Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
problem Actions of infinite 2-groups of bounded exponent on compact manifolds.
method Analyzing properties of 2-groups of bounded exponent.
result Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
Study shows a specific Carnot group violates a curvature exponent bound.
problem Understanding the curvature exponent in step-two Carnot groups.
method Examined convergence of Lie algebra structure constants.
result Found a Carnot group where curvature exponent bound is violated.
New bounds on geodesic dimension and curvature exponent in Carnot groups.
problem Characterizing geodesic dimension and curvature exponent in Carnot groups.
method Characterization and lower bound calculation for geodesic dimension and curvature exponent.
result Found an example where curvature exponent is greater than geodesic dimension.
New bounds link generalization to stochastic optimizer's lower tail exponents.
problem Understanding the impact of stochastic optimization algorithms on generalization in non-convex settings.
method Proves novel bounds linking generalization to the lower tail exponent of the transition kernel of stochastic optimizers, both discrete- and continuous-time.
result Empirical results show correlations between generalization error and lower tail exponents.
Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.
problem Asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces with ADE singularities.
method Investigates a function on the unit disc defined by fiber integrals of the forms with a smooth test function, showing a lower bound of Hölder exponent at the origin for both cscK-metrics and Ricci-flat metrics.
result Shows bounds of Hölder exponent for both cscK-metrics and Ricci-flat metrics.
Random subgroups in hyperbolic spaces have full limit sets and bounded critical exponents.
problem Understanding stationary random subgroups in hyperbolic spaces.
method Analyzing limit sets and critical exponents of random subgroups.
result Random subgroups have full limit sets and bounded critical exponents.
Extends Nash-Kuiper theorem to higher Hölder exponents.
problem Constructing isometric immersions beyond Borisov's exponent.
method Novel corrugation ansatz, integration by parts, and algebraic decomposition.
result Flexibility of C1,α isometric immersions beyond Borisov's exponent. We use a straightforward variation on a recent argument of Hezari and Rivière~\cite{HR} to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=n−12(n+1). We are able to this directly by just using the Lp-bounds for spectral projection operators from our …
In this paper, we study the Kurdyka-Łojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In …
A new flow connects manifold invariants with critical exponents.
problem Understanding invariants of non-positively curved manifolds.
method Constructing the natural flow and relating it to the critical exponent.
result Established connections between manifold invariants and critical exponents.
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
We consider the evolution of scale-free networks according to preferential attachment schemes and show the conditions for which the exponent characterizing the degree distribution is bounded by upper and lower values. Our framework is an agent model, presented in the context of economic networks of trades, which shows …
Paper analyzes error exponent in agnostic PAC learning.
problem Analyzing performance of agnostic PAC learning.
method Using error exponent from Information Theory to analyze PAC learning.
result Improved distribution-dependent error exponent for agnostic learning.
We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k exponents is greater or equal than the sum of the degree of any rank k holomorphic s…
We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in P(Rn)×P(Rn∗) is bounded between two critical exponents associated respe…
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…
New separation concepts for Anosov representations help bound Thurston asymmetric metric.
problem Understanding diverging families of Anosov representations.
method Introducing separation concepts and analyzing combinatorial invariants.
result Critical exponent asymptotic to a graph invariant.
Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
If (M,g) is a compact Riemannian manifold of dimension n≥2 we give necessary and sufficient conditions for improved Lp(M)-norms of eigenfunctions for all 2<p=pc=n−12(n+1), the critical exponent. Since improved Lpc(M) bounds imply improvement all other exponents, these conditions are nece…
Paper studies distributed learning with limited communication bits, achieving optimal error exponents.
problem Distributed hypothesis testing with constant communication bits.
method Geometric approach in distribution spaces, encoding empirical distributions to transmission bits.
result Optimal achievable error exponents and coding schemes for various communication constraints.
Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy Trivial solution proof for heat equation on certain manifolds.
problem Proving trivial solutions for semilinear heat equations on specific manifolds.
method Analyzing pointwise monotonicity and boundedness over time.
result Trivial solutions exist only for certain values of p.
In many machine learning applications, crowdsourcing has become the primary means for label collection. In this paper, we study the optimal error rate for aggregating labels provided by a set of non-expert workers. Under the classic Dawid-Skene model, we establish matching upper and lower bounds with an exact exponent …
In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.
New bounds on mapping degrees for geometric 3-manifolds.
problem Bounding the mapping degree in terms of Lipschitz constant for geometric 3-manifolds.
method Constructing Legendrian maps to prove bounds on flexible exponent.
result Complete result for flexible exponent of geometric 3-manifolds.
Given a convex representation ρ:Γ→PGL(d,R) of a convex co-compact group Γ of Hk we find upper bounds for the quantity αhρ, where hρ is the entropy of ρ and α is the Hölder exponent of the equivariant map ∂Γ→P(Rd). We also give rigidity statemen…
New convergence bounds for online learning with heavy-tailed noise.
problem Learning on streaming data with heavy-tailed noise.
method Nonlinear stochastic gradient descent (SGD) for non-convex and strongly convex costs.
result Strong convergence rates for various nonlinearities and noise distributions.
Study on curvature bounds and geodesic dimension in sub-Finsler Heisenberg groups.
problem Investigate synthetic curvature-dimension bounds in sub-Finsler geometry.
method Examine measure contraction property and geodesic dimension on Heisenberg groups with ℓp-sub-Finsler norms. result For p∈(2,∞], ℓp-Heisenberg group fails to satisfy any measure contraction property. For p∈(1,2), it satisfies MCP(K,N) under specific conditions. Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.
problem Existence of least energy solutions for nonlinear p-Laplacian problems with critical exponent.
method Proving existence of solutions through critical point theory and variational methods.
result Significant difference in existence results between p-Laplacian and Laplacian cases.
The paper analyzes error bounds and KL properties for noisy matrix recovery problems.
problem Noisy low-rank matrix recovery problems.
method Squared F-norm regularization, accelerated alternating minimization method.
result Established error bounds and KL properties for critical points and global minimizers.
We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …
We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.
New groups found with critical exponents close to but less than max.
problem Finding discrete isometry groups with critical exponents near maximum.
method Analyzing complex hyperbolic spaces to construct groups.
result Discrete isometry groups with critical exponents arbitrarily close to max but less.
Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of 21 for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. I…
Study f-Laplace bounds on gradient Ricci shrinkers, applying to Betti numbers.
problem Bounding eigenvalues of f-Laplacian on gradient Ricci shrinkers. method Upper and lower bounds established using volume growth rate; extends to vector bundles.
result Explicit upper bounds for Betti numbers derived.
Study on heat content for domains with fractal boundaries.
problem Analyzing short-time asymptotics of heat content for domains with fractal boundaries.
method Developing mathematical analysis on de Gennes' hypothesis and exploring fractal curvatures.
result Fractal curvatures and their scaling exponents may emerge in the short-time heat content asymptotics of domains with fractal boundaries.
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of PO(p,q+1) introduced by Danciger, Guéritaud and Kassel, called Hp,q-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and…
In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and genera…
Study of deep neural networks using finite-time Lyapunov exponents.
problem Understanding the geometric structures in input space formed by deep neural networks.
method Analogy with dynamical systems, computing finite-time Lyapunov exponents.
result Ridges of large positive exponents divide input space into regions associated with different classes.
In this note, we study the ultimate ruin probabilities of a real-valued L{é}vy process X with light-tailed negative jumps. It is well-known that, for such L{é}vy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to …
Adaptive algorithm identifies best arm with abstention, showing phase transition from polynomial to exponential error probability.
problem Bayesian best-arm identification with abstention to reduce undetected error.
method Adaptive algorithm PGWS that optimally uses abstention budget.
result Introducing any positive abstention budget induces an exponential decay in undetected error probability.
Study proves boundedness of operators in variable exponent Morrey spaces.
problem Boundedness of operators in global Morrey-type spaces with variable exponents.
method Analysis of Hardy-Littlewood maximal operator and potential type operator in variable exponent Morrey spaces.
result Boundedness of the Hardy-Littlewood maximal operator and potential type operator in global Morrey-type spaces with variable exponents.
Constructs free semigroups with critical exponents close to but less than ambient groups.
problem Creating free semigroups with critical exponents close to but less than ambient groups.
method Constructing finitely generated free subsemigroups with specific properties.
result Free semigroups with critical exponents arbitrarily close to but strictly less than ambient groups.
Proves critical exponent for Θ−positive representations in discrete subgroups.
problem Determining the critical exponent for Θ−positive representations. method Analyzes discrete subgroups Γ⊂PSL(2,R) and their geometric properties. result Equality of critical exponent holds if and only if Γ is a lattice for geometrically finite Γ. We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family.…
Lower bound found for volumes of modular link complements.
problem Finding a lower bound for the volumes of modular link complements.
method Analyzing the volumes of link complements associated with geodesics in the modular surface.
result First linear lower volume bound in terms of exponents of code words.
Study critical exponents in normal subgroups of higher rank Lie groups.
problem Understanding critical exponents in normal subgroups of higher rank Lie groups.
method Analyzing subgroups and their critical exponents in a higher rank semi-simple Lie group.
result Critical exponents of normal subgroups coincide under certain conditions.