Gaussian BP algorithm converges exponentially under walk summability for cyclic graphs.
problem Convergence rate of Gaussian BP for cyclic graphs.
method Extending known results on walk summability, proving exponential convergence rate.
result Gaussian BP converges exponentially under walk summability for cyclic graphs.
New algorithms learn GGMs without condition number bounds, even with strong dependencies.
problem Learning Gaussian Graphical Models without condition number bounds.
method Polynomial-time algorithms for attractive and walk-summable GGMs.
result Efficient recovery of graph structure with logarithmic number of samples.
Two heuristic algorithms improve Gaussian graphical model neighborhood selection.
problem Neighborhood selection for Gaussian graphical models.
method Forward-backward greedy algorithm and threshold-based algorithm.
result Both algorithms are structurally consistent and efficient.
The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.
problem Understanding ergodicity of geodesic flows on infinite Riemann surfaces.
method Analyzing random walks on the dual graph of pants decompositions.
result Equivalence between ergodicity of geodesic flows and recurrence of random walks.
Paper analyzes Min-Sum scheme for solving Laplacian systems and flow problems.
problem Solving systems of linear equations and computing electric flows in graphs.
method Develops a framework to analyze Min-Sum message passing for voltage and flow problems.
result Characterizes error and convergence of Min-Sum algorithm on general and regular graphs.
Paper proves non-equivalence of RKHS stability and kernel absolute summability.
problem Equivalence of RKHS stability and kernel absolute summability.
method Analyzes Reproducing Kernel Hilbert spaces and positive semidefinite kernels.
result Stable RKHSs can be induced by non-absolutely summable kernels.
Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
Study p-parabolicity on graphs using various energy functionals.
problem Characterize p-parabolicity on infinite locally summable graphs. method Analyze p-energy functionals and use approximation by finite graphs. result Prove various characterizations of p-parabolicity. Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are kn…
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
Paper analyzes convergence of distributed inference using BP in linear Gaussian models.
problem Distributed inference convergence in linear Gaussian models.
method Factor graphs, Gaussian belief propagation, local computation, message passing.
result Message information matrix converges to a unique positive definite limit matrix at a doubly exponential rate.
In the article a strenthened version of the 'Fundamental Theorem of asset Pricing' for one-period market model is proven. The principal role in this result play total and nonanihilating cones.
Extends Chern character theory to dg algebras, proving index theorems and constructing path integrals.
problem Constructing Chern character for θ-summable Fredholm modules over dg algebras.
method Introduced θ-summable Fredholm modules, constructed Chern character as a cocycle, proved index theorem.
result Rigorous construction of path integral for N=1/2 supersymmetry satisfying localization formula.
Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
To a finite, connected, unoriented graph of Betti-number g>=2 and valencies >=3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another example where non-commutative geometry provides a rigid framework for classification.
New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with p-rotation of inscribed polygonals, focusing on geometric curvature distribution. result Energy finite if and only if curve's arc-length parameterization has second order summability.
Computing Chern-Simons action for perturbed Dirac triples
problem Computing Chern-Simons action for perturbed Dirac triples
method Computing Chern-Simons action for perturbed Dirac triples
result Computing Chern-Simons action for perturbed Dirac triples
New quantum walks on simplicial complexes exhibit linear spreading and geometric localization.
problem Intrinsic difficulty in exhibiting nontrivial behavior in quantum walks.
method Constructing a new type of quantum walks on simplicial complexes as an extension of Szegedy walk.
result Localization of quantum walks reflects both topological and geometric structures.
We compute the homotopy type of the space of proper d-dimensional submanifolds of Rn with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the h…
New mathematical foundations for stable RKHSs improve system identification.
problem Improving stability tests and modeling of impulse responses.
method Providing new structural properties and stability conditions for stable RKHSs.
result Any stable kernel admits feature maps induced by orthogonal eigenvectors in l2.
Improved graph clustering for sparse graphs using non-backtracking random walks.
problem Improving graph clustering performance for sparse graphs.
method VEC-NBT uses a non-backtracking random walk to modify VEC, a graph embedding technique.
result VEC-NBT achieves comparable or greater accuracy with shorter walks than VEC for sparser graphs.
Decouples homotopy quotients of generalised configuration spaces on surfaces.
problem Homological stability of generalised configuration spaces on surfaces.
method Analyzes actions of diffeomorphism groups and uses homotopy quotients.
result Decouples theorem for homology of homotopy quotients on surfaces.
New MCMC algorithms speed up sampling from polytope distributions.
problem Sampling from uniform distributions over polytopes efficiently.
method Vaidya walk and John walk based on interior point methods.
result Vaidya walk mixes significantly faster than Dikin walk.
Researchers analyze record statistics in correlated random walks and Lévy flights.
problem Understanding record statistics in correlated time series.
method Review of random walk models and Lévy flights, focusing on number of records and record ages.
result Effects of correlations on record statistics were observed and analyzed.
Study large deviations in random walks on Lie groups.
problem Large deviations in sub-Riemannian random walks.
method Prove large deviation principle for random walks on stratified Lie groups.
result Proved a large deviation principle with a rate function adapted to sub-Riemannian geometry.
The paper introduces walks with jumps for modeling neuron activity in hyperbolic space.
problem Encoding neuron activity sequences in hyperbolic space.
method Introducing walks with jumps in hyperbolic geometry to model neuron activity.
result Endpoints of walks with jumps do not fully encode the sequence of jump times.
Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.
problem Improving graph kernel methods for better classification accuracy.
method Define and analyze walk-based node refinement methods, relate to Weisfeiler-Leman test, and introduce new walk-based kernels.
result Walk-based kernels are as expressive as Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison.
Quantum walks blend patterns into splines when averaged.
problem Understanding the asymptotic patterns of quantum random walks.
method Averaging over quantum coins using the Haar measure.
result Patterns blend into splines, showing a unified behavior.
Geodesic walk improves polytope sampling in high dimensions.
problem Generating uniform random points from polytopes in high dimensions.
method Discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold with the metric induced by the Hessian of a convex function.
result The geodesic walk mixes in O*(mn^{3/4}) steps, breaking the quadratic barrier.
Consistent estimation of constrained autoregressive processes.
problem Estimating autoregressive processes with coefficients constrained to an ellipsoid.
method Use of constrained and penalized estimators under different norms.
result Provide consistency results for estimation of constrained autoregressive processes.
Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
Random walks on convergence groups are studied, extending properties from hyperbolic groups.
problem Properties of random walks on hyperbolic groups are extended to convergence groups.
method Extending properties of random walks from hyperbolic groups to convergence groups with specific conditions.
result Random walks on convergence groups can be analyzed with a compact topology, leading to new insights into the Poisson boundary.
Study diffusions and random walks on hyperbolic spaces, focusing on their Martin boundaries.
problem Understanding diffusions and random walks on hyperbolic spaces.
method Analyzing specific diffusions and random walks on hyperbolic spaces, examining their Martin boundaries.
result Characterized the Martin boundaries of diffusions and random walks on hyperbolic spaces.
Random walks on mapping class groups have topological entropy that matches drift.
problem Understanding the topological entropy of random walks on mapping class groups.
method Defined topological entropy and proved it almost surely matches drift.
result Topological entropy of random walks on mapping class groups almost surely equals drift.
The paper studies identities for Cantor sets and their Hausdorff dimension.
problem Understanding the Hausdorff dimension of Cantor sets.
method Analyzes Basmajian-type series identities for holomorphic families of Cantor sets.
result The series is absolutely summable if and only if the Hausdorff dimension is less than 1.
New proof shows rapid mixing for random walks on nilmanifolds.
problem Proving rapid mixing for random walks on nilmanifolds.
method Proved rapid mixing for almost all random walks generated by m translations on nilmanifolds under mild assumptions.
result For several classical classes of nilmanifolds, m=2 suffices for rapid mixing.
The paper finds braid representatives minimizing simple walks for knots.
problem Finding efficient braid representatives for knots.
method Developed methods to minimize the number of simple walks in braids.
result Computed the colored Jones polynomial for specific knots.
Quantum walks model financial returns with flexibility and asymmetry.
problem Accurate modeling of financial asset price dynamics.
method Discrete-time quantum walks to model asset price evolution.
result Quantum walk models can generate asymmetric return distributions and higher probabilities for extreme events.
Random walks on metric spaces embed quasi-isometrically into the space.
problem Embedding random subgroups of metric spaces quasi-isometrically.
method Analyzing random walks and contracting elements in metric spaces.
result Random subgroups of isometry groups are quasi-isometrically embedded.
The paper develops a spectral theory for hypergraphs with edge-dependent vertex weights using random walks.
problem Lack of spectral theory for hypergraphs with edge-dependent vertex weights.
method Random walks on hypergraphs with edge-dependent vertex weights, deriving a random walk-based hypergraph Laplacian.
result Random walks on hypergraphs with edge-dependent vertex weights can capture higher-order relationships in data.
Develops Schouten-Nijenhuis bracket on infinite-dimensional manifolds.
problem Defining the Schouten-Nijenhuis bracket on smooth infinite-dimensional manifolds.
method Two-step approach: first for summable multivector fields, then for sections of a specific sheaf.
result Formalizes Schouten-Nijenhuis bracket on infinite-dimensional manifolds.
Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
New walk extraction strategies improve node embeddings in KGs.
problem Improving node embeddings in knowledge graphs.
method Proposed five different walk extraction strategies to complement basic random walks.
result The n-gram strategy performs best on average for node classification tasks.
A scalable framework preserves personalized higher-order network proximities.
problem Lack of expressive methods to preserve personalized higher-order network proximities.
method Incorporates random walk into a sound objective to preserve arbitrary higher-order proximities and introduces random walk with restart for personalized-weighted preservation.
result Consistently and substantially outperforms state-of-the-art methods on real-world networks.
A new method for embedding heterogeneous networks using spacey random walks.
problem Stationarity issues in meta-path guided random walks for HIN embedding.
method Heterogeneous personalized spacey random walk.
result Substantial improvement over existing network embedding algorithms.