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Datasets with sheer volume have been generated from fields including computer vision, medical imageology, and astronomy whose large-scale and high-dimensional properties hamper the implementation of classical statistical models. To tackle the computational challenges, one of the efficient approaches is subsampling which draws subsamples from the original large datasets according to a carefully-design task-specific probability distribution to form an informative sketch. The computation cost is reduced by applying the original algorithm to the substantially smaller sketch. Previous studies associated with subsampling focused on non-regularized regression from the computational efficiency and theoretical guarantee perspectives, such as ordinary least square regression and logistic regression. In this article, we introduce a randomized algorithm under the subsampling scheme for the Elastic-net regression which gives novel insights into L1-norm regularized regression problem. To effectively conduct consistency analysis, a smooth approximation technique based on alpha absolute function is firstly employed and theoretically verified. The concentration bounds and asymptotic normality for the proposed randomized algorithm are then established under mild conditions. Moreover, an optimal subsampling probability is constructed according to A-optimality. The effectiveness of the proposed algorithm is demonstrated upon synthetic and real data datasets.

We present an alternative approach to decompose non-negative tensors, called many-body approximation. Traditional decomposition methods assume low-rankness in the representation, resulting in difficulties in global optimization and target rank selection. We avoid these problems by energy-based modeling of tensors, where a tensor and its mode correspond to a probability distribution and a random variable, respectively. Our model can be globally optimized in terms of the KL divergence minimization by taking the interaction between variables, i.e. modes, into account that can be tuned more intuitively than ranks. Furthermore, we visualize interactions between modes as tensor networks and reveal a nontrivial relationship between many-body approximation and low-rank approximation. We demonstrate the effectiveness of our approach in tensor completion and approximation.

The randomized singular value decomposition (R-SVD) is a popular sketching-based algorithm for efficiently computing the partial SVD of a large matrix. When the matrix is low-rank, the R-SVD produces its partial SVD exactly; but when the rank is large, it only yields an approximation. Motivated by applications in data science and principal component analysis (PCA), we analyze the R-SVD under a low-rank signal plus noise measurement model; specifically, when its input is a spiked random matrix. The singular values produced by the R-SVD are shown to exhibit a BBP-like phase transition: when the SNR exceeds a certain detectability threshold, that depends on the dimension reduction factor, the largest singular value is an outlier; below the threshold, no outlier emerges from the bulk of singular values. We further compute asymptotic formulas for the overlap between the ground truth signal singular vectors and the approximations produced by the R-SVD. Dimensionality reduction has the adverse affect of amplifying the noise in a highly nonlinear manner. Our results demonstrate the statistical advantage -- in both signal detection and estimation -- of the R-SVD over more naive sketched PCA variants; the advantage is especially dramatic when the sketching dimension is small. Our analysis is asymptotically exact, and substantially more fine-grained than existing operator-norm error bounds for the R-SVD, which largely fail to give meaningful error estimates in the moderate SNR regime. It applies for a broad family of sketching matrices previously considered in the literature, including Gaussian i.i.d. sketches, random projections, and the sub-sampled Hadamard transform, among others. Lastly, we derive an optimal singular value shrinker for singular values and vectors obtained through the R-SVD, which may be useful for applications in matrix denoising.

This paper provides a comprehensive error analysis of learning with vector-valued random features (RF). The theory is developed for RF ridge regression in a fully general infinite-dimensional input-output setting, but nonetheless applies to and improves existing finite-dimensional analyses. In contrast to comparable work in the literature, the approach proposed here relies on a direct analysis of the underlying risk functional and completely avoids the explicit RF ridge regression solution formula in terms of random matrices. This removes the need for concentration results in random matrix theory or their generalizations to random operators. The main results established in this paper include strong consistency of vector-valued RF estimators under model misspecification and minimax optimal convergence rates in the well-specified setting. The parameter complexity (number of random features) and sample complexity (number of labeled data) required to achieve such rates are comparable with Monte Carlo intuition and free from logarithmic factors.

Echo State Networks (ESN) are a type of Recurrent Neural Network that yields promising results in representing time series and nonlinear dynamic systems. Although they are equipped with a very efficient training procedure, Reservoir Computing strategies, such as the ESN, require high-order networks, i.e., many neurons, resulting in a large number of states that are magnitudes higher than the number of model inputs and outputs. A large number of states not only makes the time-step computation more costly but also may pose robustness issues, especially when applying ESNs to problems such as Model Predictive Control (MPC) and other optimal control problems. One way to circumvent this complexity issue is through Model Order Reduction strategies such as the Proper Orthogonal Decomposition (POD) and its variants (POD-DEIM), whereby we find an equivalent lower order representation to an already trained high dimension ESN. To this end, this work aims to investigate and analyze the performance of POD methods in Echo State Networks, evaluating their effectiveness through the Memory Capacity (MC) of the POD-reduced network compared to the original (full-order) ESN. We also perform experiments on two numerical case studies: a NARMA10 difference equation and an oil platform containing two wells and one riser. The results show that there is little loss of performance comparing the original ESN to a POD-reduced counterpart and that the performance of a POD-reduced ESN tends to be superior to a normal ESN of the same size. Also, the POD-reduced network achieves speedups of around $80\%$ compared to the original ESN.

The limitations of turbulence closure models in the context of Reynolds-averaged NavierStokes (RANS) simulations play a significant part in contributing to the uncertainty of Computational Fluid Dynamics (CFD). Perturbing the spectral representation of the Reynolds stress tensor within physical limits is common practice in several commercial and open-source CFD solvers, in order to obtain estimates for the epistemic uncertainties of RANS turbulence models. Recent research revealed, that there is a need for moderating the amount of perturbed Reynolds stress tensor tensor to be considered due to upcoming stability issues of the solver. In this paper we point out that the consequent common implementation can lead to unintended states of the resulting perturbed Reynolds stress tensor. The combination of eigenvector perturbation and moderation factor may actually result in moderated eigenvalues, which are not linearly dependent on the originally unperturbed and fully perturbed eigenvalues anymore. Hence, the computational implementation is no longer in accordance with the conceptual idea of the Eigenspace Perturbation Framework. We verify the implementation of the conceptual description with respect to its self-consistency. Adequately representing the basic concept results in formulating a computational implementation to improve self-consistency of the Reynolds stress tensor perturbation

Determining the sensitivity of the posterior to perturbations of the prior and likelihood is an important part of the Bayesian workflow. We introduce a practical and computationally efficient sensitivity analysis approach using importance sampling to estimate properties of posteriors resulting from power-scaling the prior or likelihood. On this basis, we suggest a diagnostic that can indicate the presence of prior-data conflict or likelihood noninformativity and discuss limitations to this power-scaling approach. The approach can be easily included in Bayesian workflows with minimal effort by the model builder and we present an implementation in our new R package priorsense. We further demonstrate the workflow on case studies of real data using models varying in complexity from simple linear models to Gaussian process models.

We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the $\mathcal{O}(1/\sqrt{\text{width}})$ fluctuations of the DMFT order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For CNNs trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.

We study the problem of decentralized power allocation in a multi-access channel (MAC) with non-cooperative users, additive noise of arbitrary distribution and a generalized power constraint, i.e., the transmit power constraint is modeled by an upper bound on $\mathbb{E}[\phi(|S|)]$, where $S$ is the transmit signal and $\phi(.)$ is some non-negative, increasing and bounded function. The generalized power constraint captures the notion of power for different wireless signals such as RF, optical, acoustic, etc. We derive the optimal power allocation policy when there a large number of non-cooperative users in the MAC. Further, we show that, once the number of users in the MAC crosses a finite threshold, the proposed power allocation policy of all users is optimal and remains invariant irrespective of the actual number of users. We derive the above results under the condition that the entropy power of the MAC, $e^{2h(S)+c}$, is strictly convex, where $h(S)$ is the maximum achievable entropy of the transmit signal and $c$ is a finite constant corresponding to the entropy of the additive noise.

We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.