We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an estimator observes $n$ samples of a $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$, multiplied by a random sign $ S_i $ ($1\le i\le n$), and corrupted by isotropic standard Gaussian noise. The sequence of signs $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with flip probability $\delta\in[0,1/2]$. As $\delta$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $\delta=0$ and the Gaussian Mixture Model for which $\delta=1/2$. Assuming that the estimator knows $\delta$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|\theta_{*}\|,\delta,d,n$. We then provide an upper bound to the case of estimating $\delta$, assuming a (possibly inaccurate) knowledge of $\theta_{*}$. The bound is proved to be tight when $\theta_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $\theta_{*}$ with $\delta$ unknown a priori, and theoretical guarantees on its error are stated.
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Mark-point dependence plays a critical role in research problems that can be fitted into the general framework of marked point processes. In this work, we focus on adjusting for mark-point dependence when estimating the mean and covariance functions of the mark process, given independent replicates of the marked point process. We assume that the mark process is a Gaussian process and the point process is a log-Gaussian Cox process, where the mark-point dependence is generated through the dependence between two latent Gaussian processes. Under this framework, naive local linear estimators ignoring the mark-point dependence can be severely biased. We show that this bias can be corrected using a local linear estimator of the cross-covariance function and establish uniform convergence rates of the bias-corrected estimators. Furthermore, we propose a test statistic based on local linear estimators for mark-point independence, which is shown to converge to an asymptotic normal distribution in a parametric $\sqrt{n}$-convergence rate. Model diagnostics tools are developed for key model assumptions and a robust functional permutation test is proposed for a more general class of mark-point processes. The effectiveness of the proposed methods is demonstrated using extensive simulations and applications to two real data examples.
In Value of Information (VoI) analysis, the unit normal loss integral (UNLI) frequently emerges as a solution for the computation of various VoI metrics. However, one limitation of the UNLI has been that its closed-form solution is available for only one dimension, and thus can be used for comparisons involving only two strategies (where it is applied to the scalar incremental net benefit). We derived a closed-form solution for the two-dimensional UNLI, enabling closed-form VoI calculations for three strategies. We verified the accuracy of this method via simulation studies. A case study based on a three-arm clinical trial was used as an example. VoI methods based on the closed-form solutions for the UNLI can now be extended to three-decision comparisons, taking a fraction of a second to compute and not being subject to Monte Carlo error. An R implementation of this method is provided as part of the predtools package (//github.com/resplab/predtools/).
We study the problem of learning nonparametric distributions in a finite mixture, and establish tight bounds on the sample complexity for learning the component distributions in such models. Namely, we are given i.i.d. samples from a pdf $f$ where $$ f=\sum_{i=1}^k w_i f_i, \quad\sum_{i=1}^k w_i=1, \quad w_i>0 $$ and we are interested in learning each component $f_i$. Without any assumptions on $f_i$, this problem is ill-posed. In order to identify the components $f_i$, we assume that each $f_i$ can be written as a convolution of a Gaussian and a compactly supported density $\nu_i$ with $\text{supp}(\nu_i)\cap \text{supp}(\nu_j)=\emptyset$. Our main result shows that $(\frac{1}{\varepsilon})^{\Omega(\log\log \frac{1}{\varepsilon})}$ samples are required for estimating each $f_i$. Unlike parametric mixtures, the difficulty does not arise from the order $k$ or small weights $w_i$, and unlike nonparametric density estimation it does not arise from the curse of dimensionality, irregularity, or inhomogeneity. The proof relies on a fast rate for approximation with Gaussians, which may be of independent interest. To show this is tight, we also propose an algorithm that uses $(\frac{1}{\varepsilon})^{O(\log\log \frac{1}{\varepsilon})}$ samples to estimate each $f_i$. Unlike existing approaches to learning latent variable models based on moment-matching and tensor methods, our proof instead involves a delicate analysis of an ill-conditioned linear system via orthogonal functions. Combining these bounds, we conclude that the optimal sample complexity of this problem properly lies in between polynomial and exponential, which is not common in learning theory.
Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of $L_2$Boosting, which is tailored for regression, in a high-dimensional setting. Moreover, we introduce so-called \textquotedblleft post-Boosting\textquotedblright. This is a post-selection estimator which applies ordinary least squares to the variables selected in the first stage by $L_2$Boosting. Another variant is \textquotedblleft Orthogonal Boosting\textquotedblright\ where after each step an orthogonal projection is conducted. We show that both post-$L_2$Boosting and the orthogonal boosting achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting. We show that the rate of convergence of the classical $L_2$Boosting depends on the design matrix described by a sparse eigenvalue constant. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of $L_2$Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, post-$L_2$Boosting clearly outperforms LASSO.
Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, Cheridito proved in [Bernoulli 7 (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on a second surprise in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.
We establish the minimax risk for parameter estimation in sparse high-dimensional Gaussian mixture models and show that a constrained maximum likelihood estimator (MLE) achieves the minimax optimality. However, the optimization-based constrained MLE is computationally intractable due to non-convexity of the problem. Therefore, we propose a Bayesian approach to estimate high-dimensional Gaussian mixtures whose cluster centers exhibit sparsity using a continuous spike-and-slab prior, and prove that the posterior contraction rate of the proposed Bayesian method is minimax optimal. The mis-clustering rate is obtained as a by-product using tools from matrix perturbation theory. Computationally, posterior inference of the proposed Bayesian method can be implemented via an efficient Gibbs sampler with data augmentation, circumventing the challenging frequentist nonconvex optimization-based algorithms. The proposed Bayesian sparse Gaussian mixture model does not require pre-specifying the number of clusters, which is allowed to grow with the sample size and can be adaptively estimated via posterior inference. The validity and usefulness of the proposed method is demonstrated through simulation studies and the analysis of a real-world single-cell RNA sequencing dataset.
Inference of the marginal probability distribution is defined as the calculation of the probability of a subset of the variables and is relevant for handling missing data and hidden variables. While inference of the marginal probability distribution is crucial for various problems in machine learning and statistics, its exact computation is generally not feasible for categorical variables in Bayesian networks due to the NP-hardness of this task. We develop a divide-and-conquer approach using the graphical properties of Bayesian networks to split the computation of the marginal probability distribution into sub-calculations of lower dimensionality, thus reducing the overall computational complexity. Exploiting this property, we present an efficient and scalable algorithm for calculating the marginal probability distribution for categorical variables. The novel method is compared against state-of-the-art approximate inference methods in a benchmarking study, where it displays superior performance. As an immediate application, we demonstrate how our method can be used to classify incomplete data against Bayesian networks and use this approach for identifying the cancer subtype of kidney cancer patient samples.
In cell line perturbation experiments, a collection of cells is perturbed with external agents (e.g. drugs) and responses such as protein expression measured. Due to cost constraints, only a small fraction of all possible perturbations can be tested in vitro. This has led to the development of computational (in silico) models which can predict cellular responses to perturbations. Perturbations with clinically interesting predicted responses can be prioritized for in vitro testing. In this work, we compare causal and non-causal regression models for perturbation response prediction in a Melanoma cancer cell line. The current best performing method on this data set is Cellbox which models how proteins causally effect each other using a system of ordinary differential equations (ODEs). We derive a closed form solution to the Cellbox system of ODEs in the linear case. These analytic results facilitate comparison of Cellbox to regression approaches. We show that causal models such as Cellbox, while requiring more assumptions, enable extrapolation in ways that non-causal regression models cannot. For example, causal models can predict responses for never before tested drugs. We illustrate these strengths and weaknesses in simulations. In an application to the Melanoma cell line data, we find that regression models outperform the Cellbox causal model.
We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is a homogeneous quasi-arithmetic mean, and under some conditions, several limit theorems hold for the power mean as well as for the arithmetic mean of i.i.d. integrable random variables. We establish integrabilities and a limit theorem for the variances of the power mean of i.i.d. non-integrable random variables. We also consider behaviors of the power mean when the parameter of the power varies. Our feature is that the generator of the power mean is allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. The complex-valued power mean is an unbiased strongly-consistent estimator for the joint of the location and scale parameters of the Cauchy distribution.
We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.
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