arXiv research

Digital trees have approximate fixed point property, and conditions for products are explored.

Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.

A faster method for estimating effects in large data using fixed-point trees.

This paper studies fixed points of graph selfmaps and iwip endomorphisms of free groups.

We construct examples of finitely generated groups L that have non-trivial actions on $\mathbb{R}$-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have a fixed point-free action on some simplicial tree.

It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there is no global fixed point, then $G$ has a non-trivial action on a simplcial $\R$…

A number of problems in statistical physics and computer science can be expressed as the computation of marginal probabilities over a Markov random field. Belief propagation, an iterative message-passing algorithm, computes exactly such marginals when the underlying graph is a tree. But it has gained its popularity as …

We consider the class non-surjective irreducible endomorphisms of the free group $F_n$. We show that such an endomorphism $φ$ is topologically represented by a simplicial immersion $f:G \rightarrow G$ of a marked graph $G$; along the way we classify the dynamics of $\partial φ$ acting on $\partial F_n$: there are at mo…

L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.

Convex message passing algorithms converge to a fixed point.

Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, th…

EM algorithm converges to global max in latent Gaussian tree models.

New method solves tree-structured Schrödinger Bridge problems.

Stable commutator length scl_G(g) of an element g in a group G is an invariant for group elements sensitive to the geometry and dynamics of G. For any group G acting on a tree, we prove a sharp bound scl_G(g)>=1/2 for any g acting without fixed points, provided that the stabilizer of each edge is relatively torsion-fre…

A new method infers causal structures and generates data without DAGs.

We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that -- …

The sum-product or belief propagation (BP) algorithm is a widely used message-passing technique for computing approximate marginals in graphical models. We introduce a new technique, called stochastic orthogonal series message-passing (SOSMP), for computing the BP fixed point in models with continuous random variables.…

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank latti…

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp…

Finding the most probable assignment (MAP) in a general graphical model is known to be NP hard but good approximations have been attained with max-product belief propagation (BP) and its variants. In particular, it is known that using BP on a single-cycle graph or tree reweighted BP on an arbitrary graph will give the …

The paper classifies circle actions on 6D manifolds with isolated fixed points.

Groups with special properties always have fixed points.

Let $Γ$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $Γ$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $δ(C)$ of C satisfies $δ(C) \geq n-2$. The equality happens if …

Quantized neural networks can represent all fixed-point functions under certain conditions.

Study circle actions on unitary manifolds with discrete fixed points.

Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.

New proof for 6D symplectic manifold with 4 fixed points.

The paper highlights issues with fixed point claims in digital images.

The paper highlights issues in fixed point claims in digital topology.

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 $S^1$-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.

The paper introduces fixed-point centralities for networks and graphons.

Improved convergence of fixed-point methods using windowed Anderson acceleration.

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.

Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.

Incorrect fixed point assertions in digital topology are discussed.

The paper addresses 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 $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

Fixed point assertions in digital topology are often incorrect or poorly stated.

Incorrect fixed point assertions in digital topology are discussed.

This paper provides a new proof of the Lefschetz fixed point formula using groupoids.

Classifies circle actions on 6D manifolds with 4 fixed points.

Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geode…

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…

Study fixed-point sets of $S^{1}$-actions on quaternionic manifolds.

This systemic risk paper introduces inhomogeneous random financial networks (IRFNs). Such models are intended to describe parts, or the entirety, of a highly heterogeneous network of banks and their interconnections, in the global financial system. Both the balance sheets and the stylized crisis behaviour of banks are …