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Globally injective ReLU networks

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2022年第1期23卷 4544-4598页

作者： Michael Puthawala Konik Kothari Matti Lassas Ivan Dokmanić Maarten De Hoop Department of Computational and Applied Mathematics Rice University Houston TX Department of Computer Science University of Illinois at Urbana-Champaign Champaign IL Department of Mathematics and Statistics University of Helsinki Finland Department of Mathematics and Computer Science University of Basel Basel Switzerland

Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end--rather than layerwise--doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.

关键词： bilipschitz networks injective generative models random neural networks deep learning universal approximation

Deformations of acid-mediated invasive tumors in a model with Allee effect

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arXiv 2024年

作者： Carter, Paul Doelman, Arjen van Heijster, Peter Levy, Daniel Maini, Philip Okey, Erin Yeung, Paige Department of Mathematics University of California Irvine United States Mathematical Institute Leiden University Leiden Netherlands Mathematical and Statistical Methods—Biometris Wageningen University & Research Wageningen Netherlands Program in Applied and Computational Mathematics Princeton University Princeton United States Mathematical Institute University of Oxford Oxford United Kingdom Division of Applied Mathematics Brown University ProvidenceRI United States Massachusetts Institute of Technology Cambridge United States

We consider a Gatenby–Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and/or different invasion speeds. © 2024, CC BY.

关键词： Perturbation techniques

Critical nonlinear aspects of hopping transport for reconfigurable logic in disordered dopant networks

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Physical Review applied 2024年第2期22卷 024063页

作者： Henri Tertilt Jonas Mensing Marlon Becker Wilfred G. van der Wiel Peter A. Bobbert Andreas Heuer Institute for Physical Chemistry Department of Physics NanoElectronics Group Faculty of Electrical Engineering Mathematics and Computer Science MESA+ Institute for Nanotechnology and Center for Brain-Inspired Nano Systems (BRAINS) Department of Applied Physics Center for Computational Energy Research (CCER)

Nonlinear behavior in the hopping transport of interacting charges enables reconfigurable logic in disordered dopant network devices, where voltages applied at control electrodes tune the relation between voltages applied at input electrodes and the current measured at an output electrode. From kinetic Monte Carlo simulations we analyze the critical nonlinear aspects of variable-range hopping transport for realizing Boolean logic gates in these devices on three levels. First, we quantify the occurrence of individual gates for random choices of control voltages. We find that linearly inseparable gates such as the xor gate are less likely to occur than linearly separable gates such as the and gate, despite the fact that the number of different regions in the multidimensional control voltage space for which and or xor gates occur is comparable. Second, we use principal-component analysis to characterize the distribution of the output current vectors for the (00,10,01,11) logic input combinations in terms of eigenvectors and eigenvalues of the output covariance matrix. This allows a simple and direct comparison of the behavior of different simulated devices and a comparison to experimental devices. Third, we quantify the nonlinearity in the distribution of the output current vectors necessary for realizing Boolean functionality by introducing three nonlinearity indicators. The analysis provides a physical interpretation of the effects of changing the hopping distance and temperature and is used in a comparison with data generated by a deep neural network trained on a physical device.

关键词： Carrier dynamics Dynamics of networks Electrical conductivity Fitness Hopping transport Network optimization Optimization problems Temperature

Invertibility of Discrete-Time Linear Systems with Sparse Inputs 63

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Invertibility of Discrete-Time Linear Systems with Sparse In...

63rd IEEE Conference on Decision and Control, CDC 2024

作者： Poe, Kyle Mallada, Enrique Vidal, Rene University of Pennsylvania Applied Mathematics and Computational Science Group PA19104 United States Johns Hopkins University Mallada Is with the Department of Electrical and Computer Engineering MD21218 United States University of Pennsylvania Radiology Computer and Information Science Statistics and Data Science Departments of Electrical and Systems Engineering PA19104 United States

ISBN: (纸本)9798350316339

One of the fundamental problems of interest for discrete-time linear systems is whether its input sequence may be recovered given its output sequence, a.k.a. the left inversion problem. Many conditions on the state space geometry, dynamics, and spectral structure of a system have been used to characterize the well-posedness of this problem, without assumptions on the inputs. However, certain structural assumptions, such as input sparsity, have been shown to translate to practical gains in the performance of inversion algorithms, surpassing classical guarantees. Establishing necessary and sufficient conditions for left invertibility of systems with sparse inputs is therefore a crucial step toward understanding the performance limits of system inversion under structured input assumptions. In this work, we provide the first necessary and sufficient characterizations of left invertibility for linear systems with sparse inputs, echoing classic characterizations for standard linear systems. The key insight in deriving these results is in establishing the existence of two novel geometric invariants unique to the sparseinput setting, the weakly unobservable and strongly reachable subspace arrangements. By means of a concrete example, we demonstrate the utility of these characterizations. We conclude by discussing extensions and applications of this framework to several related problems in sparse control. © 2024 IEEE.

关键词： State space methods

Bound-preserving discontinuous Galerkin methods for compressible two-phase flows in porous media

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arXiv 2023年

作者： Joshaghani, M.S. Riviere, B. Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States Rice University United States

This paper presents a numerical study of immiscible, compressible two-phase flows in porous media, that takes into account heterogeneity, gravity, anisotropy and injection/production wells. We formulate a fully implicit stable discontinuous Galerkin solver for this system that is accurate, that respects maximum principle for the approximation of saturation, and that is locally mass conservative. To completely eliminate the overshoot and undershoot phenomena, we construct a flux limiter that produces bound-preserving elementwise average of the saturation. The addition of a slope limiter allows to recover a pointwise bound-preserving discrete saturation. Numerical results show that both maximum principle and monotonicity of the solution are satisfied. The proposed flux limiter does not impact the local mass error and the number of nonlinear solver iterations. Copyright © 2023, The Authors. All rights reserved.

关键词： Maximum principle

STEFAN PROBLEM WITH SURFACE TENSION: GLOBAL EXISTENCE OF PHYSICAL SOLUTIONS UNDER RADIAL SYMMETRY

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arXiv 2022年

作者： Nadtochiy, Sergey Shkolnikov, Mykhaylo Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University Princeton NJ08544 United States

We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d ⩾ 3, in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface *** Codes 80A22, 60H30, 35B44 Copyright © 2022, The Authors. All rights reserved.

关键词： Surface tension

Looking through the mind's eye via multimodal encoder-decoder networks

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arXiv 2024年

作者： Afrasiyabi, Arman Busch, Erica Singh, Rahul Bhaskar, Dhananjay Caplette, Laurent Turk-Browne, Nicholas Krishnaswamy, Smita Department of Computer Science Applied Mathematics Program Department of Psychology O Department of Genetics Wu Tsai Institute United States Yale University United States

In this work, we explore the decoding of mental imagery from subjects using their fMRI measurements. In order to achieve this decoding, we first created a mapping between a subject's fMRI signals elicited by the videos the subjects watched. This mapping associates the high dimensional fMRI activation states with visual imagery. Next, we prompted the subjects textually, primarily with emotion labels which had no direct reference to visual objects. Then to decode visual imagery that may have been in a person's mind's eye, we align a latent representation of these fMRI measurements with a corresponding video-fMRI based on textual labels given to the videos themselves. This alignment has the effect of overlapping the video fMRI embedding with the text-prompted fMRI embedding, thus allowing us to use our fMRI-to-video mapping to decode. Additionally, we enhance an existing fMRI dataset, initially consisting of data from five subjects, by including recordings from three more subjects gathered by our team. We demonstrate the efficacy of our model on this augmented dataset both in accurately creating a mapping, as well as in plausibly decoding mental imagery. © 2024, CC BY.

关键词： Mapping

The Numerical Solution of the External Dirichlet Generalized Harmonic Problem for a Sphere by the Method of Probabilistic Solution

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arXiv 2024年

作者： Zakradze, Mamuli Tabagari, Zaza Koblishvili, Nana Davitashvili, Tinatin Criado-Aldeanueva, Francisco Department of Computational Methods Muskhelishvili Institute of Computational Mathematics Georgian Technical University Georgia Iv.Javakhishvili Tbilisi State University FENS Georgia Department of Applied Physics II Polytechnic School Malaga University Spain

In the present paper an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given. Under a generalized problem is meant the problem when a boundary function has a finite number of curves with the first kind of discontinuity. The algorithm consists of the following main stages: 1) transition from an infinite domain to a finite domain by an inversion;2) consideration of a new Dirichlet generalized harmonic problem on the basis of Kelvin’s theorem for the obtained finite domain;3) application of the MPS for the numerical solution a new problem which in turn is based on a computer simulation of the Wiener process;4) finding the probabilistic solution of the posed generalized problem at any fixed points of the infinite domain by the solution of the new problem. For illustration numerical examples are considered and results are *** Codes 35J05, 35J25, 65C30, 65N75 © 2024, CC BY-NC-ND.

关键词： Spheres

Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

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arXiv 2022年

作者： Han, Jiequn Hu, Ruimeng Long, Jihao Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Department of Mathematics Department of Statistics and Applied Probability University of California Santa BarbaraCA93106-3080 United States The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus is inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics (IPMs) previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward *** Codes 65C30, 68T07, 49N80, 68Q25 © 2022, CC BY.

关键词： Stochastic systems