Old and new challenges in Hadamard spaces

DOI10.1007/s11537-023-1826-0arXiv1807.01355OpenAlexW4387369654MaRDI QIDQ6052560

Miroslav Bačák

Publication date: 17 October 2023

Published in: Japanese Journal of Mathematics. 3rd Series (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1807.01355

zbMATH Keywords

weak convergencestrongly continuous semigroupMosco convergenceconvex functiongradient flownon-positive curvaturesubmodular functionHadamard spaceharmonic mappingproximal point algorithmbi-Lipschitz embeddingproximal mapping

Mathematics Subject Classification ID

Semigroups of nonlinear operators (47H20) Set-valued and variational analysis (49J53) Applications of functional analysis in optimization, convex analysis, mathematical programming, economics (46N10) Decomposition methods (49M27)

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Basic Convex Analysis in Metric Spaces with Bounded Curvature ⋮ A metric fixed point theorem and some of its applications

Cites Work

• Convergence and approximation of semigroups of nonlinear operators in Banach spaces
• Integrals which are convex functionals. II
• Generalized Dirichlet forms and harmonic maps
• Submodular functions and optimization.
• Isoperimetric inequalities of Euclidean type in metric spaces
• Combinatorial harmonic maps and discrete-group actions on Hadamard spaces
• A Riemannian framework for tensor computing
• Banach space theory. The basis for linear and nonlinear analysis
• Spaces with non-positive curvature
• LIPSCHITZ RETRACTION OF FINITE SUBSETS OF HILBERT SPACES
• A Second Order Nonsmooth Variational Model for Restoring Manifold-Valued Images
• The consistency and convergence of K-energy minimizing movements
• Erratum: A CAT(0)-valued pointwise ergodic theorem
• Fruit Flies and Moduli: Interactions between Biology and Mathematics
• Extreme points in non-positive curvature
• Convex analysis and optimization in Hadamard spaces
• Approximation for convex functionals on non-positively curved spaces and the Trotter–Kato product formula
• Computing Medians and Means in Hadamard Spaces
• A new proof of the Lie–Trotter–Kato formula in Hadamard spaces
• A CAT(0)-VALUED POINTWISE ERGODIC THEOREM
• Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time
• Groups acting on spaces of non-positive curvature
• The composition of projections onto closed convex sets in Hilbert space is asymptotically regular
• Proximal minimization in CAT$(\kappa)$ spaces
• A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds
• Moving Robots Efficiently Using the Combinatorics of CAT(0) Cubical Complexes
• Computing Geodesic Distances in Tree Space
• Convex Sets and Nearest Points
• On the Product of Semi-Groups of Operators
• Nonexpansive iterations in hyperbolic spaces
• Weakly Modular Graphs and Nonpositive Curvature
• Uncertainty in Phylogenetic Tree Estimates
• Markov Type of Alexandrov Spaces of Non‐Negative Curvature Shin‐Ichi Ohta
• Open map theorem for metric spaces
• Monotone vector fields and the proximal point algorithm on Hadamard manifolds
• On the Convergence of the Proximal Point Algorithm for Convex Minimization
• Monotone Operators and the Proximal Point Algorithm
• Monotone approximation of energy functionals for mappings into metric spaces, I.
• On symmetric products of topological spaces
• Proximal Point Algorithm On Riemannian Manifolds
• The logarithm map, its limits and Fréchet means in orthant spaces
• Snowflake universality of Wasserstein spaces
• The proximal point algorithm in geodesic spaces with curvature bounded above
• Quantitative results on Fejér monotone sequences
• The Configuration Space of a Robotic Arm in a Tunnel
• Relative Optimality Conditions and Algorithms for Treespace Fréchet Means
• L-CONVEXITY ON GRAPH STRUCTURES
• An Invitation to Alexandrov Geometry
• A new proximal point iteration that converges weakly but not in norm
• Total Generalized Variation for Manifold-Valued Data
• On Type of Metric Spaces
• UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES
• An L¹ ergodic theorem with values in a non-positively curved space via a canonical barycenter map
• Computing the nc-Rank via Discrete Convex Optimization on CAT(0) Spaces
• A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes
• Variational regularization of inverse problems for manifold-valued data
• Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery
• Improvements of upper curvature bounds
• Existence and approximation of fixed points of vicinal mappings in geodesic spaces
• A variational approach to stochastic minimization of convex functionals
• Second Order Differences of Cyclic Data and Applications in Variational Denoising
• An Application of Proof Mining to the Proximal Point Algorithm in CAT(0) Spaces
• Confidence Sets for Phylogenetic Trees
• Morphing of Manifold-Valued Images Inspired by Discrete Geodesics in Image Spaces
• Convergence of nonlinear semigroups under nonpositive curvature
• Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups
• Gradient flows and a Trotter-Kato formula of semi-convex functions on CAT(1)-spaces
• Convexity in Tree Spaces
• Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees
• Variational convergence over metric spaces
• Nonpositive Curvature and the Ptolemy Inequality
• Applied Proof Theory: Proof Interpretations and Their Use in Mathematics
• Proximité et dualité dans un espace hilbertien
• The solution by iteration of nonlinear functional equations in Banach spaces
• Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
• The construction of an asymptotic center with a fixed-point property
• Spherical nonspreadingness of resolvents of convex functions in geodesic spaces
• Weak convergence of infinite products of operators in Hadamard spaces
• The 6-strand braid group is \(\mathrm{CAT}(0)\).
• A second-order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data
• The space of ultrametric phylogenetic trees
• Restoration of manifold-valued images by half-quadratic minimization
• Convex functions and barycenter on CAT(1)-spaces of small radii
• Central limit theorems for Fréchet means in the space of phylogenetic trees
• CAT(0) spaces and expanders.
• Braids, posets and orthoschemes
• Alternating projections in CAT(0) spaces
• Incremental proximal methods for large scale convex optimization
• Resolvents of set-valued monotone vector fields in Hadamard manifolds
• Geodesics in CAT(0) cubical complexes
• A Trotter product formula for gradient flows in metric spaces
• Shortest path problem in rectangular complexes of global nonpositive curvature
• Effective metastability of Halpern iterates in \(CAT(0)\) spaces
• Discrete-time gradient flows and law of large numbers in Alexandrov spaces
• Inexact proximal point methods in metric spaces
• Approximation of semi-groups of operators
• 5-point CAT(0) spaces after Tetsu Toyoda
• The asymptotic behavior of the composition of firmly nonexpansive mappings
• A concept of convergence in geodesic spaces
• Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions
• Homotopy problems for harmonic maps to spaces of nonpositive curvature
• CAT\((k)\)-spaces, weak convergence and fixed points
• An alternative proof of Berg and Nikolaev's characterization of \(CAT(0)\)-spaces via quadrilateral inequality
• A complement to Trotter's product formula for nonlinear semigroups generated by the subdifferentials of convex functionals
• Solutions of two problems of H. Brezis
• The metrical interpretation of superreflexivity in Banach spaces
• On Lipschitz embedding of finite metric spaces in Hilbert space
• Product formulas, nonlinear semigroups, and accretive operators
• The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball
• Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups
• Asymptotic convergence of nonlinear contraction semigroups in Hilbert space
• Trotter's product formula for nonlinear semigroups generated by the subdifferentials of convex functionals
• Un exemple concernant le comportement asymptotique de la solution du problème \(du/dt+\partial\varphi(\mu)\ni=0\)
• Produits infinis de resolvantes
• Semigroups and generators on convex domains with the hyperbolic metric
• Gradient flows on nonpositively curved metric spaces and harmonic maps
• Equilibrium maps between metric spaces
• An introduction to \(\Gamma\)-convergence
• Kirszbraun's theorem and metric spaces of bounded curvature
• Global existence theorems for harmonic maps to nonlocally compact spaces
• Random walk in random groups.
• Projection and proximal point methods: Convergence results and counterexamples.
• Geometry of the space of phylogenetic trees
• Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces
• An abstract proximal point algorithm
• Convergence of functions and their Moreau envelopes on Hadamard spaces
• Convex functions and \(p\)-barycenter on CAT(1)-spaces of small radii
• Limiting behaviour of Fréchet means in the space of phylogenetic trees
• A polynomial rate of asymptotic regularity for compositions of projections in Hilbert space
• Law of large numbers in CAT(1)-spaces of small radii
• Superrigidity for irreducible lattices and geometric splitting
• Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one
• The Euclidean distortion of complete binary trees
• An alternating projection that does not converge in norm
• Harmonicity of totally geodesic maps into nonpositively curved metric spaces
• Sobolev spaces and harmonic maps for metric space targets
• Convex functionals and generalized harmonic maps into spaces of non positive curvature
• \(N\)-step energy of maps and the fixed-point property of random groups.
• The proximal point algorithm in metric spaces
• On embedding trees into uniformly convex Banach spaces
• Random walks and Brownian motion on cubical complexes
• A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow
• Yet another short proof of Bourgain's distortion estimate for embedding of trees into uniformly convex Banach spaces
• Cyclic projections in Hadamard spaces
• Rademacher type and Enflo type coincide
• An intrinsic characterization of five points in a CAT(0) space
• A nonpositive curvature property of modular semilattices
• On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows
• Isometric embedding of Busemann surfaces into \(L_1\)
• Symmetric orthogonality and non-expansive projections in metric spaces
• Rigidity of Busemann convex Finsler metrics
• Self-contracted curves in CAT(0)-spaces and their rectifiability
• Nonpositive curvature is not coarsely universal
• Backward-backward splitting in Hadamard spaces
• Rectifiability of self-contracted curves in the Euclidean space and applications
• The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces
• Polyhedral computational geometry for averaging metric phylogenetic trees
• Combined first and second order variational approaches for image processing
• Packing and covering with balls on Busemann surfaces
• Convexity of the extended K-energy and the large time behavior of the weak Calabi flow
• Two modified proximal point algorithms in geodesic spaces with curvature bounded above
• Quasilinearization and curvature of Aleksandrov spaces
• Plateau's problem for integral currents in locally non-compact metric spaces
• Addendum to: ``Effective metastability of Halpern iterates in CAT(0) spaces
• Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature bounded from above
• Superrigidity, generalized harmonic maps and uniformly convex spaces
• Roundness properties of groups.
• Metric characterization of Berwald spaces of non-positive flag curvature
• On the density theorem for the subdifferential of convex functions on Hadamard spaces
• Note on product formulas for operator semigroups
• Integrals which are convex functionals
• Nonlinear semigroups and evolution equations
• Convergence of convex sets and of solutions of variational inequalities
• Approximation of semi-groups of nonlinear operators
• On the nonexistence of uniform homeomorphisms between \(L^ p\)-spaces
• Semigroups on nonlinear contractions on convex sets
• A note on Mosco convergence in spaces
• Convex analysis and monotone operator theory in Hilbert spaces
• Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu
• Expanders with respect to Hadamard spaces and random graphs
• Currents in metric spaces
• Generalized harmonic maps and representations of discrete groups
• Extensions of Lipschitz maps into Hadamard spaces
• Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces
• Weak topology on CAT(0) spaces
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