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.
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48 results for “Intrinsic Dimension”
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.
Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimationstat.ML
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.
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.
Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. …
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.
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 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.
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.
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.
New research shows that the dimension gap between intrinsic and ambient dimensions affects adversarial vulnerability of machine learning models.
problem The mystery of adversarial attacks on machine learning models.
method Introducing two types of adversarial attacks and proving their relationship to the dimension gap.
result The dimension gap between intrinsic and ambient dimensions makes clean-trained models more vulnerable to off-manifold adversarial perturbations.
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.
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…
Contrastive learning adapts to data intrinsic dimensions, learning low-dimensional representations.
problem Learning high-dimensional representations from multi-modal data.
method Multi-modal contrastive learning with temperature optimization.
result Contrastive learning adapts to intrinsic dimensions of data, not specified dimensions.
New algorithm estimates intrinsic dimension of discrete datasets.
problem Inaccuracies in using continuous methods for discrete datasets.
method Introduced an algorithm to infer intrinsic dimension of discrete spaces.
result Demonstrated accuracy on benchmark datasets and found a small intrinsic dimension in a metagenomic dataset.
Low-dimensional structure in images helps deep learning models generalize better.
problem Understanding the intrinsic dimensionality of images for better model performance.
method Applied dimension estimation tools to popular image datasets and used GANs to manipulate intrinsic dimensionality.
result Natural image datasets have very low intrinsic dimensionality, which aids neural networks in learning and generalizing.
We consider non-parametric estimation and inference of conditional moment models in high dimensions. We show that even when the dimension of the conditioning variable is larger than the sample size , estimation and inference is feasible as long as the distribution of the conditioning variable has small intrinsic…
New method estimates intrinsic dimensionality using angles, not distances.
problem Estimating local intrinsic dimensionality accurately.
method Introduces a new estimator using the distribution of angles between neighbor points.
result New estimator behaves similarly but complementarily to existing measures of intrinsic dimensionality.
New method estimates deep neural network's intrinsic dimension for better generalization.
problem Estimating intrinsic dimension of deep neural networks for generalization.
method Topological data analysis (TDA) and persistent homology (PHD).
result Efficient algorithm to estimate PHD in deep neural networks.
Robust estimation of the intrinsic dimension of data sets with quantum cognition machine learningstat.ML
Estimates intrinsic dimension of data sets robustly to noise.
problem Estimating intrinsic dimension of noisy data sets.
method Quantum Cognition Machine Learning for data representation and spectral gap detection.
result Robust estimation of intrinsic dimension in the presence of Gaussian noise.
Paper examines global Covid-19 data complexity and finds low intrinsic dimensions.
problem Understanding the complexity of Covid-19 data across countries.
method Used a Bayesian mixture model (Hidalgo) to estimate intrinsic dimensionality.
result Covid-19 data projects onto two low-dimensional manifolds without significant loss of information.
Paper improves deep learning convergence rates for low-dimensional data.
problem Sub-optimal rates in deep learning due to unrealistic assumptions on intrinsic dimension.
method Introduced an entropic notion of intrinsic dimension for exponential families and demonstrated improved convergence rates.
result Test error scales as , improving on best-known rates.
New infinite family of 2-complexes intrinsically linked in 4D.
problem Intrinsic linking of 2-complexes in 4D.
method Examining suspensions of graphs containing K6 as a minor.
result Embeddings of suspensions contain intrinsically linked cycles.
Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer th…
Estimating the effective dimension of large biological datasets using Fisher separability analysiscs.LG
Modern large-scale datasets are frequently said to be high-dimensional. However, their data point clouds frequently possess structures, significantly decreasing their intrinsic dimensionality (ID) due to the presence of clusters, points being located close to low-dimensional varieties or fine-grained lumping. We test a…
The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.
problem The manifold hypothesis fails to capture the structure of image data.
method Empirical verification of the union of manifolds hypothesis on image datasets.
result Image data lies on a disconnected set with varying intrinsic dimensions.
Develops deep generative models for stratified learning.
problem Challenges in learning distributions on stratified spaces with varying dimensions and singularities.
method Two generative frameworks: sieve maximum likelihood and diffusion-based.
result Establishes convergence rates and consistency for estimating intrinsic dimensions and number of strata.
For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…
New score matching method estimates local intrinsic dimension efficiently.
problem Quantifying the local intrinsic dimension of complex data.
method Denoising score matching loss and equivalent implicit score matching loss.
result Denoising score matching loss is a highly competitive and scalable LID estimator.
The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown support leads to the polynomial correction to the standard maximum likelihood est…
DPA autoencoders learn data distribution and intrinsic dimensionality with guarantees.
problem Learning data distribution and intrinsic dimensionality in unsupervised learning.
method Combines distributionally correct reconstruction with principal-component-like interpretability.
result Exact theoretical guarantees on disentangling factors of variation and intrinsic dimensionality.
The paper proposes a least squares method for binary compressive sampling with low intrinsic dimension signals.
problem Recovering signals from binary measurements with noise and sign flips.
method Least squares decoder for signals with low generative intrinsic dimension.
result The least squares decoder achieves a sharp estimation error of under certain conditions.
The paper examines Wiener process for LID estimation methods.
problem Estimating local intrinsic dimension in high-dimensional datasets.
method Investigates recent LID estimation methods from a Wiener process perspective.
result Explains how methods behave under non-ideal conditions.
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
TOFU-POV tackles partially observed linear bandits, achieving sublinear regret with low-dimensional action vectors.
problem Stochastic linear bandits with partially observed actions in settings like recommendation and healthcare.
method TOFU-POV estimates latent action subspace, imputes missing actions, and runs OFUL in low-dimensional coordinates.
result TOFU-POV achieves regret scaling with intrinsic subspace dimension, improving upon natural baselines.
Adaptive framework for learning latent space dimensions in GANs.
problem Inadequate latent space dimensions lead to poor generative models for complex data.
method Proposes a novel framework (LWGAN) that adaptively learns latent dimensions of data manifolds.
result Proves that the estimated intrinsic dimension is a consistent estimate of the true data manifold dimension.
We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system leads to a remarkable Weingarten type equation for surfaces on hyperbolic 3-spa…
Deep networks can approximate high-dimensional distributions from low-dimensional ones.
problem Approximating high-dimensional distributions from low-dimensional ones.
method Proved neural networks can transform low-dimensional distributions to high-dimensional ones with arbitrary closeness measured by Wasserstein distances and maximum mean discrepancy.
result Upper bounds of the approximation error are obtained in terms of the width and depth of neural network.
Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees…
This paper analyzes deep federated learning for low-dimensional data, revealing intrinsic dimensionality's role in convergence rates.
problem Insufficient investigation of generalization error in heterogeneous federated learning, especially for low-dimensional data.
method Statistical analysis of deep federated regression in a two-stage sampling model.
result Intrinsic dimensionality, characterized by entropic dimension, determines convergence rates for deep learners.
Scale-adaptive and robust intrinsic dimension estimation via optimal neighbourhood identificationstat.ML
This work introduces a protocol to automatically select the correct range of scales for meaningful Intrinsic Dimension estimation.
problem The Intrinsic Dimension (ID) varies with scale in real-world datasets, leading to erroneous results.
method The protocol selects the correct range of scales by ensuring constant density of data points.
result The method provides a robust and scale-adaptive approach to estimating meaningful Intrinsic Dimension.
Adaptive Approximation and Generalization of Deep Neural Network with Intrinsic Dimensionalitystat.ML
In this study, we prove that an intrinsic low dimensionality of covariates is the main factor that determines the performance of deep neural networks (DNNs). DNNs generally provide outstanding empirical performance. Hence, numerous studies have actively investigated the theoretical properties of DNNs to understand thei…
Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.
problem Regularity of boundaries in plentiful groups.
method Lipschitz approximation of boundaries.
result Boundary of almost minimizers can be approximated by intrinsic Lipschitz graphs.
Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionsstat.ML
Diffusion models learn multi-modal distributions with optimal efficiency.
problem Learning high-dimensional distributions with low-dimensional multi-modal structures.
method Score-based diffusion models, focusing on subgaussian distributions within subspaces.
result Diffusion models require samples for 1-Wasserstein error, improving over prior guarantees.
New research shows common ID estimators in neural representations are inaccurate.
problem Inaccurate estimation of intrinsic dimensions in neural representations.
method Theoretical and empirical investigation of ID estimators in neural representations.
result Common ID estimators do not accurately reflect the true underlying ID of neural representations.
Study intrinsic volume forms on complex hypersurfaces.
problem Computing volume functionals on pseudoconvex hypersurfaces.
method Compute first and second variation formulae, explore infinite dimensional aspects.
result Discuss possible analogues of the affine isoperimetric inequality.
Neural networks' performance scales with data size, explained by data manifold dimensionality.
problem Understanding the scaling of neural network performance with the number of parameters.
method Explained by the intrinsic dimension of the data manifold, confirmed through teacher/student framework and various datasets.
result The scaling exponent α is approximately 4 divided by the intrinsic dimension d of the data manifold.
New research shows DDPM can adapt to data's intrinsic low dimensionality efficiently.
problem Theoretical inefficiency of DDPM in high-dimensional data.
method Investigates how DDPM can exploit intrinsic low dimensionality of data.
result Proves DDPM's iteration complexity scales nearly linearly with intrinsic dimension .