While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by uncertainty quantification. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Accordingly, we develop method-of-moment estimators of network subgraph densities and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on the asymptotic normality. To construct the confidence intervals, a new and non-standard bootstrap method is proposed to compute asymptotic variances, which is infeasible otherwise. We illustrate the proposed methods in the context of gene coexpression networks.
Stanford University（斯坦福大学）；Australian National University（澳大利亚国立大学）；The University of Hong Kong（香港大学）；Rutgers University（罗格斯大学）；Massachusetts Institute of Technology（麻省理工学院）
Small subgraphs (graphlets) are important features to describe fundamental units of a large network. The calculation of the subgraph frequency distributions has a wide application in multiple domains including biology and engineering. Unfortunately due to the inherent complexity of this task, most of the existing methods are computationally intensive and inefficient. In this work, we propose GNNS, a novel representational learning framework that utilizes graph neural networks to sample subgraphs efficiently for estimating their frequency distribution. Our framework includes an inference model …
Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (201…
A tuning-free procedure is proposed to estimate the covariate-adjusted Gaussian graphical model. For each finite subgraph, this estimator is asymptotically normal and efficient. As a consequence, a confidence interval can be obtained for each edge. The procedure enjoys easy implementation and efficient computation through parallel estimation on subgraphs or edges. We further apply the asymptotic normality result to perform support recovery through edge-wise adaptive thresholding. This support recovery procedure is called ANTAC, standing for Asymptotically Normal estimation with Thresholding af…
In a parametric framework, the paper is devoted to the study of a new estimation procedure for the inverse filter and the level noise in a complex noisy blind discrete deconvolution model. Our estimation method is a consequence of the sharp exploitation of the specifical properties of the Hankel forms. The distribution of the input signal is also estimated. The strong consistency and the asymptotic distribution of all estimates are established. A consistent simulation study is added in order to demonstrate empirically the computational performance of our estimation procedures.
We consider the problem of multivariate density estimation when the unknown density is assumed to follow a particular form of dimensionality reduction, a noisy independent factor analysis (IFA) model. In this model the data are generated by a number of latent independent components having unknown distributions and are observed in Gaussian noise. We do not assume that either the number of components or the matrix mixing the components are known. We show that the densities of this form can be estimated with a fast rate. Using the mirror averaging aggregation algorithm, we construct a density est…
We investigate a semiparametric regression model where one gets noisy non linear non invertible functions of the observations. We focus on the application to bearings-only tracking. We first investigate the least squares estimator and prove its consistency and asymptotic normality under mild assumptions. We study the semiparametric likelihood process and prove local asymptotic normality of the model. This allows to define the efficient Fisher information as a lower bound for the asymptotic variance of regular estimators, and to prove that the parametric likelihood estimator is regular and asym…
In the framework of noisy quantum homodyne tomography with efficiency parameter $0 < η\leq 1$, we propose two estimators of a quantum state whose density matrix elements $ρ_{m,n}$ decrease like $e^{-B(m+n)^{r/ 2}}$, for fixed known $B>0$ and $0<r\leq 2$. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for $0<η\leq 1/2$), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in $\mathbb{L}_2$ risk.
Large graphs abound in machine learning, data mining, and several related areas. A useful step towards analyzing such graphs is that of obtaining certain summary statistics - e.g., or the expected length of a shortest path between two nodes, or the expected weight of a minimum spanning tree of the graph, etc. These statistics provide insight into the structure of a graph, and they can help predict global properties of a graph. Motivated thus, we propose to study statistical properties of structured subgraphs (of a given graph), in particular, to estimate the expected objective function value o…
The aim of this work is to estimate a quadratic functional of a unknown Wigner function from noisy tomographic data. The Wigner function can be seen as the representation of the quantum state of a light beam. The estimation of a quadratic functional is done from result of quantum homodyne measurement performed on identically prepared quantum systems. We start by constructing an estimator of a quadratic functional of the Wigner function. We show that the proposed estimator is optimal or nearly optimal in a minimax sense over a class of infinitely differentiable functions. Parametric rates are a…
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb R^2$ which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $\mathbb L_\infty$-risk over a class of infinitely differentiable. We compute also the lower …
With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The former (consistency) has been addressed heavily in the recent literature, however, the resulting estimator is not quite efficient. In Zhang, Mykland and Ait-Sahalia (2003), the best estimator converges to the true volatility only at the rate of n^{-1/6}. In this paper, …
A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate density. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian distribution, even though …
We consider the nonparametric estimation of the intensity function of a Poisson point process in a circular model from indirect observations $N_1,\ldots,N_n$. These observations emerge from hidden point process realizations with the target intensity through contamination with additive error. In case that the error distribution can only be estimated from an additional sample $Y_1,\ldots,Y_m$ we derive minimax rates of convergence with respect to the sample sizes $n$ and $m$ under abstract smoothness conditions and propose an orthonormal series estimator which attains the optimal rate of converg…
We estimate convex polytopes and general convex sets in $\mathbb R^d,d\geq 2$ in the regression framework. We measure the risk of our estimators using a $L^1$-type loss function and prove upper bounds on these risks. We show that, in the case of polytopes, these estimators achieve the minimax rate. For polytopes, this minimax rate is $\frac{\ln n}{n}$, which differs from the parametric rate for non-regular families by a logarithmic factor, and we show that this extra factor is essential. Using polytopal approximations we extend our results to general convex sets, and we achieve the minimax rat…
There is increasing interest in learning a set of small outcome-relevant subgraphs in network-predictor regression. The extracted signal subgraphs can greatly improve the interpretation of the association between the network predictor and the response. In brain connectomics, the brain network for an individual corresponds to a set of interconnections among brain regions and there is a strong interest in linking the brain connectome to human cognitive traits. Modern neuroimaging technology allows a very fine segmentation of the brain, producing very large structural brain networks. Therefore, a…
Many fundamental concepts in network-based epidemic modeling depend on the branching factor, which captures a sense of dispersion in the network connectivity and quantifies the rate of spreading across the network. Moreover, contact network information generally is available only up to some level of error. We study the propagation of such errors to the estimation of the branching factor. Specifically, we characterize the impact of network noise on the bias and variance of the observed branching factor for arbitrary true networks, with examples in sparse, dense, homogeneous and inhomogeneous ne…
We consider parametric estimation for ergodic diffusion processes with noisy sampled data based on the hybrid method, that is, the multi-step estimation with the initial Bayes type estimators. In order to select proper initial values for optimisation of the quasi likelihood function of ergodic diffusion processes with noisy observations, we construct the initial Bayes type estimator based on the local means of the noisy observations. The asymptotic properties of the initial Bayes type estimators and the hybrid multi-step estimators with the initial Bayes type estimators are shown, and a concre…
We address the question of estimating Kullback-Leibler losses rather than squared losses in recovery problems where the noise is distributed within the exponential family. Inspired by Stein unbiased risk estimator (SURE), we exhibit conditions under which these losses can be unbiasedly estimated or estimated with a controlled bias. Simulations on parameter selection problems in applications to image denoising and variable selection with Gamma and Poisson noises illustrate the interest of Kullback-Leibler losses and the proposed estimators.
Central to many inferential situations is the estimation of rational functions of parameters. The mainstream in statistics and econometrics estimates these quantities based on the plug-in approach without consideration of the main objective of the inferential situation. We propose the Bayesian Minimum Expected Loss (MELO) approach focusing explicitly on the function of interest, and calculating its frequentist variability. Asymptotic properties of the MELO estimator are similar to the plug-in approach. Nevertheless, simulation exercises show that our proposal is better in situations characteri…
Inferring unknown conic sections on the basis of noisy data is a challenging problem with applications in computer vision. A major limitation of the currently available methods for conic sections is that estimation methods rely on the underlying shape of the conics (being known to be ellipse, parabola or hyperbola). A general purpose Bayesian hierarchical model is proposed for conic sections and corresponding estimation method based on noisy data is shown to work even when the specific nature of the conic section is unknown. The model, thus, provides probabilistic detection of the underlying c…
In this letter, we consider two sets of observations defined as subspace signals embedded in noise and we wish to analyze the distance between these two subspaces. The latter entails evaluating the angles between the subspaces, an issue reminiscent of the well-known Procrustes problem. A Bayesian approach is investigated where the subspaces of interest are considered as random with a joint prior distribution (namely a Bingham distribution), which allows the closeness of the two subspaces to be adjusted. Within this framework, the minimum mean-square distance estimator of both subspaces is form…
We study the problem of estimating time-varying coefficients in ordinary differential equations. Current theory only applies to the case when the associated state variables are observed without measurement errors as presented in \cite{chenwu08b,chenwu08}. The difficulty arises from the quadratic functional of observations that one needs to deal with instead of the linear functional that appears when state variables contain no measurement errors. We derive the asymptotic bias and variance for the previously proposed two-step estimators using quadratic regression functional theory.
In this article we derive an unbiased expression for the expected mean-squared error associated with continuously differentiable estimators of the noncentrality parameter of a chi-square random variable. We then consider the task of denoising squared-magnitude magnetic resonance image data, which are well modeled as independent noncentral chi-square random variables on two degrees of freedom. We consider two broad classes of linearly parameterized shrinkage estimators that can be optimized using our risk estimate, one in the general context of undecimated filterbank transforms, and another in …
