arXiv subjects
Publications and source records attributed to Jingchen Hu.
In this paper, we establish the power convexity of solutions to the complex Monge-Amp\`{e}re equation $\det(u_{i\overline{j}})=1$ in a convex set in the complex 2 dimensional Euclidean space. Even the convexity result is only for 2 dimension, our approach is applicable to all dimensions.
Suppose that u is the potential function of a complete K\"ahler-Einstein metric on a bounded strictly convex domain in $\mathbb{C}^n$. We prove that u itself is strictly convex.
We introduce a new class of range restricted formal data privacy standards that condition on owner beliefs about sensitive data ranges. By incorporating this additional information, we can provide a stronger privacy guarantee (e.g. an amplification). The range restricted formal privacy standards protect only a subset (or ball) of data values and exclude ranges (or balls) believed to be already publicly known. The privacy standards are designed for the risk-weighted pseudo posterior (model) mechanism (PPM) used to generate synthetic data under an asymptotic Differential (aDP) privacy guarantee. The PPM downweights the likelihood contribution for each record proportionally to its disclosure risk. The PPM is adapted under inclusion of beliefs by adjusting the risk-weighted pseudo likelihood. We introduce two alternative adjustments. The first expresses data owner knowledge of the sensitive range as a probability, $\lambda$, that a datum value drawn from the underlying generating distribution lies outside the ball or subspace of values that are sensitive. The portion of each datum likelihood contribution deemed sensitive is then $(1-\lambda) \leq 1$ and is the only portion of the likelihood subject to risk down-weighting. The second adjustment encodes knowledge as the difference in probability masses $P(R) \leq 1$ between the edges of the sensitive range, $R$. We use the resulting conditional (pseudo) likelihood for a sensitive record, which boosts its worst case tail values away from 0. We compare privacy and utility properties for the PPM under the aDP and range restricted privacy standards.
Educating the next generation of scientists in statistical methodology is an important task. Educating their instructors in statistical content knowledge and pedagogical knowledge is as important and provides an indirect impact of students' learning. Statisticians are in a place to lead train-the-trainer (TTT) programs in different methods. We present our instructor training program in Bayesian methods as an effective case study of a TTT model. In addition to describing the details of the structure of our training program, we share our experience in designing and implementing our program including the challenges we face, the opportunities created, and our recommendations for TTT programs led by statisticians.
In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation $$Δ_pv + a|\nabla v|^qv^r =0$$ on a complete Riemannian manifold with Ricci curvature bounded from below, where $p>1$, $q$, $r$ and $a$ are some real constants. Assuming certain conditions on $a,\, p,\, q$ and $r$, we can derive universal and succinct Cheng-Yau type logarithmic gradient estimates for such solutions. In particular, we give the obvious expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see \thmref{t10}). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities and some local estimates near singularities for positive solutions. Some of our results are new even in the case the domain is an Euclidean space and $p=2$.
Suppose $Ω_0,Ω_1$ are two bounded strongly $\mathbb{C}$-convex domains in $\mathbb{C}^n$, with $n\geq 2$ and $Ω_1\supset\overline{Ω_0}$. Let $\mathcal{R}=Ω_1\backslash\overline{Ω_0}$. We call $\mathcal{R}$ a $\mathbb{C}$-convex ring. We will show that for a solution $Φ$ to the homogenous complex Monge-Ampère equation in $\mathcal{R}$, with $Φ=1$ on $\partialΩ_1$ and $Φ=0$ on $\partialΩ_0$, $\sqrt{-1}\partial\overline{\partial}Φ$ has rank $n-1$ and the level sets of $Φ$ are strongly $\mathbb{C}$-convex.
This paper introduces a new method that embeds any Bayesian model used to generate synthetic data and converts it into a differentially private (DP) mechanism. We propose an alteration of the model synthesizer to utilize a censored likelihood that induces upper and lower bounds of [$\exp(-ε/ 2), \exp(ε/ 2)$], where $ε$ denotes the level of the DP guarantee. This censoring mechanism equipped with an $ε-$DP guarantee will induce distortion into the joint parameter posterior distribution by flattening or shifting the distribution towards a weakly informative prior. To minimize the distortion in the posterior distribution induced by likelihood censoring, we embed a vector-weighted pseudo posterior mechanism within the censoring mechanism. The pseudo posterior is formulated by selectively downweighting each likelihood contribution proportionally to its disclosure risk. On its own, the pseudo posterior mechanism produces a weaker asymptotic differential privacy (aDP) guarantee. After embedding in the censoring mechanism, the DP guarantee becomes strict such that it does not rely on asymptotics. We demonstrate that the pseudo posterior mechanism creates synthetic data with the highest utility at the price of a weaker, aDP guarantee, while embedding the pseudo posterior mechanism in the proposed censoring mechanism produces synthetic data with a stronger, non-asymptotic DP guarantee at the cost of slightly reduced utility. The perturbed histogram mechanism is included for comparison.
Synthetic data generation is a powerful tool for privacy protection when considering public release of record-level data files. Initially proposed about three decades ago, it has generated significant research and application interest. To meet the pressing demand of data privacy protection in a variety of contexts, the field needs more researchers and practitioners. This review provides a comprehensive introduction to synthetic data, including technical details of their generation and evaluation. Our review also addresses the challenges and limitations of synthetic data, discusses practical applications, and provides thoughts for future work.
Probabilistic models such as logistic regression, Bayesian classification, neural networks, and models for natural language processing, are increasingly more present in both undergraduate and graduate statistics and data science curricula due to their wide range of applications. In this paper, we present a one-week course module for studnets in advanced undergraduate and applied graduate courses on variational inference, a popular optimization-based approach for approximate inference with probabilistic models. Our proposed module is guided by active learning principles: In addition to lecture materials on variational inference, we provide an accompanying class activity, an \texttt{R shiny} app, and guided labs based on real data applications of logistic regression and clustering documents using Latent Dirichlet Allocation with \texttt{R} code. The main goal of our module is to expose students to a method that facilitates statistical modeling and inference with large datasets. Using our proposed module as a foundation, instructors can adopt and adapt it to introduce more realistic case studies and applications in data science, Bayesian statistics, multivariate analysis, and statistical machine learning courses.
On a compact complex affine manifold with a constant coefficient Kähler metric $ω_0$, we introduce a concept: $(S,ω_0)$-convexity and show that $(S,ω_0)$-convexity is preserved by geodesics in the space of Kähler potentials. This implies that if two potentials are both strictly $(S,ω_0)$-convex, then the metrics along the geodesic connecting them are non-degenerate.
In this paper we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge-Amp\`ere equation. As a consequence, we find that in the space of K\"ahler potentials any two points close to each other in $C^2$ norm can be connected by a geodesic along which the associated metrics do not degenerate.
In the intelligent communication field, deep learning (DL) has attracted much attention due to its strong fitting ability and data-driven learning capability. Compared with the typical DL feedforward network structures, an enhancement structure with direct data feedback have been studied and proved to have better performance than the feedfoward networks. However, due to the above simple feedback methods lack sufficient analysis and learning ability on the feedback data, it is inadequate to deal with more complicated nonlinear systems and therefore the performance is limited for further improvement. In this paper, a novel multi-agent feedback enabled neural network (MAFENN) framework is proposed, which make the framework have stronger feedback learning capabilities and more intelligence on feature abstraction, denoising or generation, etc. Furthermore, the MAFENN framework is theoretically formulated into a three-player Feedback Stackelberg game, and the game is proved to converge to the Feedback Stackelberg equilibrium. The design of MAFENN framework and algorithm are dedicated to enhance the learning capability of the feedfoward DL networks or their variations with the simple data feedback. To verify the MAFENN framework's feasibility in wireless communications, a multi-agent MAFENN based equalizer (MAFENN-E) is developed for wireless fading channels with inter-symbol interference (ISI). Experimental results show that when the quadrature phase-shift keying (QPSK) modulation scheme is adopted, the SER performance of our proposed method outperforms that of the traditional equalizers by about 2 dB in linear channels. When in nonlinear channels, the SER performance of our proposed method outperforms that of either traditional or DL based equalizers more significantly, which shows the effectiveness and robustness of our proposal in the complex channel environment.
The large number of publicly available survey datasets of wide variety, albeit useful, raise respondent-level privacy concerns. The synthetic data approach to data privacy and confidentiality has been shown useful in terms of privacy protection and utility preservation. This paper aims at illustrating how synthetic data can facilitate the dissemination of highly sensitive information about youth risk behavior by presenting a case study of synthetic data for a sample of the Youth Risk Behavior Survey (YRBS). Given the categorical nature of almost all variables in YRBS, the Dirichlet Process mixture of products of multinomials (DPMPM) synthesizer is adopted to partially synthesize the YRBS sample. Detailed evaluations of utility and disclosure risks demonstrate that the generated synthetic data are able to significantly reduce the disclosure risks compared to the confidential YRSB sample while maintaining a high level of utility.
We address practical implementation of a risk-weighted pseudo posterior synthesizer for microdata dissemination with a new re-weighting strategy that maximizes utility of released synthetic data under at any level of formal privacy guarantee. Our re-weighting strategy applies to any vector-weighted pseudo posterior mechanism under which a vector of observation-indexed weights are used to downweight likelihood contributions for high disclosure risk records. We demonstrate our method on two different vector-weighted schemes that target high-risk records. Our new method for constructing record-indexed downeighting maximizes the data utility under any privacy budget for the vector-weighted synthesizers by adjusting the by-record weights, such that their individual Lipschitz bounds approach the bound for the entire database. Our method achieves an $(ε= 2 Δ_{\boldsymbolα})-$asymptotic differential privacy (aDP) guarantee, globally, over the space of databases. We illustrate our methods using simulated highly skewed count data and compare the results to a scalar-weighted synthesizer under the Exponential Mechanism (EM). We also apply our methods to a sample of the Survey of Doctorate Recipients and demonstrate the practicality of our methods.
When releasing record-level data containing sensitive information to the public, the data disseminator is responsible for protecting the privacy of every record in the dataset, simultaneously preserving important features of the data for users' analyses. These goals can be achieved by data synthesis, where confidential data are replaced with synthetic data that are simulated based on statistical models estimated on the confidential data. In this paper, we present a data synthesis case study, where synthetic values of price and the number of available days in a sample of the New York Airbnb Open Data are created for privacy protection. One sensitive variable, the number of available days of an Airbnb listing, has a large amount of zero-valued records and also truncated at the two ends. We propose a zero-inflated truncated Poisson regression model for its synthesis. We utilize a sequential synthesis approach to further synthesize the sensitive price variable. The resulting synthetic data are evaluated for its utility preservation and privacy protection, the latter in the form of disclosure risks. Furthermore, we propose methods to investigate how uncertainties in intruder's knowledge would influence the identification disclosure risks of the synthetic data. In particular, we explore several realistic scenarios of uncertainties in intruder's knowledge of available information and evaluate their impacts on the resulting identification disclosure risks.
We propose two synthetic microdata approaches to generate private tabular survey data products for public release. We adapt a pseudo posterior mechanism that downweights by-record likelihood contributions with weights $\in [0,1]$ based on their identification disclosure risks to producing tabular products for survey data. Our method applied to an observed survey database achieves an asymptotic global probabilistic differential privacy guarantee. Our two approaches synthesize the observed sample distribution of the outcome and survey weights, jointly, such that both quantities together possess a privacy guarantee. The privacy-protected outcome and survey weights are used to construct tabular cell estimates (where the cell inclusion indicators are treated as known and public) and associated standard errors to correct for survey sampling bias. Through a real data application to the Survey of Doctorate Recipients public use file and simulation studies motivated by the application, we demonstrate that our two microdata synthesis approaches to construct tabular products provide superior utility preservation as compared to the additive-noise approach of the Laplace Mechanism. Moreover, our approaches allow the release of microdata to the public, enabling additional analyses at no extra privacy cost.
Recent research in differential privacy demonstrated that (sub)sampling can amplify the level of protection. For example, for $ε$-differential privacy and simple random sampling with sampling rate $r$, the actual privacy guarantee is approximately $rε$, if a value of $ε$ is used to protect the output from the sample. In this paper, we study whether this amplification effect can be exploited systematically to improve the accuracy of the privatized estimate. Specifically, assuming the agency has information for the full population, we ask under which circumstances accuracy gains could be expected, if the privatized estimate would be computed on a random sample instead of the full population. We find that accuracy gains can be achieved for certain regimes. However, gains can typically only be expected, if the sensitivity of the output with respect to small changes in the database does not depend too strongly on the size of the database. We only focus on algorithms that achieve differential privacy by adding noise to the final output and illustrate the accuracy implications for two commonly used statistics: the mean and the median. We see our research as a first step towards understanding the conditions required for accuracy gains in practice and we hope that these findings will stimulate further research broadening the scope of differential privacy algorithms and outputs considered.
In many contexts, missing data and disclosure control are ubiquitous and challenging issues. In particular at statistical agencies, the respondent-level data they collect from surveys and censuses can suffer from high rates of missingness. Furthermore, agencies are obliged to protect respondents' privacy when publishing the collected data for public use. The NPBayesImputeCat R package, introduced in this paper, provides routines to i) create multiple imputations for missing data, and ii) create synthetic data for statistical disclosure control, for multivariate categorical data, with or without structural zeros. We describe the Dirichlet process mixture of products of multinomial distributions model used in the package, and illustrate various uses of the package using data samples from the American Community Survey (ACS). We also compare results of the missing data imputation to the mice R package and those of the synthetic data generation to the synthpop R package.