Generalized approximate message passing (GAMP) is a computationally efficient algorithm for estimating an unknown signal $w_0\in\mathbb{R}^N$ from a random linear measurement $y= Xw_0 + \epsilon\in\mathbb{R}^M$, where $X\in\mathbb{R}^{M\times N}$ is a known measurement matrix and $\epsilon$ is the noise vector. The salient feature of GAMP is that it can provide an unbiased estimator $\hat{r}^{\rm G}\sim\mathcal{N}(w_0, \hat{s}^2I_N)$, which can be used for various hypothesis-testing methods. In this study, we consider the bootstrap average of an unbiased estimator of GAMP for the elastic net. By numerically analyzing the state evolution of \emph{approximate message passing with resampling}, which has been proposed for computing bootstrap statistics of the elastic net estimator, we investigate when the bootstrap averaging reduces the variance of the unbiased estimator and the effect of optimizing the size of each bootstrap sample and hyperparameter of the elastic net regularization in the asymptotic setting $M, N\to\infty, M/N\to\alpha\in(0,\infty)$. The results indicate that bootstrap averaging effectively reduces the variance of the unbiased estimator when the actual data generation process is inconsistent with the sparsity assumption of the regularization and the sample size is small. Furthermore, we find that when $w_0$ is less sparse, and the data size is small, the system undergoes a phase transition. The phase transition indicates the existence of the region where the ensemble average of unbiased estimators of GAMP for the elastic net norm minimization problem yields the unbiased estimator with the minimum variance.
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We consider the optimization of a smooth and strongly convex objective using constant step-size stochastic gradient descent (SGD) and study its properties through the prism of Markov chains. We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. We also establish this convergence in Wasserstein-2 distance in a more general setting compared to previous work. Thanks to the invariance property of the limit distribution, our analysis shows that the latter inherits sub-Gaussian or sub-exponential concentration properties when these hold true for the gradient. This allows the derivation of high-confidence bounds for the final estimate. Finally, under such conditions in the linear case, we obtain a dimension-free deviation bound for the Polyak-Ruppert average of a tail sequence. All our results are non-asymptotic and their consequences are discussed through a few applications.
In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.
We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator as long as $d = o(n^{2/3})$, with an explicit bound on the rate of convergence to normality. We leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.
We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.
In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.
Cross-validation is a widely used technique for evaluating the performance of prediction models. It helps avoid the optimism bias in error estimates, which can be significant for models built using complex statistical learning algorithms. However, since the cross-validation estimate is a random value dependent on observed data, it is essential to accurately quantify the uncertainty associated with the estimate. This is especially important when comparing the performance of two models using cross-validation, as one must determine whether differences in error estimates are a result of chance fluctuations. Although various methods have been developed for making inferences on cross-validation estimates, they often have many limitations, such as stringent model assumptions This paper proposes a fast bootstrap method that quickly estimates the standard error of the cross-validation estimate and produces valid confidence intervals for a population parameter measuring average model performance. Our method overcomes the computational challenge inherent in bootstrapping the cross-validation estimate by estimating the variance component within a random effects model. It is just as flexible as the cross-validation procedure itself. To showcase the effectiveness of our approach, we employ comprehensive simulations and real data analysis across three diverse applications.
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.
Single-use anion-exchange resins can reduce hazardous chromates to safe levels in drinking water. However, since most process control strategies monitor effluent concentrations, detection of any chromate leakage leads to premature resin replacement. Furthermore, variations in the inlet chromate concentration and other process conditions make process control a challenging step. In this work, we capture the uncertainty of the process conditions by applying the Ito process of Brownian motion with drift into a stochastic optimal control strategy. The ion exchange process is modeled using the method of moments which helps capture the process dynamics, later formulated into mathematical objectives representing desired chromate removal. We then solved our developed models as an optimal control problem via Pontryagin's maximum principle. The objectives enabled a successful control via flow rate adjustments leading to higher chromate extraction. Such an approach maximized the capacity of the resin and column efficiency to remove toxic compounds from water while capturing deviations in the process conditions.
We study sampling problems associated with potentials that lack smoothness. The potentials can be either convex or non-convex. Departing from the standard smooth setting, the potentials are only assumed to be weakly smooth or non-smooth, or the summation of multiple such functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling for both non-convex and convex potentials that are not necessarily smooth. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
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