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. …
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48 results for “Intrinsic dependence”
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.
Most exact methods for k-nearest neighbour search suffer from the curse of dimensionality; that is, their query times exhibit exponential dependence on either the ambient or the intrinsic dimensionality. Dynamic Continuous Indexing (DCI) offers a promising way of circumventing the curse and successfully reduces the dep…
The mathematical problem concerning intrinsic storage optimisation is formulated and solved by means of variational analysis. The solution, though obtained in implicit form, still sheds light on many important features of the optimal exercise strategy. It is shown how the solution depends on different constraint types …
New method adapts DLMs to intrinsic data dependence without prior knowledge.
problem Understanding how unmasking schedules affect DLM generation quality.
method Adapts unmasking schedule to target data distribution's dependence structure.
result Sampling convergence guarantees improve for low-complexity distributions.
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.
Study nearest-neighbor radii under dependent sampling, finding they remain informative.
problem Analyzing nearest-neighbor radii under dependent sampling.
method Consider strong mixing dependent observations, establish distribution-free almost sure convergence and sharp non-asymptotic moment bounds.
result Nearest-neighbor geometry remains informative under dependence sampling.
The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this family of curves explicit formulas of curvature, torsion and geodetic curvature…
The success of modern Artificial Intelligence (AI) technologies depends critically on the ability to learn non-linear functional dependencies from large, high dimensional data sets. Despite recent high-profile successes, empirical evidence indicates that the high predictive performance is often paired with low robustne…
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.
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} , as the divergence of the horizontal gradient, once a volume is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of rando…
Paper analyzes a new Hopf-Lax semigroup in metric spaces.
problem Analyzing a new Hopf-Lax semigroup in metric spaces.
method Using continuous sections of quotient maps and variational problems.
result The 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation.
This paper introduces intrinsic time, a new measure of time for complex systems.
problem Traditional time measures fail to capture the dynamic nature of real-world phenomena.
method Intrinsic time uses an event-based, algorithmic framework to analyze time series data.
result Intrinsic time reveals novel structures and regularities in financial markets.
Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equationmath.DG
In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…
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.
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.
Deep Intrinsic Coregionalization Multi-Output Gaussian Process Surrogate with Active Learningstat.ML
DeepICMGP surrogate models multiple outputs efficiently.
problem Challenges in modeling dependencies between multiple outputs using traditional multi-output GPs.
method Introduces hierarchical coregionalization structures across layers in DGPs.
result Demonstrates competitive performance and active learning strategies.
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…
Study of Brown--York mass for four-dimensional asymptotically flat manifolds.
problem Calculating mass for hypersurfaces in four-dimensional asymptotically flat manifolds.
method Intrinsic definition of mean curvature, expansion analysis for large uniformly convex hypersurfaces.
result Shape-dependent correction to ADM mass for nearly round surfaces vanishes under certain conditions.
Study reveals LLM personas have two distinct components: frame-robust aggregated traits and frame-dependent geometric features.
problem Evaluation of LLM personas via psychometric questionnaires discards within-instance correlation structure.
method Constructed within-instance correlation matrices from IPIP-50 responses and analyzed geometry on SPD manifolds under manipulated question orderings.
result Persona expression comprises two dissociable components: aggregated features (Big Five scores) and geometric features (SPD manifold).
The paper classifies intrinsic torsion in various spacetime structures.
problem Classifying intrinsic torsion in different spacetime structures.
method Review and classification of intrinsic torsion in galilean, Carrollian, Aristotelian, and Bargmannian spacetime structures.
result Found 16 classes for Aristotelian structures and 27 for Bargmannian structures.
A new measure of causal influence quantifies intrinsic contributions in DAGs.
problem Quantifying intrinsic causal contributions in Directed Acyclic Graphs (DAGs).
method Recursive decomposition of node contributions, structure-preserving interventions, Shapley symmetrization.
result A measure of intrinsic causal contribution that is invariant to node relabeling.
Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionalitystat.ML
Bayesian neural networks achieve optimal posterior contraction rates in Besov spaces with intrinsic dimensionality.
problem High-dimensional structured estimation problems with unknown smoothness levels.
method Sparse Bayesian neural networks with either sparse or continuous shrinkage priors.
result Optimal posterior contraction rates are achieved, adapting to the unknown smoothness level of the true function.
Estimates scalar curvature of point clouds without embedding.
problem Estimating scalar curvature of data sets without embedding.
method Intrinsic estimator based on metric structure, consistent and stable.
result Estimator converges to scalar curvature as sample size increases.
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.
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…
Let be a submersion equipped with a horizontal connection over a Riemannian manifold . We present an intrinsic curvature condition that only depends on the pair . By studying a set of relative flat planes, we prove that a certain class of pairs a…
For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanif…
We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its …
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.
Introduces intrinsic Riemannian cross-covariance for manifold-valued random objects.
problem Covariance estimation for random objects on Riemannian manifolds.
method Defines covariance and correlation via parallel transport.
result Proposed covariance is independent of coordinate choices.
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.
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…
Differential privacy of Gaussian process posterior sampling
problem Privacy of posterior sample paths from Gaussian process
method Intrinsic randomness yields DP guarantees
result Intrinsic randomness yields DP guarantees
We present the first tree-based regressor whose convergence rate depends only on the intrinsic dimension of the data, namely its Assouad dimension. The regressor uses the RPtree partitioning procedure, a simple randomized variant of k-d trees.
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…
Deep networks can adapt to intrinsic dimensionality beyond domain constraints.
problem Approximating functions on low-dimensional manifolds with high-dimensional data.
method Two-layer compositions with ReLU activation, using dimensionality reducing feature maps.
result Near optimal approximation rates depend on the complexity of the dimensionality reducing map, not the ambient dimension.
A simple and computationally efficient scheme for tree-structured vector quantization is presented. Unlike previous methods, its quantization error depends only on the intrinsic dimension of the data distribution, rather than the apparent dimension of the space in which the data happen to lie.
A new method estimates Schrödinger bridges without iterative simulations or neural networks.
problem Estimating the time-dependent drift between two probability distributions.
method Solving the static entropic optimal transport problem and modifying the potentials.
result The Sinkhorn bridge method provably estimates Schrödinger bridges with a rate of convergence dependent on the target measure's intrinsic dimensionality.
High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.
problem Training score-based generative models on complex, low-dimensional manifolds.
method Proves optimal rates for SGMs on smooth manifolds, separating into noise regimes and using ReLU nearest-projection coordinates.
result Optimal intrinsic Wasserstein rates are achieved, with polynomial ambient dependence for families with controlled geometry and density.
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.
Asset-asset interactions and clustering in financial marketscond-mat.dis-nn
The collective phenomena of a liquid market is characterized in terms of a particle system scenario. This physical analogy enables us to disentangle intrinsic features from purely stochastic ones. The latter are the result of environmental changes due to a `heat bath' acting on the many-asset system, quantitatively des…
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
Geometrically classifies maps from R^0|2 to any manifold, unifying theories.
problem Classifying maps from R^0|2 to any manifold without auxiliary structures.
method Relates maps to pullback of decomposable bivector bundle over S via algebraic constraints.
result Reduced manifold has fiber dimension dim(S) + 1, unifying topological and algebraic views.
We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic pr…
Let be a ruled surface over a curve of genus . We prove that has a scalar-flat Hermitian metric if and only if and where is an intrinsic number depends on the complex structure of .
SAEs struggle with curved activation manifolds, revealing layer-dependent scaling laws.
problem Sparse autoencoders' reconstruction error varies across layers, not fitting existing scaling laws.
method Cross-layer study of 844 SAE checkpoints, fitting and regressing on manifold geometry.
result Manifold geometry predicts layer-dependent width exponents in SAEs, with transferable coefficients.
New method detects anomalies in systems influenced by their environment.
problem Detecting anomalies in systems under environmental influence.
method Adversarial learning and time series representation learning.
result Successfully addresses label sparsity and subjectivity in anomaly detection.