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In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

Approximating a function with a finite series, e.g., involving polynomials or trigonometric functions, is a critical tool in computing and data analysis. The construction of such approximations via now-standard approaches like least squares or compressive sampling does not ensure that the approximation adheres to certain convex linear structural constraints, such as positivity or monotonicity. Existing approaches that ensure such structure are norm-dissipative and this can have a deleterious impact when applying these approaches, e.g., when numerical solving partial differential equations. We present a new framework that enforces via optimization such structure on approximations and is simultaneously norm-preserving. This results in a conceptually simple convex optimization problem on the sphere, but the feasible set for such problems can be very complex. We establish well-posedness of the optimization problem through results on spherical convexity and design several spherical-projection-based algorithms to numerically compute the solution. Finally, we demonstrate the effectiveness of this approach through several numerical examples.

In this contribution, we discuss the modeling and model reduction framework known as the Loewner framework. This is a data-driven approach, applicable to large-scale systems, which was originally developed for applications to linear time-invariant systems. In recent years, this method has been extended to a number of additional more complex scenarios, including linear parametric or nonlinear dynamical systems. We will provide here an overview of the latter two, together with time-domain extensions. Additionally, the application of the Loewner framework is illustrated by a collection of practical test cases. Firstly, for data-driven complexity reduction of the underlying model, and secondly, for dealing with control applications of complex systems (in particular, with feedback controller design).

Deep Neural Networks are actively being used in the design of autonomous Cyber-Physical Systems (CPSs). The advantage of these models is their ability to handle high-dimensional state-space and learn compact surrogate representations of the operational state spaces. However, the problem is that the sampled observations used for training the model may never cover the entire state space of the physical environment, and as a result, the system will likely operate in conditions that do not belong to the training distribution. These conditions that do not belong to training distribution are referred to as Out-of-Distribution (OOD). Detecting OOD conditions at runtime is critical for the safety of CPS. In addition, it is also desirable to identify the context or the feature(s) that are the source of OOD to select an appropriate control action to mitigate the consequences that may arise because of the OOD condition. In this paper, we study this problem as a multi-labeled time series OOD detection problem over images, where the OOD is defined both sequentially across short time windows (change points) as well as across the training data distribution. A common approach to solving this problem is the use of multi-chained one-class classifiers. However, this approach is expensive for CPSs that have limited computational resources and require short inference times. Our contribution is an approach to design and train a single $\beta$-Variational Autoencoder detector with a partially disentangled latent space sensitive to variations in image features. We use the feature sensitive latent variables in the latent space to detect OOD images and identify the most likely feature(s) responsible for the OOD. We demonstrate our approach using an Autonomous Vehicle in the CARLA simulator and a real-world automotive dataset called nuImages.

DFT is the numerical implementation of Fourier transform (FT), and it has many forms. Ordinary DFT (ODFT) and symmetric DFT (SDFT) are the two main forms of DFT. The most widely used DFT is ODFT, and the phase spectrum of this form is widely used in engineering applications. However, it is found ODFT has the problem of phase aliasing. Moreover, ODFT does not have many FT properties, such as symmetry, integration, and interpolation. When compared with ODFT, SDFT has more FT properties. Theoretically, the more properties a transformation has, the wider its application range. Hence, SDFT is more suitable as the discrete form of FT. In order to promote SDFT, the unique nature of SDFT is demonstrated. The time-domain of even-point SDFT is not symmetric to zero, and the author corrects it in this study. The author raises a new issue that should the signal length be odd or even when performing SDFT. The answer is odd. However, scientists and engineers are accustomed to using even-numbered sequences. At the end of this study, the reasons why the author advocates odd SDFT are given. Besides, even sampling function, discrete frequency Fourier transform, and the Gibbs phenomenon of the SDFT are introduced.

We show that density models describing multiple observables with (i) hard boundaries and (ii) dependence on external parameters may be created using an auto-regressive Gaussian mixture model. The model is designed to capture how observable spectra are deformed by hypothesis variations, and is made more expressive by projecting data onto a configurable latent space. It may be used as a statistical model for scientific discovery in interpreting experimental observations, for example when constraining the parameters of a physical model or tuning simulation parameters according to calibration data. The model may also be sampled for use within a Monte Carlo simulation chain, or used to estimate likelihood ratios for event classification. The method is demonstrated on simulated high-energy particle physics data considering the anomalous electroweak production of a $Z$ boson in association with a dijet system at the Large Hadron Collider, and the accuracy of inference is tested using a realistic toy example. The developed methods are domain agnostic; they may be used within any field to perform simulation or inference where a dataset consisting of many real-valued observables has conditional dependence on external parameters.

We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter of multiple Gaussian distributions. We prove that the barycenter is Gaussian distribution when the entropic regularization parameter is small. We further derive closed-form expressions for the covariance matrix of the barycenter.

Adversarial training is among the most effective techniques to improve the robustness of models against adversarial perturbations. However, the full effect of this approach on models is not well understood. For example, while adversarial training can reduce the adversarial risk (prediction error against an adversary), it sometimes increase standard risk (generalization error when there is no adversary). Even more, such behavior is impacted by various elements of the learning problem, including the size and quality of training data, specific forms of adversarial perturbations in the input, model overparameterization, and adversary's power, among others. In this paper, we focus on \emph{distribution perturbing} adversary framework wherein the adversary can change the test distribution within a neighborhood of the training data distribution. The neighborhood is defined via Wasserstein distance between distributions and the radius of the neighborhood is a measure of adversary's manipulative power. We study the tradeoff between standard risk and adversarial risk and derive the Pareto-optimal tradeoff, achievable over specific classes of models, in the infinite data limit with features dimension kept fixed. We consider three learning settings: 1) Regression with the class of linear models; 2) Binary classification under the Gaussian mixtures data model, with the class of linear classifiers; 3) Regression with the class of random features model (which can be equivalently represented as two-layer neural network with random first-layer weights). We show that a tradeoff between standard and adversarial risk is manifested in all three settings. We further characterize the Pareto-optimal tradeoff curves and discuss how a variety of factors, such as features correlation, adversary's power or the width of two-layer neural network would affect this tradeoff.

Neural waveform models such as the WaveNet are used in many recent text-to-speech systems, but the original WaveNet is quite slow in waveform generation because of its autoregressive (AR) structure. Although faster non-AR models were recently reported, they may be prohibitively complicated due to the use of a distilling training method and the blend of other disparate training criteria. This study proposes a non-AR neural source-filter waveform model that can be directly trained using spectrum-based training criteria and the stochastic gradient descent method. Given the input acoustic features, the proposed model first uses a source module to generate a sine-based excitation signal and then uses a filter module to transform the excitation signal into the output speech waveform. Our experiments demonstrated that the proposed model generated waveforms at least 100 times faster than the AR WaveNet and the quality of its synthetic speech is close to that of speech generated by the AR WaveNet. Ablation test results showed that both the sine-wave excitation signal and the spectrum-based training criteria were essential to the performance of the proposed model.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.