Given a metric space of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal if there is a linear dimension function in this dimension. We prove that if is a tree-graded space …
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48 results for “dimension functions”
Functional dimension varies in ReLU networks, with implications for symmetry and connectivity.
problem Understanding the functional dimension of ReLU neural networks.
method Careful definition and analysis of functional dimension, study of quotient space and fibers.
result Functional dimension is inhomogeneous and can be non-constant, with implications for symmetry and connectivity.
New insights on eluder dimension for function approximation in machine learning.
problem Complexity measure for online bandits and reinforcement learning with function approximation.
method Study the relationship between eluder dimension and generalized rank for different activation functions.
result Eluder dimension can be exponentially smaller or larger than generalized rank depending on the activation function.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the -Laplacian, proving estimates under curvature assumptions.
result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
Study representation rings and dimension functions for fusion systems.
problem Understanding the structure of representations in fusion systems.
method Define representation ring, use Grothendieck ring, study dimension functions via transfer map.
result Find conditions for dimension functions of stable representations.
Estimates fat-shattering dimension of aggregated function classes.
problem Understanding the complexity of aggregated function classes.
method Analyzes fat-shattering dimension of -fold aggregations of real-valued function classes.
result Provides upper and lower bounds on fat-shattering dimension for linear and affine function classes.
Proves strong Morse inequalities for area functional in low dimensions.
problem Proving Morse inequalities for area functional in specific dimensions.
method Analyzes area functional in codimension one, proving inequalities under given dimension constraints.
result Strong Morse inequalities for area functional in specified dimensions.
Paper infers intrinsic dimension from quasi-convex measurements.
problem Inferring intrinsic dimension from measurements by quasi-convex functions.
method Developed a method using filtration of Dowker complexes based on discrete data of point orderings.
result Correct intrinsic dimension can be inferred in the limit of large data under generic assumptions.
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.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for -harmonic functions in manifolds with curvature conditions.
result Established a quantitative second order Sobolev estimate for -harmonic functions.
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions bel…
We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than , then it grows at least as fast as a linear function. This generalizes a resu…
Gutkin billiard tables studied in higher dimensions, rigidity proven.
problem Characterizing billiard tables with constant angle invariants.
method New generating function for billiards, rigidity proof.
result In higher dimensions, only spheres have Gutkin billiard tables with constant angle invariants.
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.
Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.
problem No specific problem stated; extension of functional.
method Extension to Calabi-Yau manifolds of arbitrary dimension.
result Extension of the Kodaira Spencer functional.
Projection pursuit model improves Gaussian process regression for high-dimensional data.
problem Scalability issues with traditional Gaussian process models in high dimensions.
method Additive Gaussian process regression with dimension expansion and gradient descent.
result The proposed method approximates more complex functions and outperforms traditional models.
Characterizes learnability of forgiving 0-1 loss functions in multiclass settings.
problem Understanding when multiclass learning with forgiving 0-1 loss functions is possible.
method Introduces a new combinatorial dimension based on Natarajan Dimension to determine learnability.
result A hypothesis class is learnable if and only if the Generalized Natarajan Dimension is finite.
Develops robust methods for infinite-dimensional stochastic processes.
problem Measuring covariations in stochastic evolution equations in infinite dimensions.
method Asymptotic theory for jump robust measurement of covariations.
result Identifies scaling limits for realized covariations.
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
Composition of Fractalsmath.MG
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with segments of equal size, the composite dimension can be expressed as a function …
Locally Sasakian Manifoldsmath.DG
We show that every Sasakian manifold in dimension is locally generated by a free real function of variables. This function is a Sasakian analogue of the Kähler potential for Kähler geometry. It is also shown that every locally Sasakian-Einstein manifold in dimensions is generated by a locally Kähler-…
Estimates dimension of subsets from random samples, proving consistency.
problem Estimating the dimension of a compact subset from random samples.
method Consistency proofs for Minkowski, correlation, and pointwise dimensions using empirical volume function.
result Statistical consistency of estimators for various dimension notions.
Smooth manifolds have functions with exactly two critical values.
problem Characterizing manifolds with specific Reeb functions.
method Proving existence of Reeb functions with prescribed critical values.
result Characterization of manifolds in dimensions 3 and n≥5 using Reeb functions.
Higher order Yang-Mills flowmath.DG
We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of…
Estimates box dimension of fractal interpolation surfaces using oscillation vectors.
problem Estimating the complexity of fractal interpolation surfaces.
method Defined vertical scaling matrices and used them to relate oscillation vectors of different levels.
result Obtained the box dimension of generalized affine fractal interpolation surfaces.
Reduces function approximation dimensions from high to low with sparse data.
problem Function approximation from sparse data.
method Nonlinear Level Set Learning (NLL) with geometric information.
result Reduces input dimension to theoretical lower bound with minor accuracy loss.
Study extends GNN VC dimension bounds to Pfaffian activation functions.
problem Bounding GNN VC dimension for new activation functions.
method Pfaffian function theory applied to GNNs with sigmoid and hyperbolic tangent activations.
result Bounds on GNN VC dimension for various architectures and graph properties.
A result is given to find points where a real valued function on the plane is not smooth. Provided this function is induced by a smooth mapping from three dimensions to the plane, from a function on surfaces in three dimensions. This has applications to numerical methods such as image processing.
Paper studies CR holomorphic functions in Sasakian manifolds.
problem Sharp dimension estimate of CR holomorphic functions in Sasakian manifolds.
method Focuses on CR analogue of Yau's uniformization conjecture in Sasakian manifolds.
result Establishes the sharp dimension estimate of CR holomorphic functions.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an -rectifiable measure associated with a stationary varifold.
It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…
This paper tackles high-dimensional Bayesian optimization using supervised dimension reduction.
problem Challenges in extending Bayesian optimization to high dimensions.
method Introduces Sliced Inverse Regression (SIR) for high-dimensional Bayesian optimization.
result Demonstrates computational benefits and theoretical regret bounds for high-dimensional Bayesian optimization.
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
Random Function Descentmath.OC
Random Function Descent improves optimization in high dimensions.
problem Lack of effective optimization methods in high-dimensional spaces.
method Introducing a 'random function' framework to optimize classical optimization problems.
result Random Function Descent (RFD) is a scalable optimization method that bridges Bayesian and classical optimization.
Unified method for computing modular curvature on toric noncommutative manifolds using hypergeometric functions.
problem Computing modular curvature on toric noncommutative manifolds.
method Unified pseudo-differential calculus and hypergeometric functions.
result Explicit expressions for spectral functions of modular curvature.
This research sets limits on how complex multi-class learning problems can be.
problem Understanding the complexity of multi-class classification problems.
method Established upper bounds on Natarajan dimensions for specific function classes.
result Upper bounds on Natarajan dimensions for multi-class decision trees, random forests, and neural networks.
Study shows skein module dimensions for surface times circle.
problem Understanding skein modules of surface times circle.
method Used Kauffman bracket skein module over rational functions.
result Dimension at least 2^{2g+1}+2g-1 for Sigma x S^1.
Paper introduces a nonparametric functional graphical model for random functions.
problem Estimating probabilistic conditional independence in functional graphical models.
method Functional sufficient dimension reduction to relax Gaussian or copula Gaussian assumptions.
result Enhances estimation accuracy and retains probabilistic conditional independence.
Functional inequality proves quasi-invariance in infinite dimensions.
problem Proving quasi-invariance of measures in infinite-dimensional spaces.
method Using a functional inequality to prove quasi-invariance of measures under group actions.
result Different proof of the Cameron-Martin theorem in infinite dimensions.
The paper establishes estimates for heat equations and harmonic functions on metric spaces with curvature-dimension condition.
problem Analyzing geometric properties of metric measure spaces with curvature-dimension condition.
method Establishing local Li-Yau estimates and proving sharp Yau's gradient estimates for heat equations and harmonic functions.
result Sharp Li-Yau and gradient estimates for weak solutions of heat equations and harmonic functions on spaces.
New algorithms achieve decision calibration without sample complexity dependent on feature dimension.
problem Achieving decision calibration for nonlinear loss functions with polynomial sample complexity.
method Developed smooth relaxation of decision calibration, enabling dimension-free algorithms.
result Efficient algorithms post-process predictors to satisfy decision calibration without worsening accuracy.
Minimum width for ReLU networks to approximate L^p functions is max(d_x+1, d_y).
problem Characterizing the minimum width for ReLU networks to approximate L^p functions.
method Analyzing networks with ReLU activation functions and proving the minimum width required.
result The minimum width required for the universal approximation of L^p functions is exactly max(d_x+1, d_y).
Two novel methods estimate multiple FDR directions for binary categorical responses.
problem Estimating multiple FDR directions for categorical responses.
method Information maximization and square loss mutual information.
result Statistical consistency of the proposed methods established.
Extends Gauduchon's result to higher dimensions, showing balanced metrics.
problem Understanding critical metrics in higher-dimensional Hermitian manifolds.
method Analyzes the functional of -norm of torsion -form and full Chern torsion.
result Critical metrics are balanced in all dimensions.
Eluder dimension and information gain are equivalent for reproducing kernel Hilbert spaces.
problem Complexity measures in bandit and reinforcement learning.
method Equivalence of eluder dimension and information gain for reproducing kernel Hilbert spaces.
result Eluder dimension and information gain are equivalent for reproducing kernel Hilbert spaces.
Study robust regression learning under adversarial attacks.
problem Understanding which function classes are learnable in the presence of adversarial attacks.
method Introduced a novel agnostic sample compression scheme and used fat-shattering dimension to construct adversarially robust sample compression schemes.
result Finite fat-shattering dimension classes are learnable in both realizable and agnostic settings.
Researchers found solutions to minimal surface equations in 6D.
problem Equations of minimal surface type in six dimensions.
method Constructed nonlinear entire solutions using parametric elliptic functionals.
result Found solutions to minimal surface equations in 6D.