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 “functional dimension”
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.
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.
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.
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…
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…
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.
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.
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.
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.
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.
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.
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 …
We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation leads to a unified (works for arbitrary dimension) method of computing the modula…
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-…
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 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.
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.
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.
We define the representation ring of a saturated fusion system as the Grothendieck ring of the semiring of -stable representations, and study the dimension functions of -stable representations using the transfer map induced by the characteristic idempotent of . We find a…
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.
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.
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.
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…
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.
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.
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.
The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…
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).
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.
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.
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.
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.
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.
Gradient span algorithms show consistent progress in high dimensions.
problem Understanding consistent training progress in large machine learning models.
method Proving deterministic behavior of gradient span algorithms on Gaussian random functions.
result Gradient span algorithms have asymptotically deterministic behavior in high dimensions.
Study maximizes eigenvalues in dimensions 3 and above.
problem Maximizing the k-th eigenvalue functional over measures on Riemannian manifolds.
method Generalizes previous work on first eigenvalue to higher dimensions, proving optimal bounds on singular set dimensions.
result Optimal upper bound for Hausdorff dimension of singular set is m-7.
New methods for functional data analysis improve manifold methods for continuous data.
problem Challenges in evaluating embeddings for functional data.
method Transfer manifold methods from tabular and image data to functional data, define a theoretical framework, and propose nuanced evaluation strategies.
result Manifold methods can be successfully applied to functional data, but careful evaluation is needed.
We prove that the space of -places of the field of rational functions of two variables with coefficients in a totally Archimedean field has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ M(K(x,y))=2$ and the cohomological dimension for any Abelian 2-di…
A diffusion model estimates data manifold dimension by tracking likelihood increases.
problem Estimating the intrinsic dimension of data manifolds.
method Trained diffusion model approximates score function, revealing manifold directionality.
result Diffusion model provides an approximation of the tangent space's dimension.
Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.
problem Verifying scalar-flatness for critical metrics in specific dimensions.
method Analyzing complete Riemannian manifolds with critical metrics of the -scalar curvature functional.
result The conjecture that all complete noncompact critical metrics with finite energy are scalar-flat is confirmed for dimensions 5 to 9.
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the g…