The -dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative -dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, -means clustering and vector quantization as special cases. Previous generalizat…
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48 results for “dimensional bounds”
New infinite-dimensional representations with bounded multiplicity found for Lie groups.
problem Finding representations with bounded multiplicity for Lie groups.
method Proving existence of infinite-dimensional irreducible representations with bounded multiplicity property.
result Infinite-dimensional irreducible representations with bounded multiplicity found for non-compact semisimple Lie groups.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global -regularity for 4D Ricci flow with integral scalar curvature bound.
Riemannian manifolds with bounded Ricci curvature have finite Uryson width.
problem Bounding Uryson width for manifolds with Ricci curvature constraints.
method Continuous map to a polyhedral space with controlled diameter.
result Riemannian manifolds with specific Ricci curvature bounds have finite Uryson width.
Novel bounds for logistic regression coreset construction and feature selection.
problem Efficiently summarize and reduce logistic regression inputs.
method Feature space sketching for logistic regression.
result Tight bounds for coreset construction and feature selection.
Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.
problem Vanishing of bounded cohomology for higher dimensional sphere diffeomorphism groups.
method Proved vanishing of bounded cohomology with real coefficients for and .
result Vanishing of bounded cohomology for higher dimensional spheres.
Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.
We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sph…
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
Unified framework for high-dimensional bandit problems with low-dimensional structures.
problem Stochastic high-dimensional bandit problems with low-dimensional structures.
method Proposed a simple unified algorithm and a general analysis framework for the regret upper bound.
result Unified algorithm achieves comparable regret bounds in various high-dimensional bandit problems.
Finite-dimensional spaces of biharmonic functions on manifolds are explored.
problem Characterizing biharmonic functions on open manifolds with nonnegative Ricci curvature.
method Analyzing bounded and polynomial growth biharmonic functions, deriving Weyl bounds, and studying fourth-order operators.
result Finite-dimensional spaces of biharmonic functions with polynomial growth are established.
Bounding Shortest Closed Geodesics with Diameter on compact 2-dimensional Orbifolds Homeomorphic to math.DG
The paper finds shortest geodesic bounds on orbifolds with diameter limits.
problem Bounding the shortest closed geodesic on compact orbifolds.
method Generalizing length-bounded sweepouts to orbifolds.
result Established an inequality linking shortest geodesic length to orbifold diameter.
One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted gr…
The paper studies knot quandles and their cohomology, proving infinite dimensionality results.
problem Understanding the cohomology of knot quandles and its implications in knot theory.
method Analyzing quandles and their inner automorphism groups, proving conditions for infinite dimensionality.
result The second bounded cohomology of knot quandles is infinite dimensional, detecting the unknot.
We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of t…
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…
The goal of the paper is to give an optimal transport characterization of sectional curvature lower (and upper) bounds for smooth -dimensional Riemannian manifolds. More generally we characterize, via optimal transport, lower bounds on the so called -Ricci curvature which corresponds to taking the trace of the Ri…
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orient…
5D shrinking solitons with bounded curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons.
method Proving rigidity for solitons with bounded curvature.
result 5D shrinking gradient Ricci solitons with bounded curvature are rigid.
We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.
In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we int…
Chernoff Bounds for Tensor Expanders on Riemannian Manifolds Using Graph Laplacian Approximationmath.PR
Paper extends tail bounds to high-dimensional random objects on Riemannian manifolds.
problem Need for tail bounds in high-dimensional data.
method Random walks on graph approximating the manifold, ensuring spectral similarity.
result Derived tensor Chernoff bound for Riemannian manifolds.
New method bounds high-dimensional regression without estimating design covariance.
problem High-dimensional linear regression with random design.
method Error-in-operator approach that incorporates design covariance into empirical risk minimization.
result Dimension-free bounds on excess prediction risk derived.
Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.
problem Bounding total geodesic curvature in a hyperbolic setting.
method Derives an upper bound for total geodesic curvature in terms of boundary data.
result Upper bound for total geodesic curvature expressed solely in terms of boundary data.
New method detects low-dimensional manifolds within bounds.
problem Detecting low-dimensional manifolds within bounds.
method Matrix Completion (MC) problem for partially observed distances on a manifold.
result The method provides theoretical guarantees on manifold detection and robustness to non-uniform sampling.
Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.
problem Approximating and estimating classification functions in high-dimensional spaces.
method Modified existing results to show that functions can be approximated by neural networks with bounded weights. Proved the existence of a neural network with bounded weights approximating a classification function. Leveraged these bounds to quantify estimation rates.
result Neural networks can approximate functions without the curse of dimensionality, leading to efficient estimation rates.
Paper analyzes GLM-tron for high-dimensional ReLU regression, providing upper and lower bounds.
problem Learning a single ReLU neuron in high-dimensional settings with overparameterization.
method Perceptron-type algorithm GLM-tron, with finite-sample analysis.
result Sharp characterization of high-dimensional ReLU regression problems via GLM-tron, contrasting with SGD.
The paper explores high-dimensional learning in finance, proving key aspects and setting lower bounds.
problem Understanding when and how large, over-parameterized models achieve predictive success in finance.
method Theoretical foundations and empirical validation of two key aspects: standardization and information-theoretic lower bounds.
result Empirical validation shows that high-dimensional learning in finance often relies on lower-complexity artefacts rather than the intended mechanism.
Kato meets Bakry-Émerymath.DG
The study connects Kato bounds to finite-dimensional RCD spaces.
problem Understanding the limits of complete Riemannian manifolds with Kato bounds.
method Using the transformation rule of the Bakry-Émery condition under time change.
result Bi-Lipschitz equivalence to finite-dimensional RCD spaces.
Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.
PCA-Net combines PCA and neural networks for operator approximation, with new bounds on complexity.
problem Developing approximation theory for PCA-Net architecture.
method Combines PCA and neural networks, derives universal approximation results and lower bounds on complexity.
result PCA-Net can overcome the curse of parametric complexity for specific operators.
We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of ed…
New algorithms improve machine learning performance with explicit regret bounds.
problem Improving machine learning performance with explicit regret bounds.
method Projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance.
result Established a priori regret bounds with explicit λ-dependence.
pHMC converges on infinite-dimensional spaces with bounds.
problem Convergence of pHMC on Hilbert spaces.
method Coupling of two pHMC copies, adapted from arXiv:1805.00452.
result Proven convergence bounds in 1-Wasserstein distance.
Upper bounds for volumes of hyperbolic polyhedra and links are derived.
problem Finding upper limits for volumes of generalized hyperbolic polyhedra and links.
method Application of Belletti's theorem and analysis of polyhedra with triangular faces and trivalent vertices.
result Improved upper bounds for volumes of hyperbolic polyhedra and links are derived.
New bounds for adaptive control in high dimensions without fixed state space.
problem Adaptive control of linear systems in high or infinite dimensions.
method Novel perturbation bound for certainty equivalence, scaling with prediction error.
result First regret bounds for LQR in infinite dimensional systems, independent of ambient dimension.
Improved location estimation for high-dimensional data with finite sample size.
problem Estimating the shift in high-dimensional data with limited samples.
method Smoothed estimators and bounds on subgamma vectors.
result Convergence to Cramér-Rao bound for finite sample sizes.
Study dynamic batch learning in high-dimensional sparse linear bandits.
problem Dynamic batch learning in high-dimensional sparse linear contextual bandits under batch constraints.
method Characterized fundamental learning limits via regret lower bound and provided matching upper bound.
result Prescribed an optimal scheme for dynamic batch learning in high-dimensional sparse linear contextual bandits.
New bounds for high-dimensional sparse linear bandits, balancing information and regret.
problem Stochastic linear bandits with high-dimensional sparse features.
method Derivation of minimax regret lower and upper bounds for explore-then-commit algorithm.
result Optimal rate of for data-poor regime, complemented by under signal magnitude assumption.
Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these man…
We bound the size of -dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.
We introduce and define "oriented framed measured lamination links" in a 3-manifold . These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links boun…
Proposes bounds on bias from low-dimensional representations in CATE estimation.
problem Bias in CATE estimation due to low-dimensional representations.
method Proposes a refutation framework to estimate bounds on representation-induced confounding bias.
result Demonstrates effectiveness of refutation framework in practice.
In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
Study shows bound on Uryson width for specific 3D manifolds.
problem Bounding Uryson width for 3D manifolds with non-negative Ricci curvature and strictly mean convex boundary.
method Proved existence of a Morse function with uniform diameter bounds on level sets.
result Upper bound on Uryson width for the specified 3D manifolds.