The paper explores learning metrics in low dimensions with bounds and complexities.
problem Learning metrics in low dimensions with bounds and complexities.
method Develops upper and lower bounds on generalization error, quantifies sample complexity, and bounds accuracy relative to the true metric.
result Novel mathematical approaches to metric learning and insights into ordinal embedding.
The paper proves compactness of scalar-flat metrics on low-dimensional manifolds with umbilic boundary.
problem Finding scalar-flat metrics with specific boundary conditions.
method Analyzing compact Riemannian manifolds with umbilic boundaries and proving compactness of scalar-flat metrics under certain conditions.
result Scalar-flat metrics are a compact set in low-dimensional manifolds (n=6,7,8) when the Weyl tensor is non-zero on the boundary.
Study metric learning from limited preference comparisons, showing how low-dimensional structure can still reveal metric information.
problem Learning metric from limited pairwise preference comparisons.
method Ideal point model, divide-and-conquer approach for low-dimensional structure.
result Metric can be jointly identified even with limited comparisons when items exhibit low-dimensional structure.
Paper presents a new method for learning hyperbolic representations using tree structures.
problem Learning faithful low-dimensional hyperbolic embeddings of data.
method Metric-first approach to learn tree structure, then embed into hyperbolic manifold.
result Novel fast algorithm TreeRep learns tree approximating original metric.
The input data features set for many data driven tasks is high-dimensional while the intrinsic dimension of the data is low. Data analysis methods aim to uncover the underlying low dimensional structure imposed by the low dimensional hidden parameters by utilizing distance metrics that consider the set of attributes as…
Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.
problem Existence of invariant metrics with positive intermediate Ricci curvature on low-dimensional cohomogeneity one manifolds.
method Construction of invariant metrics with positive intermediate Ricci curvature on specific manifolds.
result Invariant metrics with positive 4th-intermediate Ricci curvature exist but not for 3rd-intermediate Ricci curvature on certain manifolds.
A new metric captures shape information in manifold learning.
problem Capturing shape information in high-dimensional data.
method Metric based on angular changes along geodesic lines.
result Feasibility and merits of proposed dimensionality reduction scheme.
In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions,…
Low-rank approach to metric learning from data.
problem Learning a Mahalanobis metric from data.
method Low-rank geometric mean metric learning (GMML) approach.
result Competes effectively with GMML at lower ranks.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
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.
Paper provides statistical guarantees for GANs estimating Hölder space densities.
problem Statistical properties and theoretical guarantees for GANs.
method Approximation and statistical guarantees for GANs using Hölder space densities.
result GANs are consistent estimators of data distributions under strong discrepancy metrics.
Article proves effective conditions for existence of Kähler metrics.
problem Existence of extremal Kähler metrics on fibrations.
method Weighted uniform K-stability conditions derived from moment polytopes.
result Various effective conditions for K-stability verified.
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA f…
New metrics for high-dimensional data improve on energy distance.
problem Testing equality of distributions and independence in high dimensions.
method Proposed new metrics inheriting properties of energy distance and others.
result Improved metrics detect homogeneity and independence in high dimensions.
Proposes dynamic graph and node feature learning in GCNNs for better adaptability.
problem Fixed graphs for all GCNN layers limit adaptability to node feature structures.
method Dynamic graph and node feature learning using Mahalanobis distance metric.
result Superior performance in point clouds and citation networks.
This study analyzes how well GANs approximate distributions from small samples.
problem Understanding how well GANs approximate distributions from limited data.
method Analysis of GANs using integral probability metrics and Hölder classes.
result GANs can adaptively learn low-dimensional structures or Hölder densities.
Mercat preserves angles to create accurate low-dimensional embeddings.
problem Reconstructing global relationships in low-dimensional embeddings.
method Reconstructing angles between data points to preserve both local and global structures.
result Mercat yields good reconstruction across various experiments and metrics.
This research improves classification performance by learning a distance metric from balanced data.
problem Data imbalance in learning methods.
method Extracts a low-dimensional manifold, learns local neighborhood relationships, and optimizes distance metric.
result The proposed method outperforms other approaches, especially in imbalanced datasets.
MVTV improves interpretability in low-dimensional regression.
problem Estimating regression functions with few features and high interpretability needs.
method MVTV divides space into blocks, fits values jointly, and optimizes automatically.
result MVTV outperforms CART and CRISP in both complexity and human interpretability studies.
Course notes on Lie groups and Riemannian geometry, focusing on applications and low-dimensional examples.
problem Exploring Lie groups and their representations in Riemannian geometry.
method Review of well-known topics and recent advances in Riemannian geometry with symmetries.
result First construction of exceptional holonomy metrics.
New complex non-Kähler manifolds with specific properties are constructed.
problem Constructing complex non-Kähler manifolds with special properties.
method Using families of compact solvmanifolds and properties of the ∂ ∂ ˉ \partial\bar{\partial} ∂ ∂ ˉ -Lemma. result Provided families of compact ( n + 1 ) (n + 1) ( n + 1 ) -dimensional complex non-Kähler manifolds with specific properties. Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine wh…
In this short note we apply methods introduced by B. Hanke and T. Shick to prove the vanishing of (low dimensional) higher A A A -genera for spin manifolds admitting a positive scalar curvature metric. Our aim is to provide a short and unified proof for this beautiful result without using the strong Novikov conjecture.
Researchers add random metrics to data models to enable operations.
problem Lack of meaningful operations in learned low-dimensional representations.
method Endow latent space of generative models with a random Riemannian metric.
result Derived tight error bounds on expected distances in deterministic approximations.
Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
Visual observations of dynamic phenomena, such as human actions, are often represented as sequences of smoothly-varying features . In cases where the feature spaces can be structured as Riemannian manifolds, the corresponding representations become trajectories on manifolds. Analysis of these trajectories is challengin…
New algorithm MR-MISSING learns low-dimensional representations from data with missing entries.
problem Learning low-dimensional representations from data with missing entries.
method Extends matrix completion techniques to handle missing data and non-linear manifold structure.
result Demonstrates effectiveness on synthetic and real data sets, providing theoretical guarantees.
Estimates metric tensor on neuromanifolds using Fisher information and random methods.
problem Computing the metric tensor on high-dimensional neuromanifolds efficiently and accurately.
method Deterministic bounds and unbiased random estimators based on Hutchinson's trace method.
result An efficient random estimator with bounded standard deviation.
We consider invariant Riemannian metrics on compact homogeneous spaces G / H G/H G / H where an intermediate subgroup K K K between G G G and H H H exists. In this case, the homogeneous space G / H G/H G / H is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics o…
This work refines Cover's theory for binary classification on low-dimensional data.
problem The challenge of analyzing how low-dimensional data structures affect classification models.
method Refines Cover's function-counting theory to account for low-dimensional data structure.
result Derives dichotomy counts and analyzes the impact of data structure on classification models.
We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for som…
Upper bounds on Einstein metrics on homogeneous spaces.
problem Counting isolated homogeneous Einstein metrics on compact spaces.
method Combinatorial volume computation of polytopes, algebraic statistics, numerical algebraic geometry.
result Explicit upper bounds confirmed for Einstein metrics on specific spaces.
A new method for manifold learning robust to irregular sampling.
problem Nonlinear dimensionality reduction of high-dimensional datasets with poor sampling and arbitrary topology.
method Parallel transport unfolding (PTU) for quasi-isometric low-dimensional mapping.
result Improved robustness to irregularity and voids in sampling compared to Isomap.
Global minima found for multidimensional scaling with penalties.
problem Finding global minima in multidimensional scaling.
method Combining stress loss function with a quadratic penalty term to find minimizers.
result Trajectory of minimizers leads to global minima.
Proposes tests for comparing high-dimensional manifold samples.
problem Determining if two manifold samples come from the same distribution.
method Integral Probability Metric (IPM) with neural network approximations.
result Tests achieve type-II risk in specific orders of n n n . Deep learning method improves radiographic similarity detection.
problem Learning a distance metric for radiographs to capture radiological similarity.
method Deep convolutional neural networks (DCNs) learn a low-dimensional embedding with a distance metric for radiographs.
result The learned metric effectively distinguishes normal from abnormal radiographs.
Diffusion models adapt to low-dimensional data regardless of coefficient choices.
problem Understanding how diffusion models adapt to low-dimensional data structures.
method Analysis of diffusion models with flexible coefficient choices.
result Proven that O ~ ( k / ε ) \widetilde{O}(k/\varepsilon) O ( k / ε ) iterations suffice for accurate sampling in total variation distance. Improved likelihood-free inference by localizing and refining low-dimensional approximations.
problem Poor performance of common likelihood-free methods in high-dimensional models.
method Localisation followed by refinement of low-dimensional summaries.
result Improved accuracy in marginal posteriors through localized and refined approximations.
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
The paper explores low-dimensional solenoidal manifolds and their properties.
problem Characterizing and understanding solenoidal manifolds of dimensions 1, 2, and 3.
method Survey and new results about solenoidal manifolds, using theorems of A. Clark and S. Hurder.
result Topologically homogeneous, compact solenoidal manifolds are McCord solenoids and behave like laminated versions of compact manifolds.
Deep ReLU networks estimate Hölder functions on low-dimensional manifolds with fast convergence.
problem Estimating Hölder functions on low-dimensional manifolds with noisy data.
method Deep ReLU network architecture designed for nonparametric regression.
result Empirical estimator convergence rate of n − 2 ( s + α ) 2 ( s + α ) + d log 3 n n^{-\frac{2(s+α)}{2(s+α) + d}}\log^3 n n − 2 ( s + α ) + d 2 ( s + α ) log 3 n . We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let g g g be an asymptotically flat Lipschitz metric on a smooth manifold M n M^n M n , such that n < 8 n<8 n < 8 or M M M is spin. As long as g g g has bounded $C^…
Study benchmarks 19 survival models on 34 datasets, finding Cox model still best.
problem Quantitative comparison of survival models on low-dimensional data.
method Comprehensive benchmarking of 19 models on 34 datasets, tuning and evaluating using 6 metrics.
result Cox Proportional Hazards model remains best overall for low-dimensional, right-censored data.
The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adap…
Classifies geodesics for Carathéodory metric on Teichmüller spaces.
problem Distinguishing Carathéodory and Teichmüller metrics on Teichmüller disks.
method Dynamical results of Minsky, Smillie, and Weiss; complex-analytic criterion.
result Proves conjecture for specific surfaces, extending result to punctured surfaces.