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The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyse a chaotic partial differential equation, the Kuramoto-Sivashinsky (KS), and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data.

In the context of finite sums minimization, variance reduction techniques are widely used to improve the performance of state-of-the-art stochastic gradient methods. Their practical impact is clear, as well as their theoretical properties. Stochastic proximal point algorithms have been studied as an alternative to stochastic gradient algorithms since they are more stable with respect to the choice of the stepsize but a proper variance reduced version is missing. In this work, we propose the first study of variance reduction techniques for stochastic proximal point algorithms. We introduce a stochastic proximal version of SVRG, SAGA, and some of their variants for smooth and convex functions. We provide several convergence results for the iterates and the objective function values. In addition, under the Polyak-{\L}ojasiewicz (PL) condition, we obtain linear convergence rates for the iterates and the function values. Our numerical experiments demonstrate the advantages of the proximal variance reduction methods over their gradient counterparts, especially about the stability with respect to the choice of the step size.

Optimal transport has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. Recently, Bredies et al. (ESAIM:M2AN 54:2351-2382, 2020) and Schmitzer et al. (IEEE T MED IMAGING 39:1626-1635, 2019) established the framework of optimal transport regularization for dynamic inverse problems. In this paper, we incorporate Wasserstein distance, together with total variation, into static inverse problems as a prior regularization. The Wasserstein distance formulated by Benamou-Brenier energy measures the similarity between the given template and the reconstructed image. Also, we analyze the existence of solutions of such variational problem in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a specific grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation.

Time-dependent basis reduced order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values, and (iv) error accumulation due to fixed rank. To this end, we present a scalable method based on oblique projections for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive, and highly parallelizable. These favorable properties are achieved via low-rank approximation of the time discrete MDE. Using the discrete empirical interpolation method (DEIM), a low-rank decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. We coin the new approach TDB-CUR since it is equivalent to a CUR decomposition based on sparse row and column samples of the MDE. We also propose a rank-adaptive procedure to control the error on-the-fly. Numerical results demonstrate the accuracy, efficiency, and robustness of the new method for a diverse set of problems.

A population-averaged additive subdistribution hazards model is proposed to assess the marginal effects of covariates on the cumulative incidence function and to analyze correlated failure time data subject to competing risks. This approach extends the population-averaged additive hazards model by accommodating potentially dependent censoring due to competing events other than the event of interest. Assuming an independent working correlation structure, an estimating equations approach is outlined to estimate the regression coefficients and a new sandwich variance estimator is proposed. The proposed sandwich variance estimator accounts for both the correlations between failure times and between the censoring times, and is robust to misspecification of the unknown dependency structure within each cluster. We further develop goodness-of-fit tests to assess the adequacy of the additive structure of the subdistribution hazards for the overall model and each covariate. Simulation studies are conducted to investigate the performance of the proposed methods in finite samples. We illustrate our methods using data from the STrategies to Reduce Injuries and Develop confidence in Elders (STRIDE) trial.

Several methods in survival analysis are based on the proportional hazards assumption. However, this assumption is very restrictive and often not justifiable in practice. Therefore, effect estimands that do not rely on the proportional hazards assumption are highly desirable in practical applications. One popular example for this is the restricted mean survival time (RMST). It is defined as the area under the survival curve up to a prespecified time point and, thus, summarizes the survival curve into a meaningful estimand. For two-sample comparisons based on the RMST, previous research found the inflation of the type I error of the asymptotic test for small samples and, therefore, a two-sample permutation test has already been developed. The first goal of the present paper is to further extend the permutation test for general factorial designs and general contrast hypotheses by considering a Wald-type test statistic and its asymptotic behavior. Additionally, a groupwise bootstrap approach is considered. Moreover, when a global test detects a significant difference by comparing the RMSTs of more than two groups, it is of interest which specific RMST differences cause the result. However, global tests do not provide this information. Therefore, multiple tests for the RMST are developed in a second step to infer several null hypotheses simultaneously. Hereby, the asymptotically exact dependence structure between the local test statistics is incorporated to gain more power. Finally, the small sample performance of the proposed global and multiple testing procedures is analyzed in simulations and illustrated in a real data example.

We propose an end-to-end Automatic Speech Recognition (ASR) system that can be trained on transcribed speech data, text-only data, or a mixture of both. The proposed model uses an integrated auxiliary block for text-based training. This block combines a non-autoregressive multi-speaker text-to-mel-spectrogram generator with a GAN-based enhancer to improve the spectrogram quality. The proposed system can generate a mel-spectrogram dynamically during training. It can be used to adapt the ASR model to a new domain by using text-only data from this domain. We demonstrate that the proposed training method significantly improves ASR accuracy compared to the system trained on transcribed speech only. It also surpasses cascade TTS systems with the vocoder in the adaptation quality and training speed.

Long-span bridges are subjected to a multitude of dynamic excitations during their lifespan. To account for their effects on the structural system, several load models are used during design to simulate the conditions the structure is likely to experience. These models are based on different simplifying assumptions and are generally guided by parameters that are stochastically identified from measurement data, making their outputs inherently uncertain. This paper presents a probabilistic physics-informed machine-learning framework based on Gaussian process regression for reconstructing dynamic forces based on measured deflections, velocities, or accelerations. The model can work with incomplete and contaminated data and offers a natural regularization approach to account for noise in the measurement system. An application of the developed framework is given by an aerodynamic analysis of the Great Belt East Bridge. The aerodynamic response is calculated numerically based on the quasi-steady model, and the underlying forces are reconstructed using sparse and noisy measurements. Results indicate a good agreement between the applied and the predicted dynamic load and can be extended to calculate global responses and the resulting internal forces. Uses of the developed framework include validation of design models and assumptions, as well as prognosis of responses to assist in damage detection and structural health monitoring.

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under very weak assumptions, and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals. To elaborate, our methods take the form of confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time. CSs provide valid inference at arbitrary stopping times, incurring no penalties for "peeking" at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, and hence do not enjoy the aforementioned broad applicability of asymptotic confidence intervals. Our work bridges the gap by giving a definition for "asymptotic CSs", and deriving a universal asymptotic CS that requires only weak CLT-like assumptions. While the CLT approximates the distribution of a sample average by that of a Gaussian at a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen and improvements by Koml\'os, Major, and Tusn\'ady) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration of our theory, we derive asymptotic CSs for the average treatment effect using efficient estimators in observational studies (for which no nonasymptotic bounds can exist even in the fixed-time regime) as well as randomized experiments, enabling causal inference that can be continuously monitored and adaptively stopped.

In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.