Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. …
This survey reviews dimension estimation methods for datasets.
problem Understanding the intrinsic dimension of high-dimensional datasets.
method Categorizes dimension estimation methods by geometric information: tangential, parametric, and topological.
result Many dimension estimation methods may overfit and not generalize well.
CA-PCA improves manifold dimension estimation by accounting for curvature.
problem Estimating the dimension of manifolds in high-dimensional data.
method Develops CA-PCA, a local PCA method calibrated with a quadratic embedding to account for curvature.
result Improves manifold dimension estimation in various settings.
Estimates manifold dimension from random samples.
problem Estimating the dimension of a manifold from random samples.
method Explicit theoretical and heuristic bounds for data set size.
result Data set needs to be sufficiently large for accurate dimension estimation.
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.
Python package for estimating intrinsic dimensionality of datasets.
problem Estimating intrinsic dimensionality for machine learning applications.
method Implementation of various intrinsic dimension estimators in Python.
result Benchmarking of ID estimation methods on real and synthetic data.
Develops a computationally tractable high-dimensional differential privacy estimator.
problem Differential privacy in high dimensions is computationally intractable.
method Combines high-dimensional robust statistics with differential privacy techniques.
result A computationally tractable algorithm with dimension-independent privacy loss.
LIDL estimates local intrinsic dimension in high dimensions.
problem Estimating local intrinsic dimension in high-dimensional data.
method Approximate likelihood using parametric neural density estimation.
result LIDL scales to thousands of dimensions and yields competitive results.
Estimates intrinsic dimension of data for GANs.
problem Estimating intrinsic dimension of high-dimensional data.
method Uses Wasserstein distances for estimation.
result Provides sample complexity bounds for GANs.
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.
The paper corrects biases in estimating intrinsic dimension and differential entropy.
problem Systematic bias in estimating intrinsic dimension and differential entropy.
method A bias-corrected estimator for both measures is proposed, highlighting shared steps and useful consequences.
result Simultaneous estimation of differential entropy and intrinsic dimension provides complementary perspectives on underlying manifolds.
We consider non-parametric estimation and inference of conditional moment models in high dimensions. We show that even when the dimension D of the conditioning variable is larger than the sample size n, estimation and inference is feasible as long as the distribution of the conditioning variable has small intrinsic…
A new method reduces dimensionality for better likelihood-free parameter estimation.
problem Estimating parameters from data with no closed-form likelihood.
method Combines reconstruction map estimation with dimension-reduction techniques.
result The proposed method outperforms existing techniques in accuracy and efficiency.
New robust estimators achieve subgaussian bounds using VC-dimension.
problem Robust estimation of sparse and corrupted data.
method Use of VC-dimension to measure statistical complexity.
result First robust estimators for sparse estimation with subgaussian rate.
This paper reviews SDR methods for multivariate response regression.
problem Handling sufficient dimension reduction for multivariate response regression.
method Characterizes SDR estimators as inverse or forward regression methods.
result Pooled marginal, projective resampling, distance-based, ordinary least squares, partial least squares, and semiparametric SDR estimators are discussed.
Neural estimator improves mutual information estimation in high dimensions.
problem Estimating mutual information in high dimensions is challenging.
method Parametrizing conditional densities with normalizing flows and using block autoregressive structure.
result Improved mutual information estimation on benchmark tasks.
Estimates manifold dimension using local graph structure.
problem Estimating the intrinsic dimension of manifolds from data.
method Regression on local PCA coordinates, focusing on local graph structure.
result Proposed QE and TLS estimators outperform existing methods.
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
eDCF estimates intrinsic dimension using local connectivity.
problem Challenges in estimating intrinsic dimension due to scale dependence.
method eDCF: a novel, scalable, and parallelizable method based on Connectivity Factor (CF).
result eDCF consistently matches leading estimators with comparable MAE and higher exact intrinsic dimension match rates.
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…
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.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Estimates expected information gain using density approximations and dimension reduction.
problem Estimating expected information gain in nonlinear and non-Gaussian settings.
method Flexible transport-based schemes for EIG estimation, optimal sample allocation, and gradient-based upper bounds on mutual information.
result Optimal sample allocation and dimension reduction schemes improve EIG estimation accuracy and convergence rate.
Estimates latent dimensionality for prediction tasks using mutual information.
problem Estimating the latent dimensionality needed for accurate prediction.
method Formulates the problem as an Information Bottleneck question and uses neural mutual information estimators with a hybrid critic to preserve latent geometry.
result The hybrid critic method provides a more accurate estimation of task-relevant dimensionality.
Unified model for interactive estimation with improved learnability measure.
problem Improving learnability in interactive estimation models.
method Introducing a combinatorial measure (dissimilarity dimension) and a general algorithm with polynomial bounds.
result Unified model subsumes statistical-query learning and structured bandits.
Paper studies estimating asset correlations across sectors.
problem Estimating correlations between different asset sectors.
method Separates cross-sectional and time dimensions for estimation.
result Developed method for better asset correlation estimation.
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.
Rdimtools simplifies DR and IDE for high-dimensional data analysis.
problem Discovering patterns in complex high-dimensional data.
method Provides an R package with 133 DR and 17 IDE algorithms.
result Facilitates geometric understanding of high-dimensional data.
Uniform consistency proven for spatial distribution and depth estimators in any dimension.
problem Uniform consistency of spatial distribution and depth estimators in arbitrary dimensions.
method Proof of uniform L1-consistency using sample size n as the only dependency. result Consistency rate is independent of dimension d and sample size n. A new neural network for efficient density estimation.
problem Efficient density estimation for high-dimensional data.
method Triangular neural network implementation of neural autoregressive flow (NAF).
result Achieves state-of-the-art bits-per-dimension indices on MNIST and CIFAR-10.
We derive a priori interior Hessian estimates for the special Lagrangian equation σ2=1 in dimension three.
We propose a new method for estimating the intrinsic dimension of a dataset by applying the principle of regularized maximum likelihood to the distances between close neighbors. We propose a regularization scheme which is motivated by divergence minimization principles. We derive the estimator by a Poisson process appr…
POTD estimates SDR subspace using optimal transport for binary response.
problem Insufficient performance of existing SDR methods for categorical responses.
method Principal optimal transport direction (POTD) using optimal transport coupling.
result POTD exclusively estimates SDR subspace for error-free class labels.
In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…
We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.
New scalable algorithm estimates barycenters of measures in high dimensions.
problem Estimating barycenters of measures in high-dimensional settings.
method Optimizes generative models to estimate barycenters, scaling by introducing inductive biases.
result First scalable method to estimate barycenters in thousands of dimensions.
Efficient method estimates intrinsic dimension for big data.
problem Estimating intrinsic dimension for large datasets is costly and complex.
method Proposes a matrix-vector product-based approach for efficient intrinsic dimension estimation.
result Demonstrates superior performance compared to state-of-the-art methods.
Study on dimensions of Killing vector fields on gradient Ricci solitons.
problem Estimating dimensions of Killing vector fields on gradient Ricci solitons.
method Analyzes the structure of gradient Ricci solitons to estimate dimensions of Killing vector fields.
result Maximal dimension of Killing vector fields on irreducible non-trivial gradient Ricci solitons.
New methods estimate curvature, tangent spaces, and dimension of noisy data.
problem Estimating geometric properties of noisy or sparse data.
method Diffusion geometry tools for Riemannian manifold analysis.
result Significantly outperforms existing methods in noisy or sparse data.
The paper applies potential theory to conformal geometry, proving theorems and dimension estimates.
problem Understanding the behavior of solutions near singularities in conformal geometry.
method Linear and nonlinear potential theory applied to conformal geometry problems.
result Established Huber's type theorems and Hausdorff dimension estimates for conformal geometry.
New method estimates robust mean in high dimensions with minimized outliers.
problem Estimating the mean in high dimensions when a fraction of data is corrupted.
method Formulating the problem as ℓ0-norm minimization under second moment constraints, and using ℓ1 and ℓp minimization techniques. result The proposed method achieves order optimal robust mean estimation and significantly outperforms existing methods.
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
New method estimates Gaussian vector functions more efficiently.
problem Estimating functions of Gaussian vectors with high dimensions.
method Combines randomized dimension reduction and PCA.
result Algorithm outperforms Monte Carlo method by a factor of d.
A typical goal of supervised dimension reduction is to find a low-dimensional subspace of the input space such that the projected input variables preserve maximal information about the output variables. The dependence maximization approach solves the supervised dimension reduction problem through maximizing a statistic…
Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.
problem Estimating the homological dimensions of Riemann surfaces with boundary and marked points.
method Developed an estimate for the rational homological dimension of Riemann surfaces with possible boundary and marked points.
result Provided an estimate for the rational homological dimension of Riemann surfaces with boundary and marked points.
Removes singularities for Yang-Mills-Higgs fields in higher dimensions.
problem Yang-Mills-Higgs fields with isolated singularities.
method Establishes decay estimates and conformally invariant energy bounds.
result Removable singularity theorem for Yang-Mills-Higgs fields.
Improves MARS for nonparametric multivariate regression with dimension reduction.
problem High number of basis functions in MARS for high-order interactions.
method Linear combinations of covariates for dimension reduction, facilitating gradient calculation and eigen-analysis for estimation.
result Asymptotic theory and numerical studies show improved performance over MARS.
Improved Sparse Polyak for high-dimensional M-estimation with sparser solutions.
problem High-dimensional M-estimation problems with potential loss of sparsity and accuracy.
method Variant of Sparse Polyak with optimal thresholding operators.
result Retains desirable scaling properties while achieving sparser and more accurate solutions.