sequential Monte Carlo approximations for a variety of target distributions…

异质幂律数据的极值推断AI 解读

We extend extreme value statistics to independent data with possibly very different distributions. In particular, we present novel asymptotic normality results for the Hill estimator, which now estimates the extreme value index of the average distribution. Due to the heterogeneity, the asymptotic variance can be substantially smaller than that in the i.i.d. case. As a special case, we consider a heterogeneous scales model where the asymptotic variance can be calculated…

低秩模型的统计推断AI 解读

This paper studies inference in linear models with a high-dimensional parameter matrix that can be well approximated by a “spiked low-rank matrix.” A spiked low-rank matrix has rank that grows slowly compared to its dimensions and nonzero singular values that diverge to infinity. We show that this framework covers a broad class of models of latent variables, which can accommodate matrix completion problems, factor models, varying coefficient models and heterogeneous…

对抗损失下未知子流形上的分布估计极小极大速率AI 解读

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted a lot of attention. In machine learning, deep generative modelling approaches implicitly estimate distributions of complex objects by creating new samples from the underlying distribution, and have achieved great success in generating synthetic realistic-looking images and texts. A key step in these approaches is the extraction of latent features or representations…

跨尺度的扩散密度估计AI 解读

We consider a space structured population model generated by two-point clouds: a homogeneous Poisson process M with intensity n→∞ as a model for a parent generation together with a Cox point process N as offspring generation, with conditional intensity given by the convolution of M with a scaled dispersal density σ−1f(·/σ). Based on a realisation of M and N, we study the nonparametric estimation of f and the estimation of the physical scale parameter σ>0 simultaneously for…

高维潜在因子模型中的最优判别分析AI 解读

In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower-dimensional space, and base the classification on the resulting lower-dimensional projections. In this paper, we formulate a latent-variable model with a hidden low-dimensional structure to justify this two-step procedure and to guide which projection to choose. We propose a computationally efficient classifier that takes certain principal…

非平稳时间序列的自回归近似、推断与应用AI 解读

Understanding the time-varying structure of complex temporal systems is one of the main challenges of modern time-series analysis. In this paper, we show that every uniformly-positive-definite-in-covariance and sufficiently short-range dependent nonstationary and nonlinear time series can be well approximated globally by a white-noise-driven autoregressive (AR) process of slowly diverging order. To our best knowledge, it is the first time such a structural approximation…

稠密正则图上 Ising 模型的统计推断AI 解读

In this paper, we derive the limit of experiments for one-parameter Ising models on dense regular graphs. In particular, we show that the limiting experiment is Gaussian in the “low temperature” regime, and non-Gaussian in the “critical” regime. We also derive the limiting distributions of the maximum likelihood and maximum pseudolikelihood estimators, and study limiting power for tests of hypothesis against contiguous alternatives. To the best of our knowledge, this is the…

利用基展开估计平稳 Gibbs 点过程的成对交互函数AI 解读

The class of Gibbs point processes (GPP) is a large class of spatial point processes able to model both clustered and repulsive point patterns. They are specified by their conditional intensity, which for a point pattern x and a location u, is roughly speaking the probability that an event occurs in an infinitesimal ball around u given the rest of the configuration is x. The most simple and natural class of models is the class of pairwise interaction point processes where…

残差子抽样的通用秩推断及其大型网络应用AI 解读

Determining the precise rank is an important problem in many large-scale applications with matrix data exploiting low-rank plus noise models. In this paper, we suggest a universal approach to rank inference via residual subsampling (RIRS) for testing and estimating rank in a wide family of models, including many popularly used network models such as the degree corrected mixed membership model as a special case. Our procedure constructs a test statistic via subsampling…

可达半径的最优估计与度量学习AI 解读

We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown d-dimensional Ck-smooth submanifold M of RD, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand and geodesic metric distortion on the other. The derived rates are adaptive, with rates depending on whether the…

高维有向无环图模型及其等价类的贝叶斯学习复杂度AI 解读

Structure learning via MCMC sampling is known to be very challenging because of the enormous search space and the existence of Markov equivalent DAGs. Theoretical results on the mixing behavior are lacking. In this work, we prove the rapid mixing of a random walk Metropolis–Hastings algorithm, which reveals that the complexity of Bayesian learning of sparse equivalence classes grows only polynomially in n and p, under some high-dimensional assumptions. A series of…

使用 ℓp 正则化统计量的 Model-X knockoffs 功效分析AI 解读

Variable selection properties of procedures utilizing penalized-likelihood estimates is a central topic in the study of high-dimensional linear regression problems. Existing literature emphasizes the quality of ranking of the variables by such procedures as reflected in the receiver operating characteristic curve or in prediction performance. Specifically, recent works have harnessed modern theory of approximate message-passing (AMP) to obtain, in a particular setting, exact…

众包问题中的最优排列估计AI 解读

Motivated by crowdsourcing applications, we consider a model where we have partial observations from a bivariate isotonic n×d matrix with an unknown permutation π∗ acting on its rows. Focusing on the twin problems of recovering the permutation π∗ and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way we establish that,…

通信约束下最优高维与非参数分布式检验AI 解读

We derive minimax testing errors in a distributed framework where the data is split over multiple machines and their communication to a central machine is limited to b bits. We investigate both the d- and infinite-dimensional signal detection problem under Gaussian white noise. We also derive distributed testing algorithms reaching the theoretical lower bounds. Our results show that distributed testing is subject to fundamentally different phenomena that are not observed in…

维数与样本量之比趋于无穷时样本协方差特征值统计量的渐近正态性及应用AI 解读

The asymptotic normality for a large family of eigenvalue statistics of a general sample covariance matrix is derived under the ultrahigh-dimensional setting, that is, when the dimension to sample size ratio p/n→∞. Based on this CLT result, we extend the covariance matrix test problem to the new ultra-high-dimensional context, and apply it to test a matrix-valued white noise. Simulation experiments are conducted for the investigation of finite-sample properties of the…

局部渐近正态量子统计模型的估计效率AI 解读

We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators, such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020)…

多元极值中的全正性AI 解读

Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular, the notion of multivariate total positivity of order 2 (MTP2) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce EMTP2, an extremal version of MTP2. This notion turns out to appear prominently in…

正交统计学习AI 解读

We provide nonasymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input arbitrary estimation algorithms for the target parameter and nuisance parameter. We show that if the population risk satisfies a condition called Neyman…

高维向量自回归模型的速率最优稳健估计AI 解读

High-dimensional time series data appear in many scientific areas in the current data-rich environment. Analysis of such data poses new challenges to data analysts because of not only the complicated dynamic dependence between the series, but also the existence of aberrant observations, such as missing values, contaminated observations, and heavy-tailed distributions. For high-dimensional vector autoregressive (VAR) models, we introduce a unified estimation procedure that is…

利用高频数据推断时变协方差矩阵的最大秩AI 解读

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(ΣX(t))≤r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends…

通过经验熵刻画条件密度估计的极小极大速率AI 解读

We consider the task of estimating a conditional density using i.i.d. samples from a joint distribution, which is a fundamental problem with applications in both classification and uncertainty quantification for regression. For joint density estimation, minimax rates have been characterized for general density classes in terms of uniform (metric) entropy, a well-studied notion of statistical capacity. When applying these results to conditional density estimation, the use of…

有效泛函估计与超越自然预言机的现象AI 解读

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of…

低秩高斯混合模型的最优估计与计算极限AI 解读

Structural matrix-variate observations routinely arise in diverse fields such as multilayer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the fundamental questions like its statistical optimality and computational limit are largely under-explored. In this paper, we propose a low-rank Gaussian mixture model (LrMM) assuming each matrix-valued observation has a planted low-rank…