In 2D, synthetic curvature notions match.
problem Matching synthetic curvature notions in 2D.
method Synthetic curvature notions in 2D.
result Synthetic curvature notions match in 2D.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,291 papers · 148 categories
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48 results for “synthetic dimension”
The paper extends Lorentzian splitting theorems under curvature-dimension bounds.
problem Analyzing Lorentzian spacetimes with curvature-dimension bounds.
method Using Bakry-Émery-Ricci tensor, extending singularity and splitting theorems.
result Theorems hold for all synthetic dimensions, including negative and positive.
The paper examines how curvature-dimension conditions transform under time change for diffusions.
problem Transforming curvature-dimension conditions for diffusions under time change.
method Derives precise transformation formulas for synthetic lower Ricci bounds and curvature-dimension conditions.
result Precise formulas for curvature-dimension conditions under time change for diffusions and metric measure spaces.
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
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.
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.
Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.
problem Understanding causal bubbling in spacetimes.
method Example of a globally hyperbolic spacetime with a continuous metric, splitting orthogonally into timelike and spacelike parts.
result The synthetic timelike curvature-dimension (TCD) condition does not prevent causal bubbling.
Study shows flat torus is identified by first cohomology group's dimension.
problem Identifying flat torus among spaces.
method Analysis of the first cohomology group.
result Proves flat torus when first cohomology group dimension equals N.
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.
Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the condition.
method Developed a new strategy to contradict the 1-dimensional condition.
result 2D and strongly regular almost-Riemannian manifolds do not satisfy for any and .
Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong -timelike Brunn-Minkowski condition and proving equivalence to curvature conditions.
result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
Privacy is enhanced by synthetic data release even with unlimited data.
problem Improving privacy guarantees for synthetic data release.
method Analyzing a bounded-parameter assumption to show privacy amplification persists with unlimited synthetic records.
result Privacy amplification is possible even with an unbounded number of synthetic records.
Uniformly Euclidean metrics with isolated singularities on certain manifolds are Ricci flat and have nonnegative synthetic Ricci curvature.
problem Proving the existence of Ricci flat metrics with isolated singularities on specific manifolds.
method Demonstrating nonnegative synthetic Ricci curvature using the RCD(0, n) condition.
result Uniformly Euclidean metrics with isolated singularities on are Ricci flat and extend smoothly over the singularity.
Paper defines new curvature measures for foliated manifolds and proves new theorems.
problem Estimating the diameter and splitting theorems for foliated manifolds.
method Introduced weighted mixed curvatures and new conditions to update estimates.
result Updated estimates of diameter and proved new splitting theorems.
MNIST-Nd offers synthetic datasets to benchmark clustering across dimensions.
problem Clustering performance degrades with high-dimensional data.
method Training mixture variational autoencoders on MNIST to create synthetic datasets with varying latent dimensions.
result Leiden clustering algorithm is most robust as dimensionality grows.
Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicitymath.GT
We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, …
Framework evaluates quality of synthetic data generated with differential privacy.
problem Ensuring synthetic data retains statistical quality after applying differential privacy.
method Developed a framework to evaluate synthetic data quality from a practical researcher's viewpoint.
result Synthetic data can be evaluated against training data or underlying populations, and for specific tasks like inference or prediction.
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.
New findings show different cost functions yield equivalent curvature bounds.
problem Establishing equivalence of curvature bounds under various transport costs.
method Needle decomposition and localization technique for optimal transport.
result All conditions are equivalent for .
Sharp isoperimetric inequality proven for specific metric measure spaces.
problem Proving isoperimetric inequality in metric measure spaces with synthetic conditions.
method Synthetic condition called Measure-Contraction property; Lévy-Gromov inequality.
result Sharp isoperimetric inequality holds true for spaces with synthetic conditions.
We use geometric algebra techniques to give a synthetic and computationally efficient approach to Fierz identities in arbitrary dimensions and signatures, thus generalizing previous work. Our approach leads to a formulation which displays the underlying real, complex or quaternionic structure in an explicit and concept…
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.
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 method addresses crowding in high-dimensional data visualization.
problem Crowding issue in visualizing high-dimensional data.
method Adjusting capacity of high-dimensional balls and estimating correlation dimension.
result Mitigates crowding in various distance metrics.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
Motion Classification using Kinematically Sifted ACGAN-Synthesized Radar Micro-Doppler Signatureseess.SP
Paper generates synthetic radar signatures for motion classification.
problem Lack of large training datasets for radar-based human activity recognition.
method Adversarial learning for synthetic data generation, kinematic sifting for consistency.
result 93% overall accuracy achieved on diverse aspect angles.
The paper proves stability properties for quotients of spaces with synthetic Ricci curvature bounds.
problem Stability properties of quotients of spaces with synthetic Ricci curvature bounds.
method Analyzes quotients of Riemannian manifolds with isometric group actions and metric-measure foliations/submersions.
result Quotients of Riemannian manifolds with synthetic Ricci curvature bounds are also Riemannian manifolds with synthetic Ricci curvature bounds.
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.
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geo…
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.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Additive Gaussian process framework handles monotonicity constraints in high dimensions.
problem Handling monotonicity constraints in high-dimensional data.
method Additive Gaussian process framework with MaxMod algorithm for dimension reduction.
result Framework enables to satisfy monotonicity constraints everywhere in the input space.
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.
New encoding improves volatility surface generation and risk management.
problem Generating accurate synthetic volatility surfaces from limited data.
method PCA variational auto-encoder model to encode surface descriptors into a latent space.
result Better scenario generation, volatility extrapolation, and direct stock surface inference.
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
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.
GeoMLE improves intrinsic dimension estimation for nonlinearly embedded data.
problem Inaccurate estimation of intrinsic dimension for nonlinearly embedded data.
method GeoMLE uses geometric properties to correct the standard MLE for flat manifolds.
result GeoMLE achieves state-of-the-art performance and is computationally efficient.
Study evaluates RKHS choices for assessing graph models using KSD tests.
problem Effect of RKHS choice on KSD tests for graph model assessment.
method Investigated power performance and computational runtime of KSD tests for ERGMs and synthetic graph generators.
result Different RKHS choices affect KSD test performance and computational runtime.
Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groupsmath.MG
The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.
problem Investigating synthetic curvature-dimension bounds in sub-Finsler Heisenberg groups.
method Study of measure contraction property (MCP) and curvature-dimension condition (CD).
result Sub-Finsler Heisenberg groups do not satisfy MCP or CD for any parameters.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
The study establishes conditions for stratified spaces to satisfy RCD(K, N) curvature-dimension condition.
problem Conditions for stratified spaces to satisfy RCD(K, N) curvature-dimension condition.
method Proves conditions for stratified spaces to satisfy RCD(K, N) using Ricci tensor bounds and cone angles.
result New examples of metric measure spaces satisfying RCD(K, N) curvature-dimension condition.
Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg groupmath.DG
Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
problem Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
method Elementary variational approach, Lorentzian isoperimetric problem, uniform estimate of causal diamonds.
result Heisenberg group has Lorentzian Hausdorff dimension 4 and satisfies neither timelike curvature-dimension nor measure contraction properties.
Study on cones over metric spaces with curvature bounds.
problem Establishing curvature bounds for cones over metric spaces.
method Developed a localization technique to prove synthetic curvature bounds.
result Riemannian and Lorentzian cones over CD-spaces satisfy MCP and vice versa.
AdaScale-TuRBO improves high-dimensional Bayesian optimization by dynamically scaling the GP lengthscale.
problem Inappropriate lengthscale design in TuRBO's local GP model causes suboptimal performance in high dimensions.
method Proposes AdaScale-TuRBO, which scales the GP lengthscale with both problem dimension and trust region size.
result AdaScale-TuRBO robustly outperforms standard TuRBO and other methods on synthetic and real-world tasks.
Study on curvature bounds and geodesic dimension in sub-Finsler Heisenberg groups.
problem Investigate synthetic curvature-dimension bounds in sub-Finsler geometry.
method Examine measure contraction property and geodesic dimension on Heisenberg groups with -sub-Finsler norms.
result For , -Heisenberg group fails to satisfy any measure contraction property. For , it satisfies under specific conditions.
Synthetic construction of Hopf fibration in 4D space.
problem Visualizing 4D objects in 3D space.
method Double orthogonal projection method to visualize 4D space.
result Direct synthetic construction of 3-sphere fibers from 2-sphere points.
Efficient and Robust Bayesian Selection of Hyperparameters in Dimension Reduction for Visualizationstat.ML
Efficient auto-tuning for DR hyperparameters with BO.
problem Hyperparameter selection in DR for large-scale datasets.
method Bayesian optimization with surrogate model, normalization, subsampling.
result Robust and efficient hyperparameter selection for DR algorithms.