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利用 Wasserstein–Fisher–Rao 梯度流学习高斯混合模型AI 解读

Gaussian mixture models form a flexible and expressive parametric family of distributions that has found a variety of applications. Unfortunately, fitting these models to data is a notoriously hard problem from a computational perspective. Currently, only moment-based methods enjoy theoretical guarantees while likelihood-based methods are dominated by heuristics such as Expectation-Maximization that are known to fail in simple examples. In this work, we propose a new…

高维加性回归的高效函数型 Lasso 核平滑AI 解读

Smooth backfitting has been proposed and proved as a powerful nonparametric estimation technique for additive regression models in various settings. Existing studies are restricted to cases with a moderate number of covariates and are not directly applicable to high dimensional settings. In this paper, we develop new kernel estimators based on the idea of smooth backfitting for high dimensional additive models. We introduce a novel penalization scheme, combining the idea of…

贝叶斯与频率学派反卷积模型中的 Wasserstein 收敛AI 解读

We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors having known distribution. For errors with ordinary smooth distribution, we recast the multidimensional problem as a one-dimensional problem leveraging the equivalence between the L1-Wasserstein and the max-sliced L1-Wasserstein metrics and derive an inversion inequality relating the…

Gromov–Wasserstein距离：熵正则化、对偶与样本复杂度AI 解读

The Gromov–Wasserstein (GW) distance, rooted in optimal transport (OT) theory, quantifies dissimilarity between metric measure spaces and provides a framework for aligning heterogeneous datasets. While computational aspects of the GW problem have been widely studied, a duality theory and fundamental statistical questions concerning empirical convergence rates remained obscure. This work closes these gaps for the quadratic GW distance over Euclidean spaces of different…

连续处理效应的非参数双重稳健检验AI 解读

The vast majority of literature on evaluating the significance of a treatment effect based on observational data has been confined to discrete treatments. These methods are not applicable to drawing inference for a continuous treatment, which arises in many important applications. To adjust for confounders when evaluating a continuous treatment, existing inference methods often rely on discretizing the treatment or using (possibly misspecified) parametric models for the…

稀疏序列多重检验的精确边界AI 解读

This work investigates multiple testing by considering minimax separation rates in the sparse sequence model, when the testing risk is measured as the sum FDR+FNR (False Discovery Rate plus False Negative Rate). First, using the popular beta-min separation condition, with all nonzero signals separated from 0 by at least some amount, we determine the sharp minimax testing risk asymptotically and thereby explicitly describe the transition from “achievable multiple testing with…

可微 Hilbert 空间值参数的一步估计AI 解读

rate estimators and corresponding confidence sets. These estimators correspond to generalizations of cross-fitted one-step estimators based on Hilbert-valued efficient influence functions. We give theoretical guarantees even when arbitrary estimators of nuisance functions are used, including those based on machine learning techniques. We show that these results naturally extend to Hilbert spaces that lack a reproducing kernel, as long as the parameter has an efficient…

广义Grenander型估计量的自助法辅助推断AI 解读

Westling and Carone (Ann. Statist. 48 (2020) 1001–1024) proposed a framework for studying the large sample distributional properties of generalized Grenander-type estimators, a versatile class of nonparametric estimators of monotone functions. The limiting distribution of those estimators is representable as the left derivative of the greatest convex minorant of a Gaussian process whose monomial mean can be of unknown order (when the degree of flatness of the function of…

长宽比发散时低秩矩阵估计的基本极限AI 解读

We consider the problem of estimating the factors of a low-rank n×d matrix, when this is corrupted by additive Gaussian noise. A special example of our setting corresponds to clustering mixtures of Gaussians with equal (known) covariances. Simple spectral methods do not take into account the distribution of the entries of these factors and are therefore often suboptimal. Here, we characterize the asymptotics of the minimum estimation error under the assumption that the…

重尾先验下的贝叶斯非参数自适应AI 解读

We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy-tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed,…

高斯变分推断的近似精度AI 解读

The main computational challenge in Bayesian inference is to compute integrals against a high-dimensional posterior distribution. In the past decades, variational inference (VI) has emerged as a tractable approximation to these integrals, and a viable alternative to the more established paradigm of Markov chain Monte Carlo. However, little is known about the approximation accuracy of VI. In this work, we bound the TV error and the mean and covariance approximation error of…

利用t统计量的Edgeworth展开分析分批方法的高阶覆盖误差AI 解读

While batching methods have been widely used in simulation and statistics, their higher-order coverage behaviors and relative advantages in this regard remain open. We develop techniques to obtain higher-order coverage errors for batching methods by building Edgeworth-type expansions on t-statistics. The coefficients in these expansions are intricate analytically, but we provide algorithms to estimate the coefficients of the n−1 error terms via Monte Carlo simulation. We…

矩形矩阵 Schatten 范数的最优估计AI 解读

We consider the twin problems of estimating the effective rank and the Schatten norms ‖A‖s of a rectangular p×q matrix A from noisy observations. When s is an even integer, we introduce a polynomial-time estimator of ‖A‖s that achieves the minimax rate (pq)1/4. Interestingly, this optimal rate does not depend on the underlying rank of the matrix A. When s is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves…

基于多重测量的线性随机偏微分方程最优参数估计AI 解读

The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is nondecreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic…

粗糙波动率的统计推断：极小极大理论AI 解读

In recent years, rough volatility models have gained considerable attention in quantitative finance. In this paradigm, the stochastic volatility of the price of an asset has quantitative properties similar to that of a fractional Brownian motion with small Hurst index H<1/2. In this work, we provide the first rigorous statistical analysis of the problem of estimating H from historical observations of the underlying asset. We establish minimax lower bounds and design optimal…

高斯过程先验下数据同化的后验一致性：二维Navier–Stokes方程AI 解读

We consider a nonlinear Bayesian data assimilation model for the periodic two-dimensional Navier–Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by…

高维函数型时间序列中结构突变的检测与估计AI 解读

We consider detecting and estimating breaks in heterogenous mean functions of high-dimensional functional time series which are allowed to be cross-sectionally correlated. A new test statistic combining the functional CUSUM statistic and power enhancement component is proposed with asymptotic null distribution comparable to the conventional CUSUM theory derived for a single functional time series. In particular, the extra power enhancement component enlarges the region where…

潜在双向因子结构矩阵时间序列的在线变点检测AI 解读

This paper proposes a novel methodology for the online detection of changepoints in the factor structure of large matrix time series. Our approach is based on the well-known fact that, in the presence of a changepoint, the number of spiked eigenvalues in the second moment matrix of the data increases (e.g., in the presence of a change in the loadings, or if a new factor emerges). Based on this, we propose two families of procedures—one based on the fluctuations of partial…

非光滑随机逼近中的渐近正态性与最优性AI 解读

In their seminal work, Polyak and Juditsky showed that stochastic approximation algorithms for solving smooth equations enjoy a central limit theorem. Moreover, it has since been argued that the asymptotic covariance of the method is best possible among any estimation procedure in a local minimax sense of Hájek and Le Cam. A long-standing open question in this line of work is whether similar guarantees hold for important nonsmooth problems, such as stochastic nonlinear…

e 统计量、群不变性与任意时点有效检验AI 解读

We study worst-case-growth-rate-optimal (GROW) e-statistics for hypothesis testing between two group models. It is known that under a mild condition on the action of the underlying group G on the data, there exists a maximally invariant statistic. We show that among all e-statistics, invariant or not, the likelihood ratio of the maximally invariant statistic is GROW, both in the absolute and in the relative sense, and that an anytime-valid test can be based on it. The GROW…

利用特征函数线性状态空间表示估计期权定价模型AI 解读

We develop a novel filtering and estimation procedure for parametric option pricing models driven by general affine jump-diffusions. Our procedure is based on the comparison between an option-implied, model-free representation of the conditional log-characteristic function and the model-implied conditional log-characteristic function, which is functionally affine in the model's state vector. We formally derive an associated linear state space representation and the…

尾指数与极端分位数的一致置信区间AI 解读

This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that there exists a lower bound of the length for confidence intervals that satisfy the correct uniform coverage over a nonparametric family of tail distributions. Second, in light of the impossibility result, we construct honest confidence intervals that are uniformly valid by incorporating the worst-case bias in the nonparametric family. The…

内生二元处理的无条件政策效应：无条件 MTE 方法AI 解读

This paper studies the identification and estimation of policy effects when treatment status is binary and endogenous. We introduce a new class of marginal treatment effects (MTEs) based on the influence function of the functional underlying the policy target. We show that an unconditional policy effect can be represented as a weighted average of the newly defined MTEs over the individuals who are indifferent about their treatment status. We provide conditions for point…

多值处理效应的高维模型辅助推断AI 解读

Consider estimation of average treatment effects with multi-valued treatments using augmented inverse probability weighted (IPW) estimators, depending on outcome regression and propensity score models in high-dimensional settings. These regression models are often fitted by regularized likelihood-based estimation, while ignoring how the fitted functions are used in the subsequent inference about the treatment parameters. Such separate estimation can be associated with known…