Do ideas have shape? Idea registration as the continuous limit of artificial neural networks

DOI10.1016/j.physd.2022.133592OpenAlexW4309879846MaRDI QIDQ2111734

Houman Owhadi

Publication date: 17 January 2023

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2008.03920

zbMATH Keywords

regularizationimage registrationdeep Gaussian processesresidual neural networkHamiltonian and Lagrangian analysisinfinite depth limit

Mathematics Subject Classification ID

Numerical analysis (65-XX) Computer science (68-XX)

Related Items (3)

Learning dynamical systems from data: a simple cross-validation perspective. IV: Case with partial observations ⋮ Gaussian process hydrodynamics ⋮ Kernel methods are competitive for operator learning

Uses Software

• LDDMM

Cites Work

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• Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity
• Diffeomorphic 3D image registration via geodesic shooting using an efficient adjoint calculation
• The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind
• Brittleness of Bayesian inference and new Selberg formulas
• Existence of solutions for ordinary differential equations in Banach spaces
• Approximation of the effective conductivity of ergodic media by periodization
• Metamorphoses of functional shapes in Sobolev spaces
• On the \(\Gamma\)-convergence of discrete dynamics and variational integrators
• Deep regularization and direct training of the inner layers of neural networks with kernel flows
• Learning dynamical systems from data: a simple cross-validation perspective. I: Parametric kernel flows
• Learning ``best kernels from data in Gaussian process regression. With application to aerodynamics
• Uncertainty quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball
• Solving and learning nonlinear PDEs with Gaussian processes
• Hyperparameter estimation in Bayesian MAP estimation: parameterizations and consistency
• Kernel flows: from learning kernels from data into the abyss
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• A mean-field optimal control formulation of deep learning
• A proposal on machine learning via dynamical systems
• The momentum map representation of images
• Strong solutions for differential equations in abstract spaces
• Geodesic shooting for computational anatomy
• Metamorphoses through Lie group action
• Brittleness of Bayesian inference under finite information in a continuous world
• Computing large deformation metric mappings via geodesic flows of diffeomorphisms
• On completeness of groups of diffeomorphisms
• Computational graph completion
• Landmark matching via large deformation diffeomorphisms
• A Concise Introduction to Geometric Numerical Integration
• On the Brittleness of Bayesian Inference
• Kernels for Vector-Valued Functions: A Review
• Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
• Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games
• The Random Feature Model for Input-Output Maps between Banach Spaces
• Discrete mechanics and variational integrators
• Metric-based upscaling
• Support Vector Machines
• Local error estimates for radial basis function interpolation of scattered data
• Universal approximation bounds for superpositions of a sigmoidal function
• Computable Elastic Distances Between Shapes
• Geometric numerical integration illustrated by the Störmer–Verlet method
• Variational problems on flows of diffeomorphisms for image matching
• Computational anatomy: an emerging discipline
• Qualitative Robustness in Bayesian Inference
• Stable architectures for deep neural networks
• PCANet: A Simple Deep Learning Baseline for Image Classification?
• Bayesian Inference for Non-Stationary Spatial Covariance Structure via Spatial Deformations
• Consistency of empirical Bayes and kernel flow for hierarchical parameter estimation
• Kernel Mode Decomposition and the Programming of Kernels
• Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
• A mean field view of the landscape of two-layer neural networks
• Sparse Cholesky Factorization by Kullback--Leibler Minimization
• Statistical Numerical Approximation
• Bayesian Probabilistic Numerical Methods
• Bayesian Numerical Homogenization
• The Euler-Poincaré theory of metamorphosis
• Probabilistic numerics and uncertainty in computations
• A representer theorem for deep kernel learning
• Geodesic Image Regression with a Sparse Parameterization of Diffeomorphisms
• Geometric Numerical Integration
• Shapes and diffeomorphisms
• Shapes and diffeomorphisms

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