We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.
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We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups…
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the -phase diagram of large-dimensional kernel interpolation.
We study the concept of coarse disjointness and large scale -to- functions. As a byproduct, we obtain an Ostrand-type characterization of asymptotic dimension for coarse structures. It is shown that properties like finite asymptotic dimension, coarse finitism, large scale weak paracompactness, ect. are all invari…
New neural network method simplifies high-dimensional data.
problem Scalability issues in nonlinear sufficient dimension reduction.
method Stochastic neural network with adaptive gradient algorithm.
result Proposed method outperforms existing methods on large-scale data.
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
Introduces resemblance structure for large scale geometry.
problem Defining similarity in large scale geometry.
method Axiomatizing the concept of resemblance for subsets of a set.
result Large scale resemblance structures can induce nearness and generalize large scale properties.
Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau manifolds with high Betti numbers.
method Smoothing normal crossing varieties to create K3 fibrations over smooth projective varieties.
result Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers.
We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum , $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes where is the localization…
Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain …
In importance sampling (IS)-based reinforcement learning algorithms such as Proximal Policy Optimization (PPO), IS weights are typically clipped to avoid large variance in learning. However, policy update from clipped statistics induces large bias in tasks with high action dimensions, and bias from clipping makes it di…
Study shows large extra dimensions are hidden from view by black holes.
problem Difficulty in observing large extra dimensions in the universe.
method Analysis of conditions leading to black hole formation in lower dimensional theories.
result Extra dimensions larger than are hidden inside black holes.
We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.
Gromov's tori are optimalmath.DG
Optimal bounds found for torus curvatures in high dimensions.
problem Finding optimal bounds on normal curvatures of tori.
method Analyzing immersed n-torus in a Euclidean ball of large dimension.
result Optimal bounds on normal curvatures of tori established.
Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller dimen…
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
The current study proposes a dimension reduction method, stepwise support vector machine (SVM), to reduce the dimensions of large p small n datasets. The proposed method is compared with other dimension reduction methods, namely, the Pearson product difference correlation coefficient (PCCs), recursive feature eliminati…
A faster method for estimating effects in large data using fixed-point trees.
problem Estimating heterogeneous effects in large dimensions with computational efficiency.
method Fixed-point approximation to eliminate Jacobian estimation and speed up GRFs.
result Significant computational efficiency improvement without sacrificing statistical accuracy.
Roughly speaking, let us say that a map between metric spaces is large scale conformal if it maps packings by large balls to large quasi-balls with limited overlaps. This quasi-isometry invariant notion makes sense for finitely generated groups. Inspired by work by Benjamini and Schramm, we show that under such maps, s…
Proposes a method for evaluating multiple dimensions of organizational effectiveness using DEA.
problem Evaluating multiple dimensions of organizational effectiveness in large data sets.
method Introduces two regularized DEA models (SBM and GP-SBM) to estimate both dimension-specific and aggregate efficiency scores.
result Demonstrates improved efficiency and validity compared to conventional methods.
We obtain a tight distribution-specific characterization of the sample complexity of large-margin classification with L_2 regularization: We introduce the γ-adapted-dimension, which is a simple function of the spectrum of a distribution's covariance matrix, and show distribution-specific upper and lower bounds on the s…
New insights into tSNE for large datasets.
problem Limitations of tSNE in handling large datasets.
method Identified continuum limit of tSNE objective function, proposed rescaled model.
result Rescaled model has a consistent limit for large datasets.
Paper proposes PPMM for fast estimation of large-scale OTM.
problem Estimation of large-scale optimal transport maps (OTM) is challenging due to the curse of dimensionality.
method Combines projection pursuit regression and sufficient dimension reduction to adaptively select projection directions.
result PPMM consistently estimates the most informative projection direction and weakly converges to the target OTM.
Strong Frankel theorem for shrinkers in all dimensions.
problem Intersection of shrinkers in large balls.
method Proof using strong Bernstein theorem for stable Gaussian surfaces.
result Shrinkers are connected in all large balls.
In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
Study identifies key parameters and input dimensions making LLMs and VLMs brittle.
problem Vulnerability of large language and vision-language models to perturbations.
method Proposed FI measure based on information geometry to quantify sensitivity.
result Small subset of high FI parameters significantly contribute to brittleness.
Graphs on surfaces have a 2-dimensional large scale structure.
problem Understanding the large scale structure of graphs on surfaces.
method Proving asymptotic dimension for specific graph classes and surfaces.
result Graphs on surfaces have an asymptotic dimension of 2.
Study reveals learning curves and benign overfitting in spectral algorithms for large dimensions.
problem Understanding learning curves and benign overfitting in spectral algorithms for large-dimensional data.
method Analysis of learning curves and benign overfitting in spectral algorithms for inner-product kernels on the sphere and general domains.
result Characterization of three distinct regimes: over-regularized, under-regularized, and interpolation regimes, revealing benign overfitting across both under-regularized and interpolation regimes.
Paper uses non-linear dimension reduction for better economic forecasting.
problem Analyzing economic effects of shocks in large datasets.
method Non-linear dimension reduction in factor-augmented vector autoregressions.
result Non-linear dimension reduction techniques improve forecasting, especially in volatile data.
Efficient knockoffs for large-scale feature selection.
problem Large-scale feature selection problems.
method Gaussian model-X knockoffs with efficient methods for solving semidefinite programs.
result Efficient knockoffs can be generated with linear complexity in the dimension.
This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona of . The first …
This work improves tensor decomposition methods, especially for large datasets.
problem Lack of efficient methods for estimating Tucker decompositions.
method Applies Johnson-Lindenstrauss type guarantees to Tucker decompositions with random embeddings.
result Effective dimension reduction with minimal error for large tensors.
We prove that many features of Thurston's Dehn surgery theory for hyperbolic 3-manifolds generalize to Einstein metrics in any dimension. In particular, this gives large, infinite families of new Einstein metrics on compact manifolds.
Buyalo and Lebedeva have shown that the asymptotic dimension of a hyperbolic group is equal to the dimension of the group boundary plus one. Among the work presented here is a partial extension of that result to all groups admitting -structures; in particular, we show that …
We present the notion of asymptotically large depth for a metric space which is (a priory) weaker than having subexponential asymptotic dimension growth and (a priory) stronger than property A.
If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that every co-orientable contact structure on any closed contact manifold admits a co…
We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…
This paper explores saturation effects in spectral algorithms over large dimensions.
problem Saturation effects in spectral algorithms over large dimensions.
method Improved minimax lower bound and gradient flow with early stopping strategy.
result Exact convergence rates of spectral algorithms in large dimensional settings.
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
Study on kernel regression risk in high dimensions using Pinsker bound.
problem Kernel regression risk in high-dimensional inner product spaces.
method Investigation of Pinsker bound for kernel regression on sphere with sample size .
result Exact minimax risk and Pinsker constant identified for kernel regression.
A new, fast kernel test for large data.
problem Efficient kernel two-sample tests for high-dimensional, large-scale data.
method A new kernel-based test that is computationally efficient and robust to high dimensions.
result The new test performs well across various alternatives and dimensions.
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…
This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study th…
Recommendation problems with large numbers of discrete items, such as products, webpages, or videos, are ubiquitous in the technology industry. Deep neural networks are being increasingly used for these recommendation problems. These models use embeddings to represent discrete items as continuous vectors, and the vocab…
We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subcl…
New method corrects missing data bias in dimension reduction.
problem Missing data complicates high-dimensional data analysis.
method Developed a bias-corrected Gram matrix for heterogeneous missingness.
result Proposed method improves dimension reduction techniques significantly.
We prove that the boundary of the trapped region in an asymptotically Euclidean Riemannian manifold of dimension at least 3 is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.