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We propose a spectral method based on the implementation of Chebyshev polynomials to study a model of conservation laws on network. We avoid the Gibbs phenomenon near shock discontinuities by implementing a filter in the frequency space in order to add local viscosity able to contrast the spurious oscillations appearing in the profile of the solution and we prove the convergence of the semi-discrete method by using the compensated compactness theorem. thanks to several simulation, we make a comparison between the implementation of the proposed method with a first order finite volume scheme.

This investigation is motivated by PDE-constrained optimization problems arising in connection with electrocardiograms (ECGs) and electroencephalography (EEG). Standard sparsity regularization does not necessarily produce adequate results for these applications because only boundary data/observations are available for the identification of the unknown source, which may be interior. We therefore study a weighted $\ell^1$-regularization technique for solving inverse problems when the forward operator has a significant null space. In particular, we prove that a sparse source, regardless of whether it is interior or located at the boundary, can be exactly recovered with this weighting procedure as the regularization parameter $\alpha$ tends to zero. Our analysis is supported by numerical experiments for cases with one and several local sources. The theory is developed in terms of Euclidean spaces, and our results can therefore be applied to many problems.

Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this algorithm are mainly derived for Lipschitz continuous objective functions. In this work, we first extend the typical complexity results for the subgradient method to convex and weakly convex minimization without assuming Lipschitz continuity. Specifically, we establish $\mathcal{O}(1/\sqrt{T})$ bound in terms of the suboptimality gap ``$f(x) - f^*$'' for convex case and $\mathcal{O}(1/{T}^{1/4})$ bound in terms of the gradient of the Moreau envelope function for weakly convex case. Furthermore, we provide convergence results for non-Lipschitz convex and weakly convex objective functions using proper diminishing rules on the step sizes. In particular, when $f$ is convex, we show $\mathcal{O}(\log(k)/\sqrt{k})$ rate of convergence in terms of the suboptimality gap. With an additional quadratic growth condition, the rate is improved to $\mathcal{O}(1/k)$ in terms of the squared distance to the optimal solution set. When $f$ is weakly convex, asymptotic convergence is derived. The central idea is that the dynamics of properly chosen step sizes rule fully controls the movement of the subgradient method, which leads to boundedness of the iterates, and then a trajectory-based analysis can be conducted to establish the desired results. To further illustrate the wide applicability of our framework, we extend the complexity results to the truncated subgradient, the stochastic subgradient, the incremental subgradient, and the proximal subgradient methods for non-Lipschitz functions.

Graph neural networks generalize conventional neural networks to graph-structured data and have received widespread attention due to their impressive representation ability. In spite of the remarkable achievements, the performance of Euclidean models in graph-related learning is still bounded and limited by the representation ability of Euclidean geometry, especially for datasets with highly non-Euclidean latent anatomy. Recently, hyperbolic space has gained increasing popularity in processing graph data with tree-like structure and power-law distribution, owing to its exponential growth property. In this survey, we comprehensively revisit the technical details of the current hyperbolic graph neural networks, unifying them into a general framework and summarizing the variants of each component. More importantly, we present various HGNN-related applications. Last, we also identify several challenges, which potentially serve as guidelines for further flourishing the achievements of graph learning in hyperbolic spaces.

In the last decade or so, we have witnessed deep learning reinvigorating the machine learning field. It has solved many problems in the domains of computer vision, speech recognition, natural language processing, and various other tasks with state-of-the-art performance. The data is generally represented in the Euclidean space in these domains. Various other domains conform to non-Euclidean space, for which graph is an ideal representation. Graphs are suitable for representing the dependencies and interrelationships between various entities. Traditionally, handcrafted features for graphs are incapable of providing the necessary inference for various tasks from this complex data representation. Recently, there is an emergence of employing various advances in deep learning to graph data-based tasks. This article provides a comprehensive survey of graph neural networks (GNNs) in each learning setting: supervised, unsupervised, semi-supervised, and self-supervised learning. Taxonomy of each graph based learning setting is provided with logical divisions of methods falling in the given learning setting. The approaches for each learning task are analyzed from both theoretical as well as empirical standpoints. Further, we provide general architecture guidelines for building GNNs. Various applications and benchmark datasets are also provided, along with open challenges still plaguing the general applicability of GNNs.

Text to speech (TTS), or speech synthesis, which aims to synthesize intelligible and natural speech given text, is a hot research topic in speech, language, and machine learning communities and has broad applications in the industry. As the development of deep learning and artificial intelligence, neural network-based TTS has significantly improved the quality of synthesized speech in recent years. In this paper, we conduct a comprehensive survey on neural TTS, aiming to provide a good understanding of current research and future trends. We focus on the key components in neural TTS, including text analysis, acoustic models and vocoders, and several advanced topics, including fast TTS, low-resource TTS, robust TTS, expressive TTS, and adaptive TTS, etc. We further summarize resources related to TTS (e.g., datasets, opensource implementations) and discuss future research directions. This survey can serve both academic researchers and industry practitioners working on TTS.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

The time and effort involved in hand-designing deep neural networks is immense. This has prompted the development of Neural Architecture Search (NAS) techniques to automate this design. However, NAS algorithms tend to be slow and expensive; they need to train vast numbers of candidate networks to inform the search process. This could be alleviated if we could partially predict a network's trained accuracy from its initial state. In this work, we examine the overlap of activations between datapoints in untrained networks and motivate how this can give a measure which is usefully indicative of a network's trained performance. We incorporate this measure into a simple algorithm that allows us to search for powerful networks without any training in a matter of seconds on a single GPU, and verify its effectiveness on NAS-Bench-101, NAS-Bench-201, NATS-Bench, and Network Design Spaces. Our approach can be readily combined with more expensive search methods; we examine a simple adaptation of regularised evolutionary search. Code for reproducing our experiments is available at //github.com/BayesWatch/nas-without-training.

Machine-learning models have demonstrated great success in learning complex patterns that enable them to make predictions about unobserved data. In addition to using models for prediction, the ability to interpret what a model has learned is receiving an increasing amount of attention. However, this increased focus has led to considerable confusion about the notion of interpretability. In particular, it is unclear how the wide array of proposed interpretation methods are related, and what common concepts can be used to evaluate them. We aim to address these concerns by defining interpretability in the context of machine learning and introducing the Predictive, Descriptive, Relevant (PDR) framework for discussing interpretations. The PDR framework provides three overarching desiderata for evaluation: predictive accuracy, descriptive accuracy and relevancy, with relevancy judged relative to a human audience. Moreover, to help manage the deluge of interpretation methods, we introduce a categorization of existing techniques into model-based and post-hoc categories, with sub-groups including sparsity, modularity and simulatability. To demonstrate how practitioners can use the PDR framework to evaluate and understand interpretations, we provide numerous real-world examples. These examples highlight the often under-appreciated role played by human audiences in discussions of interpretability. Finally, based on our framework, we discuss limitations of existing methods and directions for future work. We hope that this work will provide a common vocabulary that will make it easier for both practitioners and researchers to discuss and choose from the full range of interpretation methods.