Improved convergence of fixed-point methods using windowed Anderson acceleration.
problem Improving convergence of fixed-point methods for symmetric operators.
method Windowed Anderson acceleration for symmetric fixed-point iterations.
result Windowed Anderson acceleration improves convergence over standard fixed-point methods.
Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed poi…
Interpreting gradient methods as fixed-point iterations, we provide a detailed analysis of those methods for minimizing convex objective functions. Due to their conceptual and algorithmic simplicity, gradient methods are widely used in machine learning for massive data sets (big data). In particular, stochastic gradien…
New method produces reflections with nonseparating fixed points.
problem Constructing hyperbolic manifolds with reflective symmetries.
method Standard method for constructing closed hyperbolic manifolds.
result Fixed point sets of reflections are nonseparating.
We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.
New method for robust fixed-point smoothing without state augmentation.
problem Estimating initial states in Gaussian smoothing algorithms.
method Cholesky-based formulation without state augmentation.
result Matches runtime and robustness of existing methods.
Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem li…
Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poinca…
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer's fixed point theorem (known methods use Schauder's fixed point theorem) while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These…
Paper proves new method for constructing initial data in general relativity.
problem Proving the existence of solutions for initial data in general relativity.
method Using the Banach fixed point theorem to prove existence, with guarantees of uniqueness and explicit construction.
result Guaranteed uniqueness and explicit construction of solutions to the conformal method equations.
In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with…
A faster method for estimating effects in large data using fixed-point trees.
problem Estimating heterogeneous effects in large dimensions with computational efficiency.
method Fixed-point approximation to eliminate Jacobian estimation and speed up GRFs.
result Significant computational efficiency improvement without sacrificing statistical accuracy.
Machine learning finds a compact fixed point action for SU(3) gauge theory.
problem Finding accurate and compact parametrizations of fixed point actions for SU(3) gauge theory.
method Used machine learning, specifically a gauge equivariant convolutional neural network.
result Obtained a superior parametrization of a fixed point action for SU(3) gauge theory.
The paper classifies circle actions on 6D manifolds with isolated fixed points.
problem Classifying circle actions on 6D manifolds with isolated fixed points.
method Performing equivariant connected sums at fixed points with specific manifolds.
result A sequence of operations can reduce the fixed point data to the empty collection.
New methods for federated learning reduce communication costs.
problem Efficiently solving optimization problems in a distributed setting.
method Developed two strategies for achieving consensus in federated learning: fixed number of local steps and randomized computations.
result Convergence analysis and experiments show benefits of the proposed methods.
Developed an efficient iterative algorithm for SVI model.
problem SVI model's optimizer's strong dependence on input starting point.
method Fixed-point and least-square optimizer.
result Convergence results for fixed-point iterative algorithm in certain situations.
Groups with special properties always have fixed points.
problem Groups acting on finite CW-complexes without fixed points.
method Exhibited specific groups with strong fixed-point properties.
result Groups with finite generation and torsion-freeness have global fixed points.
Given an S1-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this short survey article is, by using the language of equ…
Formula derived for Dirac operators on Lie groupoids.
problem Equivariant Dirac operators on Lie groupoids.
method Getzler rescaling method to derive a fixed-point formula.
result Reduces to standard fixed-point formula for closed manifolds.
Develops parametrised Poincaré duality for equivariant fixed points.
problem Understanding equivariant fixed points in non-presentable settings.
method Introduces parametrised Poincaré duality in parametrised higher category theory, proving basechange results.
result Generalises Cnossen's twisted ambidexterity to non-presentable settings and applies to isotropy separation methods.
Quantized neural networks can represent all fixed-point functions under certain conditions.
problem Expressive power of quantized neural networks under fixed-point arithmetic.
method Analyzing necessary and sufficient conditions for quantized networks to represent all fixed-point functions.
result Various popular activation functions satisfy the sufficient condition for representing all fixed-point functions.
Study circle actions on unitary manifolds with discrete fixed points.
problem Understanding circle actions on compact unitary manifolds with discrete fixed points.
method Prove relationships between weights at fixed points and derive results regarding the first equivariant Chern class and Hirzebruch χy-genus. result Derive a multigraph encoding fixed point data, leading to new insights into unitary S1-manifolds. Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.
problem Analyzing the growth of derivative maxima for C2 interval diffeomorphisms with parabolic fixed points. method Examining C2 diffeomorphisms with only parabolic fixed points, focusing on tangency and repelling behavior. result Maximal growth of derivative maxima is exactly quadratic for diffeomorphisms with a non-quadratic tangency to identity at a repelling fixed point.
New proof for 6D symplectic manifold with 4 fixed points.
problem Classifying the integral cohomology ring and total Chern class for 6D symplectic manifolds with 4 fixed points.
method New different argument using moment map values and weights of fixed points.
result Determined the sets of weights and global invariants for the manifold.
The paper highlights issues with fixed point claims in digital images.
problem Flaws in published assertions about fixed points in digital images.
method Continues a series of studies examining digital topology.
result Identifies and discusses problems with fixed point claims.
The paper highlights issues in fixed point claims in digital topology.
problem Flaws in published assertions about fixed points in digital metric spaces.
method Continues a series of studies examining these flaws.
result Identifies and discusses problems in fixed point claims.
Improved stochastic Halpern iteration for fixed-point approximation in normed spaces.
problem Approximating fixed-points of nonexpansive and contractive operators in normed finite-dimensional spaces.
method Stochastic Halpern iteration with minibatch, analyzing oracle complexity.
result Improved oracle complexity for nonexpansive operators, with a lower bound of Ω(ε−3). In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the S1-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be…
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
The paper introduces fixed-point centralities for networks and graphons.
problem Defining network centralities for networks and graphons.
method Fixed-point centralities defined via permutation equivariant mappings and graphons.
result Variation bounds of fixed-point centralities under mild assumptions.
A new method improves ICA performance by approximating MDI.
problem Improving F astICA's performance with nonlinear functions.
method Second-order approximation of MDI for joint maximization.
result Efficiency validated through experiments compared to other ICA algorithms.
A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regula…
Corrects incorrect assertions about fixed points in digital topology.
problem Incorrect or incorrectly proven assertions about fixed points in digital metric spaces.
method Analysis of existing assertions and proofs.
result Identifies and corrects errors in published assertions.
Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.
problem Finding the minimum number of fixed points for a circle action on a 10D almost complex manifold.
method Established a lower bound by showing the non-existence of a circle action with 4 fixed points.
result There are at least 6 fixed points for a circle action on a 10D compact almost complex manifold.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect or poorly stated fixed point assertions in digital topology.
method Discussion of problematic publications in digital metric spaces.
result Clarification of incorrect fixed point assertions.
The paper addresses flaws in fixed point assertions for digital images.
problem Deficiencies in previously published works on fixed point assertions for digital images.
method Continues a series of studies to identify and rectify issues in fixed point assertions.
result Identifies and corrects flaws in fixed point assertions for digital images.
Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set MG. We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of MG. Fixed-point homogeneous manifolds with positive sectional curvature have been c…
Fixed point assertions in digital topology are often incorrect or poorly stated.
problem Fixed points in digital metric spaces
method Discussing publications with bad assertions
result Identifying and correcting errors in fixed point assertions
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect, incorrectly proven, or trivial fixed point assertions in digital topology.
method Continues earlier work on identifying and critiquing bad fixed point assertions.
result Clarifies the nature and extent of incorrect fixed point assertions in digital topology.
Estimates distances between fixed points of certain Teichmüller actions.
problem Estimating distances between fixed points of specific Teichmüller actions.
method Uses a pants decomposition and quasi-isometry to estimate distances.
result Provides a method to estimate distances between fixed points.
Classifies circle actions on 6D manifolds with 4 fixed points.
problem Classifying circle actions on 6D manifolds with specific fixed points.
method Analyzes fixed point data and proves agreement with known actions.
result Agrees with actions on 6-spheres or CP3. Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.
problem Characterize circle actions on oriented manifolds with exactly 3 fixed points.
method Analyzes manifold dimensions, isotropy submanifolds, and uses quaternionic projective space as a reference.
result For a manifold with three fixed points, its dimension must be a multiple of 4, and specific weights are unique.
Study fixed-point sets of S1-actions on quaternionic manifolds.
problem Characterize fixed-point sets and compatible complex structures on quaternionic manifolds.
method Analyze fixed-point sets and derive equations involving first Chern classes.
result Conditions for the existence of hypercomplex structures on quaternionic manifolds.
New method stabilizes DEQ models by regularizing Jacobian of fixed-point equations.
problem Stability and performance of DEQ models.
method Jacobian regularization to stabilize DEQ models.
result Significant stabilization of fixed-point convergence in DEQ models.
Upper bounds on fixed points in PWL neural networks with hyperplane analysis.
problem Analyzing the number of fixed points in neural networks with PWL activation.
method Hyperplane arrangements to bound the number of fixed points.
result Upper bounds on the number of fixed points for PWL networks, showing exponential growth in layers.
Fixed points of nonnegative neural networks are analyzed using fixed point theory.
problem Analyzing fixed points in nonnegative neural networks.
method Fixed point theory, nonlinear Perron-Frobenius theory, monotonic and scalable mappings.
result Conditions for the existence of fixed points in nonnegative neural networks are provided.
Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper we investigate fiberwise analoga and represent a general approach e.g. to the question when two maps can be deform…
Convex message passing algorithms converge to a fixed point.
problem Understanding convergence properties of convex message passing methods.
method Proving convergence of coordinate descent applied to piecewise-affine convex objectives, and showing this applies to various message passing methods.
result The iterates converge to a fixed point of the method, and the algorithm terminates in a known number of iterations.