Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.
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Finding the conceptual difference between the two images in an industrial environment has been especially important for HSE purposes and there is still no reliable and conformable method to find the major differences to alert the related controllers. Due to the abundance and variety of objects in different environments, the use of supervised learning methods in this field is facing a major problem. Due to the sharp and even slight change in lighting conditions in the two scenes, it is not possible to naively subtract the two images in order to find these differences. The goal of this paper is to find and localize the conceptual differences of two frames of one scene but in two different times and classify the differences to addition, reduction and change in the field. In this paper, we demonstrate a comprehensive solution for this application by presenting the deep learning method and using transfer learning and structural modification of the error function, as well as a process for adding and synthesizing data. An appropriate data set was provided and labeled, and the model results were evaluated on this data set and the possibility of using it in real and industrial applications was explained.
To meet the fairly high safety and reliability requirements in practice, the state of health (SOH) estimation of Lithium-ion batteries (LIBs), which has a close relationship with the degradation performance, has been extensively studied with the widespread applications of various electronics. The conventional SOH estimation approaches with digital twin are end-of-cycle estimation that require the completion of a full charge/discharge cycle to observe the maximum available capacity. However, under dynamic operating conditions with partially discharged data, it is impossible to sense accurate real-time SOH estimation for LIBs. To bridge this research gap, we put forward a digital twin framework to gain the capability of sensing the battery's SOH on the fly, updating the physical battery model. The proposed digital twin solution consists of three core components to enable real-time SOH estimation without requiring a complete discharge. First, to handle the variable training cycling data, the energy discrepancy-aware cycling synchronization is proposed to align cycling data with guaranteeing the same data structure. Second, to explore the temporal importance of different training sampling times, a time-attention SOH estimation model is developed with data encoding to capture the degradation behavior over cycles, excluding adverse influences of unimportant samples. Finally, for online implementation, a similarity analysis-based data reconstruction has been put forward to provide real-time SOH estimation without requiring a full discharge cycle. Through a series of results conducted on a widely used benchmark, the proposed method yields the real-time SOH estimation with errors less than 1% for most sampling times in ongoing cycles.
In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most $k$ distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most $k$ thresholds. We present asymptotic results in $k$ and results for small values of $k$. For $k=2$ prices, we show an improvement of at least $11\%$ over the best fixed-price solution. Moreover, $k=5$ prices suffice to guarantee almost $99\%$ of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems.
Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.
Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix $R$ of the sensed data from the sample covariance matrix $\widehat{R}$. Existing approaches, such as that based on information theoretic criteria, rely on the closeness of the noise eigenvalues of $\widehat R$ to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The analysis presented in this report focuses on the splitting of the spectrum of $\widehat{R}$ into noise and signal eigenvalues. It is shown that, when the number of sensors is large, the number of signals can be estimated with a sample size considerably less than that required by previous approaches. The practical significance of the main result is that detection can be achieved with a number of samples comparable to the number of sensors in large dimensional array processing.
Differentially private mechanisms protect privacy by introducing additional randomness into the data. Restricting access to only the privatized data makes it challenging to perform valid statistical inference on parameters underlying the confidential data. Specifically, the likelihood function of the privatized data requires integrating over the large space of confidential databases and is typically intractable. For Bayesian analysis, this results in a posterior distribution that is doubly intractable, rendering traditional MCMC techniques inapplicable. We propose an MCMC framework to perform Bayesian inference from the privatized data, which is applicable to a wide range of statistical models and privacy mechanisms. Our MCMC algorithm augments the model parameters with the unobserved confidential data, and alternately updates each one conditional on the other. For the potentially challenging step of updating the confidential data, we propose a generic approach that exploits the privacy guarantee of the mechanism to ensure efficiency. We give results on the computational complexity, acceptance rate, and mixing properties of our MCMC. We illustrate the efficacy and applicability of our methods on a na\"ive-Bayes log-linear model as well as on a linear regression model.
The need for data privacy and security -- enforced through increasingly strict data protection regulations -- renders the use of healthcare data for machine learning difficult. In particular, the transfer of data between different hospitals is often not permissible and thus cross-site pooling of data not an option. The Personal Health Train (PHT) paradigm proposed within the GO-FAIR initiative implements an 'algorithm to the data' paradigm that ensures that distributed data can be accessed for analysis without transferring any sensitive data. We present PHT-meDIC, a productively deployed open-source implementation of the PHT concept. Containerization allows us to easily deploy even complex data analysis pipelines (e.g, genomics, image analysis) across multiple sites in a secure and scalable manner. We discuss the underlying technological concepts, security models, and governance processes. The implementation has been successfully applied to distributed analyses of large-scale data, including applications of deep neural networks to medical image data.
We develop a data-driven optimal shrinkage algorithm for matrix denoising in the presence of high-dimensional noise with separable covariance structure; that is, the nose is colored and dependent. The algorithm, coined extended OptShrink (eOptShrink), involves a new imputation and rank estimation and we do not need to estimate the separable covariance structure of the noise. On the theoretical side, we study the asymptotic behavior of singular values and singular vectors of the random matrix associated with the noisy data, including the sticking property of non-outlier singular values and delocalization of the non-outlier singular vectors with a convergence rate. We apply these results to establish the guarantee of the imputation, rank estimation and eOptShrink algorithm with a convergence rate. On the application side, in addition to a series of numerical simulations with a comparison with various state-of-the-art optimal shrinkage algorithms, we apply eOptShrink to extract fetal electrocardiogram from the single channel trans-abdominal maternal electrocardiogram.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
For deploying a deep learning model into production, it needs to be both accurate and compact to meet the latency and memory constraints. This usually results in a network that is deep (to ensure performance) and yet thin (to improve computational efficiency). In this paper, we propose an efficient method to train a deep thin network with a theoretic guarantee. Our method is motivated by model compression. It consists of three stages. In the first stage, we sufficiently widen the deep thin network and train it until convergence. In the second stage, we use this well-trained deep wide network to warm up (or initialize) the original deep thin network. This is achieved by letting the thin network imitate the immediate outputs of the wide network from layer to layer. In the last stage, we further fine tune this well initialized deep thin network. The theoretical guarantee is established by using mean field analysis, which shows the advantage of layerwise imitation over traditional training deep thin networks from scratch by backpropagation. We also conduct large-scale empirical experiments to validate our approach. By training with our method, ResNet50 can outperform ResNet101, and BERT_BASE can be comparable with BERT_LARGE, where both the latter models are trained via the standard training procedures as in the literature.
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