Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.
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48 results for “dimension bound”
Improved bounds on combining hypothesis classes for binary functions.
problem Understanding how to combine hypothesis classes for binary functions.
method Established upper bounds on Littlestone and threshold dimensions for combined classes.
result Upper bounds are nearly tight and give exponential improvements.
The paper aims to develop new combinatorial dimensions for bounded memory learning.
problem Characterize bounded memory learning using combinatorial dimensions.
method Proposes a candidate solution based on the SQ dimension of neighboring distributions and proves upper and lower bounds.
result Characterizes bounded memory learning in a specific parameter regime, matching equivalence between bounded memory and SQ learning.
The paper introduces gapped scale-sensitive dimensions to improve learning rate bounds.
problem Improving lower bounds on rates of convergence in statistical and online learning.
method Introducing and analyzing gapped scale-sensitive dimensions for function classes.
result Gapped dimensions lead to stronger lower bounds on offset Rademacher averages.
Lower bounds on cone density for nontrivial complements in low dimensions.
problem Finding density limits for minimal cones with nontrivial complements.
method Proving lower bounds on cone density for cones of dimensions less than seven with nontrivial complements.
result Established lower bounds on cone density for minimal cones with nontrivial complements in dimensions less than seven.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
Geodesics with bounded angles have zero Hausdorff dimension.
problem Understanding the geometric properties of geodesics with bounded angles.
method Analyzing the Hausdorff dimension of geodesics with specific angle constraints.
result The set of geodesics with bounded self-intersection angles has a Hausdorff dimension of zero.
Sharp bound on singular set dimension for specific geometric problems.
problem Hausdorff dimension of singular set in free boundary problems.
method Analysis of noncollapsed limits of manifolds with Ricci curvature bounds.
result Dimension bound of singular set is .
Eigenfunction maps embed manifolds with a dimension bound.
problem Bounding the embedding dimension of Riemannian manifolds.
method Analyzing Laplacian eigenfunction maps and their embedding properties.
result The maximal embedding dimension is bounded by geometric properties.
Study provides bounds for estimating intrinsic dimension using Gaussian kernels.
problem Estimating intrinsic dimension from data.
method Finite-sample concentration and anti-concentration bounds for Gaussian kernel sums.
result Explicit dependence on sample size, bandwidth, and geometric parameters.
Paper introduces new bounds linking data compressibility to generalization error.
problem Establishing data-dependent generalization bounds.
method Variable-size compressibility framework linking generalization error to compression rate of input data.
result New bounds depend on empirical data measure, subsuming existing PAC-Bayes and intrinsic dimension bounds.
New bounds for adaptive control in high dimensions without fixed state space.
problem Adaptive control of linear systems in high or infinite dimensions.
method Novel perturbation bound for certainty equivalence, scaling with prediction error.
result First regret bounds for LQR in infinite dimensional systems, independent of ambient dimension.
New bounds found for vertices of hyperbolic polyhedra in dimensions 5 to 12.
problem Determining minimum number of ideal and finite vertices in hyperbolic polyhedra.
method Geometric method of orthogonal gluings combined with double counting and recurrence relations.
result Improved lower bounds for vertices in all dimensions up to 12.
The paper extends Lorentzian splitting theorems under curvature-dimension bounds.
problem Analyzing Lorentzian spacetimes with curvature-dimension bounds.
method Using Bakry-Émery-Ricci tensor, extending singularity and splitting theorems.
result Theorems hold for all synthetic dimensions, including negative and positive.
Bound on curve complex's asymptotic dimension.
problem Understanding the complexity of curve complexes.
method Linear bound in surface complexity.
result Bound on asymptotic dimension and capacity dimension.
Bounds on conformal dimension for certain Coxeter group boundaries.
problem Determining conformal dimensions for specific Coxeter group boundaries.
method Quasi-isometric embeddings and geometric actions on CAT(-1) spaces.
result Upper and lower bounds on conformal dimension for specified Coxeter groups.
The paper proves curvature-related dimension bounds for manifolds.
problem Proving dimension bounds for manifolds with positive scalar curvature.
method Using asymptotic cones and linear growth harmonic functions.
result Upper bounds on essential and Hausdorff dimensions of manifolds.
In 2D, synthetic curvature notions match.
problem Matching synthetic curvature notions in 2D.
method Synthetic curvature notions in 2D.
result Synthetic curvature notions match in 2D.
Bounds on VC dimension for 1NN classifiers with fixed prototype sets.
problem No theoretical results for VC dimension of 1NN classifiers with fixed size prototype sets.
method Collected and used relevant theoretical results to provide explicit lower and upper bounds.
result Explicit lower and upper bounds for VC dimension of 1NN classifiers with fixed prototype set size.
New bounds on asymptotic dimension for elementary amenable groups.
problem Bounding the asymptotic dimension of box spaces for elementary amenable groups.
method Using subadditivity of asymptotic dimension in group extensions and properties of box spaces.
result For elementary amenable groups, the asymptotic dimension of box spaces is bounded by their Hirsch length, with equality for a specific subclass.
Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the generalization capacity of learning algorithms. However, apart from a few special cases, it is hard or impossible to calculate analytically. Vapnik et al. [10] proposed a technique for estimating the VC dimension empirically. While their approach behave…
New bounds on singularities for flow in higher dimensions.
problem Understanding singularities in higher-dimensional flows.
method Entropy and codimension bounds for generic singularities.
result Uniform entropy and dimension bounds for singularities.
The study proves curvature bounds for hyperkähler manifolds.
problem Proving curvature invariants of hyperkähler manifolds.
method Analytical proof in complex dimension four, experimental proof in higher dimensions, verification for known manifolds.
result The conjectured curvature invariants are proven to be positive/negative for all known hyperkähler manifolds up to dimension eight.
Gromov's tori are optimalmath.DG
Optimal bounds found for torus curvatures in high dimensions.
problem Finding optimal bounds on normal curvatures of tori.
method Analyzing immersed n-torus in a Euclidean ball of large dimension.
result Optimal bounds on normal curvatures of tori established.
We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As a…
Sharp bound on cohomology of quasiregularly elliptic manifolds.
problem Bounding cohomological dimension of manifolds with quasiregular mappings.
method Using combinatorial and geometric properties of cohomology, proving a sharp upper bound.
result A sharp upper bound on the cohomological dimension of manifolds, answering Gromov's question.
Groups act on Euclidean buildings with a minimum dimension bound.
problem Understanding the minimum dimension a group can act on properly.
method Analyzing actions on Euclidean buildings and S-arithmetic groups.
result Action dimension is at least twice the building's dimension.
New lower bounds for sampling from log-concave distributions in higher dimensions.
problem Proving lower bounds for sampling from log-concave distributions in higher dimensions.
method Multiscale construction inspired by geometric measure theory and reduction to block Krylov algorithms.
result Query lower bounds for sampling from log-concave distributions in higher dimensions are established.
Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.
New lower bounds on embedding dimensions for neural network architectures.
problem Ensuring neural networks can handle symmetries like permutations in high dimensions.
method Novel technique to prove lower bounds on embedding dimensions.
result Proves new lower bounds on embedding dimensions for Deep Sets and Janossy pooling.
We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the in…
Improved bound on groups preserving Clifford parallelism in 4 dimensions.
problem Characterizing Clifford parallelism by automorphisms.
method Improving the bound on the dimension of groups preserving Clifford parallelism.
result Improved bound to 4 dimensions for groups preserving Clifford parallelism.
Upper bounds on topological dimensions of certain graph boundaries.
problem Calculating topological dimensions of specific graph boundaries.
method Linear upper bounds in terms of rank.
result Linear bounds on topological dimensions, dependent on rank.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
Optimal bounds found for ancient caloric functions on manifolds.
problem Bounding the dimension of ancient caloric functions on manifolds with polynomial volume growth.
method Analyzing polynomial growth and using Yau's conjecture for harmonic functions.
result Sharp bound for the dimension of ancient caloric functions on spaces where Yau's conjecture holds.
Metric tangent cones have consistent dimension in limit spaces with bounded Ricci curvature.
problem Consistent dimension of tangent cones in spaces with lower Ricci curvature bounds.
method Analysis of limit spaces and geodesics, using Hölder continuity and measured Gromov-Hausdorff topology.
result Metric tangent cones are Hölder continuous and have the same dimension.
This paper proves a quadratic upper bound for the recursive teaching dimension of finite VC classes.
problem Prove an upper bound for the recursive teaching dimension of finite VC classes.
method Analyzing the relationship between recursive teaching dimension and VC dimension.
result Showed a quadratic upper bound of O(d^2) for RTD of finite VC classes.
Quantifies scalar curvature under convergence, proving a refined version in all dimensions.
problem Proving a refined quantitative bound for scalar curvature under convergence.
method Established the refined quantitative bound in all dimensions using smoothing techniques.
result Established the refined quantitative bound for scalar curvature in all dimensions.
Paper resolves open problems on VC-dimension bounds.
problem VC-dimension bounds for set systems.
method Analyzes two fundamental set systems: k-fold unions/intersections of half-spaces and simplices.
result Resolves two open problems in machine learning and geometry.
In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
Graphs on surfaces have a 2-dimensional large scale structure.
problem Understanding the large scale structure of graphs on surfaces.
method Proving asymptotic dimension for specific graph classes and surfaces.
result Graphs on surfaces have an asymptotic dimension of 2.
New method proves dimension-free convergence for ULD in KL divergence.
problem Polynomial scaling of existing convergence guarantees in high dimensions.
method Refined KL local error framework, focusing on tr(H) instead of d.
result First dimension-free KL divergence bounds for discretized ULD.
Upper bound on singular set dimension for area-minimizing currents.
problem Bounding the dimension of singular points in area-minimizing currents.
method Using upper Minkowski dimension and properties of blow-up scales.
result Upper Minkowski bound of for the interior singular set.
New bounds found for certain types of algebraic varieties.
problem Bounding properties of algebraic varieties with specific invariants.
method Analyzing the boundedness of -Fano varieties with controlled anti-canonical degrees and alpha-invariants.
result Fixed-dimensional -Fano varieties with controlled invariants form a bounded family.
This work proves lower bounds on a greedy teaching set construction algorithm.
problem Characterize the best-case teaching dimension of a concept class.
method A greedy algorithm that iteratively adds points to the teaching set to restrict the concept class the most.
result Lower bounds on the performance of the greedy approach for small k, extending up to k ≤ c*d for small constant c.
Quantitative estimate of diameter for weighted manifolds under integral curvature bounds and -rangemath.DG
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.