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Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay

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arXiv 2025年

作者： Laus, Hannah Parkinson, Suzanna Charisopoulos, Vasileios Krahmer, Felix Willett, Rebecca Department of Mathematics Technical University of Munich Germany Germany Committee on Computational and Applied Mathematics University of Chicago United States Data Science Institute University of Chicago United States Technical University of Munich Germany Department of Computer Science University of Chicago United States Department of Statistics University of Chicago United States

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few measurements generated via a known acquisition procedure. In particular, neural networks perform well empirically but have limited theoretical guarantees. In this work, we study an underdetermined linear inverse problem that admits several possible solution mappings. A standard remedy (e.g., in compressed sensing) establishing uniqueness of the solution mapping is to assume knowledge of latent low-dimensional structure in the source signal. We ask the following question: do deep neural networks adapt to this low-dimensional structure when trained by gradient descent with weight decay regularization? We prove that mildly overparameterized deep linear networks trained in this manner converge to an approximate solution that accurately solves the inverse problem while implicitly encoding latent subspace structure. To our knowledge, this is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data under practical stepsize and weight initialization schemes. Our work highlights that regularization and overparameterization improve generalization, while overparameterization also accelerates convergence during training. © 2025, CC BY.

关键词： Deep neural networks

A bivariate two-state Markov modulated Poisson process for failure modelling

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arXiv 2024年

作者： Yera, Yoel G. Lillo, Rosa E. Nielsen, Bo F. Ramírez-Cobo, Pepa Ruggeri, Fabrizio Department of Statistics Universidad Carlos III de Madrid Spain uc3m-Santander Big Data Institute Madrid Spain Department of Applied Mathematics and Computer Science Technical University of Denmark Kongens Lyngby Denmark Department of Statistics and Operations Research Universidad de Cádiz Spain IMATI-CNR Institute of Applied Mathematics and Information Technologies Milano Italy

Motivated by a real failure dataset in a two-dimensional context, this paper presents an extension of the Markov modulated Poisson process (MMPP) to two dimensions. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-failure times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence between the two sequences of inter-failure times, while at the same time preserves the MMPP properties, marginally. The generalization is based on the Marshall-Olkin exponential distribution. Inference is undertaken for the new model through a method combining a matching moments approach with an Approximate Bayesian Computation (ABC) algorithm. The performance of the method is shown on simulated and real datasets representing times and distances covered between consecutive failures in a public transport company. For the real dataset, some quantities of importance associated with the reliability of the system are estimated as the probabilities and expected number of failures at different times and distances covered by trains until the occurrence of a failure. © 2024, CC BY-NC-ND.

关键词： Reliability

Infinitely Many Non-Conservative Solutions for the Three-Dimensional Euler Equations with Arbitrary Initial data in C1 / 3 −Ε

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SSRN

SSRN 2023年

作者： Khor, Calvin Miao, Changxing Ye, Weikui Beijing Computational Science Research Center Beijing100193 China Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing100088 China

关键词： Euler equations

LOCAL BEHAVIOR FOR SOLUTIONS TO ANISOTROPIC WEIGHTED QUASILINEAR DEGENERATE PARABOLIC EQUATIONS

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arXiv 2023年

作者： Miao, Changxing Zhao, Zhiwen Beijing Computational Science Research Center Beijing100193 China Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing100088 China

This paper aims to study the local behavior of solutions to a class of anisotropic weighted quasilinear degenerate parabolic equations with the weights comprising two power-type weights of different dimensions. We first capture the asymptotic behavior of the solution near the singular or degenerate point of the weights. In particular, we find an explicit upper bound on the decay rate exponent determined by the structures of the equations and weights, which can be achieved under certain condition and meanwhile reflects the damage effect of the weights on the regularity of the solution. Furthermore, we prove the local Hölder regularity of solutions to non-homogeneous parabolic p-Laplace equations with single power-type weights. Copyright © 2023, The Authors. All rights reserved.

关键词： Anisotropy

Theoretical and Numerical Investigation of Acoustic Cavitation Bubble Based on the Impact of Ultrasound Frequency

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Bulletin of the Russian Academy of sciences: physics 2023年第Suppl 3期87卷 S341-S347页

作者： Ahmed K. Abu-Nab Abu-Bakr, Ali F. Zain Abu Shaeer Department of Mathematics and Computer Science Faculty of Science Menoufia University Shebin El-Koom32511 Egypt Phystech School of Applied Mathematics and Informatics Moscow Institute of Physics and Technology Moscow oblast Dolgoprudny141700 Russia Department of Basic Science Higher Institute of Engineering and Technology Kafr El Sheikh33511 Egypt

Abstract: Acoustic cavitation is the expansion and contraction of existing microbubbles in liquids brought on by an ultrasonic field. The dynamics of oscillations at higher pressures and temperatures when the cavitation bubbles collapse can be used to characterize cavitation dynamical behavior. This work deals with the theoretical and numerical investigation of acoustic cavitation in viscoelastic media considering the influence of ultrasound frequency. An acoustic pressure equation combined with a modified Rayleigh–Plesset equation forms the theoretical framework for ultrasound cavitation dynamics. The numerical solutions of the given cavitation bubble model were obtained taking into account the effect of acoustic pressure, using the exponential B-Spline collection method. The dynamics of the cavitation bubbles’ radius and the internal pressure of acoustic cavitation were examined at different values of viscosity and the impact of physical parameters on the monatomic, diatomic, and adiabatic gas-filled spherical bubble’s collapse dynamics for (k = 7/5, 5/3, 1) respectively. © Pleiades Publishing, Ltd. 2023. ISSN 1062-8738, Bulletin of the Russian Academy of sciences: physics, 2023, Vol. 87, Suppl. 3, pp. S341–S347. Pleiades Publishing, Ltd., 2023. ISSN 1062-8738, Bulletin of the Russian Academy of sciences: physics, 2023. Pleiades Publishing, Ltd., 2023.

关键词： Dynamics

Interactions Between Differential Geometry and Production Theory 1

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4th International Workshop on Differential Geometry, IWDG 2023

作者： Vîlcu, Alina-Daniela Vîlcu, Gabriel-Eduard Department of Information Technology Mathematics and Physics Petroleum-Gas University of Ploieşti Bd. Bucureşti 39 Ploieşti100680 Romania Department of Mathematics and Informatics National University of Science and Technology Politehnica Bucharest Splaiul Independenţei 313 Bucharest060042 Romania Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy Calea 13 Septembrie 13 Bucharest050711 Romania Research Center in Geometry Topology and Algebra University of Bucharest Str. Academiei 14 Bucharest70109 Romania

ISBN: (数字)9783031505867

ISBN: (纸本)9783031505850

The use of differential geometry in the theory of production is an interesting topic that has attracted the attention of many researchers in the last two decades, both from the fields of mathematics and economics. The purpose of this work is to provide a comprehensive survey on the properties of the homogeneous and quasi-homogeneous production models, most of these properties being of geometric nature and obtained using a differential geometric treatment. Some open problems are also discussed at the end of the article. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

关键词： Gauss-Kronecker curvature Hypersurface Mean curvature Production function

Parasite chain detection in the IOTA protocol 2

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Parasite chain detection in the IOTA protocol

2nd International Conference on Blockchain Economics, Security and Protocols, Tokenomics 2020

作者： Penzkofer, Andreas Kusmierz, Bartosz Capossele, Angelo Sanders, William Saa, Olivia IOTA Foundation Berlin Germany Department of Theoretical Physics Wroclaw University of Science and Technology Poland Department of Applied Mathematics Institute of Mathematics and Statistics University of São Paulo Brazil

ISBN: (纸本)9783959771573

In recent years several distributed ledger technologies based on directed acyclic graphs (DAGs) have appeared on the market. Similar to blockchain technologies, DAG-based systems aim to build an immutable ledger and are faced with security concerns regarding the irreversibility of the ledger state. However, due to their more complex nature and recent popularity, the study of adversarial actions has received little attention so far. In this paper we are concerned with a particular type of attack on the IOTA cryptocurrency, more specifically a Parasite Chain attack that attempts to revert the history stored in the DAG structure, also called the Tangle. In order to improve the security of the Tangle, we present a detection mechanism for this type of attack. In this mechanism, we embrace the complexity of the DAG structure by sampling certain aspects of it, more particularly the distribution of the number of approvers. We initially describe models that predict the distribution that should be expected for a Tangle without any malicious actors. We then introduce metrics that compare this reference distribution with the measured distribution. Upon detection, measures can then be taken to render the attack unsuccessful. We show that due to a form of the Parasite Chain that is different from the main Tangle it is possible to detect certain types of malicious chains. We also show that although the attacker may change the structure of the Parasite Chain to avoid detection, this is done so at a significant cost since the attack is rendered less efficient. © Andreas Penzkofer, Bartosz Kusmierz, Angelo Capossele, William Sanders, and Olivia Saa;licensed under Creative Commons License CC-BY

关键词： Distributed ledger

On a class of high dimensional linear regression methods with debiasing and thresholding

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arXiv 2025年

作者： Wang, Ying-Ao Zhang, Yunyi Zhang, Ye School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen518172 China MSU-BIT-SMBU Joint Research Center of Applied Mathematics Shenzhen MSU-BIT University Shenzhen518172 China

In this paper, we introduce a unified framework, inspired by classical regularization theory, for designing and analyzing a broad class of linear regression approaches. Our framework encompasses traditional methods like least squares regression and Ridge regression, as well as innovative techniques, including seven novel regression methods such as Landweber and Showalter regressions. Within this framework, we further propose a class of debiased and thresholded regression methods to promote feature selection, particularly in terms of sparsity. These methods may offer advantages over conventional regression techniques, including Lasso, due to their ease of computation via a closed-form expression. Theoretically, we establish consistency results and Gaussian approximation theorems for this new class of regularization methods. Extensive numerical simulations further demonstrate that the debiased and thresholded counterparts of linear regression methods exhibit favorable finite sample performance and may be preferable in certain settings. Copyright © 2025, The Authors. All rights reserved.

关键词： Linear regression

Extreme rotational events in a forced-damped nonlinear pendulum

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arXiv 2023年

作者： Pal, Tapas Kumar Ray, Arnob Chowdhury, Sayantan Nag Ghosh, Dibakar Physics and Applied Mathematics Unit Indian Statistical Institute Kolkata700108 India Department of Environmental Science and Policy University of California DavisCA95616 United States

Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when phase difference between the instantaneous phase of the system and the externally applied ac torque is observed. © 2023, CC BY.

关键词： Pendulums