Our research delves into the balance between maintaining privacy and preserving statistical accuracy when dealing with multivariate data that is subject to \textit{componentwise local differential privacy} (CLDP). With CLDP, each component of the private data is made public through a separate privacy channel. This allows for varying levels of privacy protection for different components or for the privatization of each component by different entities, each with their own distinct privacy policies. We develop general techniques for establishing minimax bounds that shed light on the statistical cost of privacy in this context, as a function of the privacy levels $\alpha_1, ... , \alpha_d$ of the $d$ components. We demonstrate the versatility and efficiency of these techniques by presenting various statistical applications. Specifically, we examine nonparametric density and covariance estimation under CLDP, providing upper and lower bounds that match up to constant factors, as well as an associated data-driven adaptive procedure. Furthermore, we quantify the probability of extracting sensitive information from one component by exploiting the fact that, on another component which may be correlated with the first, a smaller degree of privacy protection is guaranteed.
相關內容
Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers which seem sufficiently accurate for the forward problem, i.e., for obtaining an accurate simulation, may not be sufficiently accurate for the inverse problem, i.e., for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which may become jagged, causing inference algorithms to get stuck in local "phantom" optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyze an ODE changepoint model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall-runoff model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.
Understanding superfluidity remains a major goal of condensed matter physics. Here we tackle this challenge utilizing the recently developed Fermionic neural network (FermiNet) wave function Ansatz for variational Monte Carlo calculations. We study the unitary Fermi gas, a system with strong, short-range, two-body interactions known to possess a superfluid ground state but difficult to describe quantitatively. We demonstrate key limitations of the FermiNet Ansatz in studying the unitary Fermi gas and propose a simple modification that outperforms the original FermiNet significantly, giving highly accurate results. We prove mathematically that the new Ansatz, which only differs from the original Ansatz by the method of antisymmetrization, is a strict generalization of the original FermiNet architecture, despite the use of fewer parameters. Our approach shares several advantages with the FermiNet: the use of a neural network removes the need for an underlying basis set; and the flexibility of the network yields extremely accurate results within a variational quantum Monte Carlo framework that provides access to unbiased estimates of arbitrary ground-state expectation values. We discuss how the method can be extended to study other superfluids.
A series of experiments in stationary and moving passenger rail cars were conducted to measure removal rates of particles in the size ranges of SARS-CoV-2 viral aerosols, and the air changes per hour provided by existing and modified air handling systems. Such methods for exposure assessments are customarily based on mechanistic models derived from physical laws of particle movement that are deterministic and do not account for measurement errors inherent in data collection. The resulting analysis compromises on reliably learning about mechanistic factors such as ventilation rates, aerosol generation rates and filtration efficiencies from field measurements. This manuscript develops a Bayesian state space modeling framework that synthesizes information from the mechanistic system as well as the field data. We derive a stochastic model from finite difference approximations of differential equations explaining particle concentrations. Our inferential framework trains the mechanistic system using the field measurements from the chamber experiments and delivers reliable estimates of the underlying physical process with fully model-based uncertainty quantification. Our application falls within the realm of Bayesian ``melding'' of mechanistic and statistical models and is of significant relevance to environmental hygienists and public health researchers working on assessing performance of aerosol removal rates for rail car fleets.
Multiple systems estimation is a standard approach to quantifying hidden populations where data sources are based on lists of known cases. A typical modelling approach is to fit a Poisson loglinear model to the numbers of cases observed in each possible combination of the lists. It is necessary to decide which interaction parameters to include in the model, and information criterion approaches are often used for model selection. Difficulties in the context of multiple systems estimation may arise due to sparse or nil counts based on the intersection of lists, and care must be taken when information criterion approaches are used for model selection due to issues relating to the existence of estimates and identifiability of the model. Confidence intervals are often reported conditional on the model selected, providing an over-optimistic impression of the accuracy of the estimation. A bootstrap approach is a natural way to account for the model selection procedure. However, because the model selection step has to be carried out for every bootstrap replication, there may be a high or even prohibitive computational burden. We explore the merit of modifying the model selection procedure in the bootstrap to look only among a subset of models, chosen on the basis of their information criterion score on the original data. This provides large computational gains with little apparent effect on inference. Another model selection approach considered and investigated is a downhill search approach among models, possibly with multiple starting points.
This paper focuses on optimal beamforming to maximize the mean signal-to-noise ratio (SNR) for a reconfigurable intelligent surface (RIS)-aided MISO downlink system under correlated Rician fading. The beamforming problem becomes non-convex because of the unit modulus constraint of passive RIS elements. To tackle this, we propose a semidefinite relaxation-based iterative algorithm for obtaining statistically optimal transmit beamforming vector and RIS-phase shift matrix. Further, we analyze the outage probability (OP) and ergodic capacity (EC) to measure the performance of the proposed beamforming scheme. Just like the existing works, the OP and EC evaluations rely on the numerical computation of the iterative algorithm, which does not clearly reveal the functional dependence of system performance on key parameters. Therefore, we derive closed-form expressions for the optimal beamforming vector and phase shift matrix along with their OP performance for special cases of the general setup. Our analysis reveals that the i.i.d. fading is more beneficial than the correlated case in the presence of LoS components. This fact is analytically established for the setting in which the LoS is blocked. Furthermore, we demonstrate that the maximum mean SNR improves linearly/quadratically with the number of RIS elements in the absence/presence of LoS component under i.i.d. fading.
Vision Through the Veil: Differential Privacy in Federated Learning for Medical Image Classification
The proliferation of deep learning applications in healthcare calls for data aggregation across various institutions, a practice often associated with significant privacy concerns. This concern intensifies in medical image analysis, where privacy-preserving mechanisms are paramount due to the data being sensitive in nature. Federated learning, which enables cooperative model training without direct data exchange, presents a promising solution. Nevertheless, the inherent vulnerabilities of federated learning necessitate further privacy safeguards. This study addresses this need by integrating differential privacy, a leading privacy-preserving technique, into a federated learning framework for medical image classification. We introduce a novel differentially private federated learning model and meticulously examine its impacts on privacy preservation and model performance. Our research confirms the existence of a trade-off between model accuracy and privacy settings. However, we demonstrate that strategic calibration of the privacy budget in differential privacy can uphold robust image classification performance while providing substantial privacy protection.
Robust feature selection is vital for creating reliable and interpretable Machine Learning (ML) models. When designing statistical prediction models in cases where domain knowledge is limited and underlying interactions are unknown, choosing the optimal set of features is often difficult. To mitigate this issue, we introduce a Multidata (M) causal feature selection approach that simultaneously processes an ensemble of time series datasets and produces a single set of causal drivers. This approach uses the causal discovery algorithms PC1 or PCMCI that are implemented in the Tigramite Python package. These algorithms utilize conditional independence tests to infer parts of the causal graph. Our causal feature selection approach filters out causally-spurious links before passing the remaining causal features as inputs to ML models (Multiple linear regression, Random Forest) that predict the targets. We apply our framework to the statistical intensity prediction of Western Pacific Tropical Cyclones (TC), for which it is often difficult to accurately choose drivers and their dimensionality reduction (time lags, vertical levels, and area-averaging). Using more stringent significance thresholds in the conditional independence tests helps eliminate spurious causal relationships, thus helping the ML model generalize better to unseen TC cases. M-PC1 with a reduced number of features outperforms M-PCMCI, non-causal ML, and other feature selection methods (lagged correlation, random), even slightly outperforming feature selection based on eXplainable Artificial Intelligence. The optimal causal drivers obtained from our causal feature selection help improve our understanding of underlying relationships and suggest new potential drivers of TC intensification.
The Gaussian graphical model (GGM) incorporates an undirected graph to represent the conditional dependence between variables, with the precision matrix encoding partial correlation between pair of variables given the others. To achieve flexible and accurate estimation and inference of GGM, we propose the novel method FLAG, which utilizes the random effects model for pairwise conditional regression to estimate the precision matrix and applies statistical tests to recover the graph. Compared with existing methods, FLAG has several unique advantages: (i) it provides accurate estimation without sparsity assumptions on the precision matrix, (ii) it allows for element-wise inference of the precision matrix, (iii) it achieves computational efficiency by developing an efficient PX-EM algorithm and a MM algorithm accelerated with low-rank updates, and (iv) it enables joint estimation of multiple graphs using FLAG-Meta or FLAG-CA. The proposed methods are evaluated using various simulation settings and real data applications, including gene expression in the human brain, term association in university websites, and stock prices in the U.S. financial market. The results demonstrate that FLAG and its extensions provide accurate precision estimation and graph recovery.
Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients
To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.
The rate-distortion curve captures the fundamental tradeoff between compression length and resolution in lossy data compression. However, it conceals the underlying dynamics of optimal source encodings or test channels. We argue that these typically follow a piecewise smooth trajectory as the source information is compressed. These smooth dynamics are interrupted at bifurcations, where solutions change qualitatively. Sub-optimal test channels may collide or exchange optimality there, for example. There is typically a plethora of sub-optimal solutions, which stems from restrictions of the reproduction alphabet. We devise a family of algorithms that exploits the underlying dynamics to track a given test channel along the rate-distortion curve. To that end, we express implicit derivatives at the roots of a non-linear operator by higher derivative tensors. Providing closed-form formulae for the derivative tensors of Blahut's algorithm thus yields implicit derivatives of arbitrary order at a given test channel, thereby approximating others in its vicinity. Finally, our understanding of bifurcations guarantees the optimality of the root being traced, under mild assumptions, while allowing us to detect when our assumptions fail. Beyond the interest in rate distortion, this is an example of how understanding a problem's bifurcations can be translated to a numerical algorithm.
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