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We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. Moreover, we consider characterizations in terms of Fourier series, singular value decompositions and frame decompositions, as well as representations in finite dimensional settings. While many of these results are already known to researchers from different fields, a detailed and general overview or reference work containing rigorous mathematical proofs is still missing. Hence, in this paper we aim to fill this gap by collecting, introducing and generalizing a large number of characterizations of $E_s^*$ and discuss their use in regularization methods for solving inverse problems. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.

In multivariate time series analysis, the coherence measures the linear dependency between two-time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. Although quantile coherence is a more powerful tool, its estimation remains challenging due to the high level of noise. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of the vector autoregressive (VAR) model as an approximation to the quantile spectral matrix, along with nonparametric smoothing across quantiles. For each fixed quantile level, we obtain the VAR parameters from the quantile periodograms, then, using the Durbin-Levinson algorithm, we calculate the preliminary estimate of quantile coherence using the VAR parameters. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more accurate structure of diversified investment portfolios that may be used by investors to make better decisions.

Evaluating failure probability for complex engineering systems is a computationally intensive task. While the Monte Carlo method is easy to implement, it converges slowly and, hence, requires numerous repeated simulations of a complex system to generate sufficient samples. To improve the efficiency, methods based on surrogate models are proposed to approximate the limit state function. In this work, we reframe the approximation of the limit state function as an operator learning problem and utilize the DeepONet framework with a hybrid approach to estimate the failure probability. The numerical results show that our proposed method outperforms the prior neural hybrid method.

Neural implicit surface representations have recently emerged as popular alternative to explicit 3D object encodings, such as polygonal meshes, tabulated points, or voxels. While significant work has improved the geometric fidelity of these representations, much less attention is given to their final appearance. Traditional explicit object representations commonly couple the 3D shape data with auxiliary surface-mapped image data, such as diffuse color textures and fine-scale geometric details in normal maps that typically require a mapping of the 3D surface onto a plane, i.e., a surface parameterization; implicit representations, on the other hand, cannot be easily textured due to lack of configurable surface parameterization. Inspired by this digital content authoring methodology, we design a neural network architecture that implicitly encodes the underlying surface parameterization suitable for appearance data. As such, our model remains compatible with existing mesh-based digital content with appearance data. Motivated by recent work that overfits compact networks to individual 3D objects, we present a new weight-encoded neural implicit representation that extends the capability of neural implicit surfaces to enable various common and important applications of texture mapping. Our method outperforms reasonable baselines and state-of-the-art alternatives.

We consider the problem of estimating a scalar target parameter in the presence of nuisance parameters. Replacing the unknown nuisance parameter with a nonparametric estimator, e.g.,a machine learning (ML) model, is convenient but has shown to be inefficient due to large biases. Modern methods, such as the targeted minimum loss-based estimation (TMLE) and double machine learning (DML), achieve optimal performance under flexible assumptions by harnessing ML estimates while mitigating the plug-in bias. To avoid a sub-optimal bias-variance trade-off, these methods perform a debiasing step of the plug-in pre-estimate. Existing debiasing methods require the influence function of the target parameter as input. However, deriving the IF requires specialized expertise and thus obstructs the adaptation of these methods by practitioners. We propose a novel way to debias plug-in estimators which (i) is efficient, (ii) does not require the IF to be implemented, (iii) is computationally tractable, and therefore can be readily adapted to new estimation problems and automated without analytic derivations by the user. We build on the TMLE framework and update a plug-in estimate with a regularized likelihood maximization step over a nonparametric model constructed with a reproducing kernel Hilbert space (RKHS), producing an efficient plug-in estimate for any regular target parameter. Our method, thus, offers the efficiency of competing debiasing techniques without sacrificing the utility of the plug-in approach.

Full waveform inversion (FWI) updates the subsurface model from an initial model by comparing observed and synthetic seismograms. Due to high nonlinearity, FWI is easy to be trapped into local minima. Extended domain FWI, including wavefield reconstruction inversion (WRI) and extended source waveform inversion (ESI) are attractive options to mitigate this issue. This paper makes an in-depth analysis for FWI in the extended domain, identifying key challenges and searching for potential remedies torwards practical applications. WRI and ESI are formulated within the same mathematical framework using Lagrangian-based adjoint-state method with a special focus on time-domain formulation using extended sources, while putting connections between classical FWI, WRI and ESI: both WRI and ESI can be viewed as weighted versions of classic FWI. Due to symmetric positive definite Hessian, the conjugate gradient is explored to efficiently solve the normal equation in a matrix free manner, while both time and frequency domain wave equation solvers are feasible. This study finds that the most significant challenge comes from the huge storage demand to store time-domain wavefields through iterations. To resolve this challenge, two possible workaround strategies can be considered, i.e., by extracting sparse frequencial wavefields or by considering time-domain data instead of wavefields for reducing such challenge. We suggest that these options should be explored more intensively for tractable workflows.

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell's equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with $Q_k$-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy. The approaches presented prove to be robust and allow the modeling of optical problems and the treatment of complex nonlinearities as well as geometries of various physical systems coupled with electromagnetic fields.

Latent variable models are powerful tools for modeling complex phenomena involving in particular partially observed data, unobserved variables or underlying complex unknown structures. Inference is often difficult due to the latent structure of the model. To deal with parameter estimation in the presence of latent variables, well-known efficient methods exist, such as gradient-based and EM-type algorithms, but with practical and theoretical limitations. In this paper, we propose as an alternative for parameter estimation an efficient preconditioned stochastic gradient algorithm. Our method includes a preconditioning step based on a positive definite Fisher information matrix estimate. We prove convergence results for the proposed algorithm under mild assumptions for very general latent variables models. We illustrate through relevant simulations the performance of the proposed methodology in a nonlinear mixed effects model and in a stochastic block model.

Despite the advancement of machine learning techniques in recent years, state-of-the-art systems lack robustness to "real world" events, where the input distributions and tasks encountered by the deployed systems will not be limited to the original training context, and systems will instead need to adapt to novel distributions and tasks while deployed. This critical gap may be addressed through the development of "Lifelong Learning" systems that are capable of 1) Continuous Learning, 2) Transfer and Adaptation, and 3) Scalability. Unfortunately, efforts to improve these capabilities are typically treated as distinct areas of research that are assessed independently, without regard to the impact of each separate capability on other aspects of the system. We instead propose a holistic approach, using a suite of metrics and an evaluation framework to assess Lifelong Learning in a principled way that is agnostic to specific domains or system techniques. Through five case studies, we show that this suite of metrics can inform the development of varied and complex Lifelong Learning systems. We highlight how the proposed suite of metrics quantifies performance trade-offs present during Lifelong Learning system development - both the widely discussed Stability-Plasticity dilemma and the newly proposed relationship between Sample Efficient and Robust Learning. Further, we make recommendations for the formulation and use of metrics to guide the continuing development of Lifelong Learning systems and assess their progress in the future.

This work addresses a novel and challenging problem of estimating the full 3D hand shape and pose from a single RGB image. Most current methods in 3D hand analysis from monocular RGB images only focus on estimating the 3D locations of hand keypoints, which cannot fully express the 3D shape of hand. In contrast, we propose a Graph Convolutional Neural Network (Graph CNN) based method to reconstruct a full 3D mesh of hand surface that contains richer information of both 3D hand shape and pose. To train networks with full supervision, we create a large-scale synthetic dataset containing both ground truth 3D meshes and 3D poses. When fine-tuning the networks on real-world datasets without 3D ground truth, we propose a weakly-supervised approach by leveraging the depth map as a weak supervision in training. Through extensive evaluations on our proposed new datasets and two public datasets, we show that our proposed method can produce accurate and reasonable 3D hand mesh, and can achieve superior 3D hand pose estimation accuracy when compared with state-of-the-art methods.