Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
The k-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k-dimensional vectors, and include non-negative matrix factorization, dictionary learning, sparse coding, k-means clustering and vector quantization as special cases. Previous generalizat…
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors --generalizing to one side Kaluza-Klein gauge fields and…
Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.
problem Proving a theorem for a specific type of Dirac operator on various manifolds.
method Extending previous results to even-dimensional almost product Riemannian spin manifolds.
result Established the general Kastler-Kalau-Walze type theorem for even-dimensional manifolds.
Generative model handles varying data dimensions using jump diffusion processes.
problem Handling data of varying dimensionality in generative models.
method Formulated as a jump diffusion process, learning to approximate the process with a novel evidence lower bound.
result Effective sampling of data of varying dimensionality, better compatibility with test-time diffusion guidance imputation tasks.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.
Generative models learn complex data from low-dimensional manifolds.
problem Theoretical justification for generative models on manifold structures.
method Prove statistical guarantees of generative networks under Wasserstein-1 loss, considering intrinsic dimensionality.
result Generative networks converge to zero at a fast rate depending on intrinsic dimensionality, not ambient data dimension.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
We construct a three-dimensional topological sigma model which is induced from a generalized complex structure on a target generalized complex manifold. This model is constructed from maps from a three-dimensional manifold X to an arbitrary generalized complex manifold M. The theory is invariant under the diffeomor…
Generative models can approximate high-dimensional data from lower dimensions without needing a latent dimension equal to or greater than the data's intrinsic dimension.
problem Theoretical limitations on the latent dimension required for generative models to approximate high-dimensional data distributions.
method Inspired by space-filling curves, the work demonstrates that generative networks can approximate distributions on d-dimensional manifolds from inputs of any arbitrary dimension, even lower than d. result Generative models can approximate high-dimensional data distributions from lower-dimensional inputs without needing a latent dimension equal to or greater than the data's intrinsic dimension.
Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower dimensionality. Learning to identify and extract task-relevant variables from this embedde…
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
New research shows CFG improves high-dimensional data generation.
problem Characterizing CFG's effect on high-dimensional distributions.
method High-dimensional analysis of CFG's impact on target distributions.
result CFG accurately reproduces the target distribution in high dimensions.
We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C3 of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\…
We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
Deep multi-task learning benefits from low intrinsic dimensionality, leading to better generalization.
problem Improving generalization in deep multi-task learning with high-dimensional models.
method Parametrizing multi-task networks in a low-dimensional space using random expansions and weight compression.
result First non-vacuous generalization bounds for deep multi-task networks are derived.
In this paper, we generalize the Hersch-Payne-Schiffer inequality for Steklov eigenvalues to higher dimensional case by extending the trick used by Hersch, Payne and Schiffer to higher dimensional manifolds.
A new framework generates high-dimensional event sequences efficiently.
problem Challenges in modeling high-dimensional marked temporal point processes.
method Conditional generator that learns from event history.
result Superior performance compared to existing methods.
Generative models speed up complex system simulations.
problem Accurately forecasting the dynamics of complex systems at reduced cost.
method Generative Learning of Effective Dynamics (G-LED) using auto-regressive attention and Bayesian diffusion models.
result Generative models can accurately forecast complex system dynamics at lower computational cost.
MsIGN tackles high-dimensional Bayesian inference using multiscale structure.
problem High-dimensional Bayesian inference challenges due to the curse of dimensionality.
method MsIGN generates samples from coarse to fine scale, minimizing Jeffreys divergence.
result MsIGN outperforms previous approaches in posterior approximation and mode capture.
A weakly Einstein manifold is a generalization of a 4-dimensional Einstein manifold, which is defined as an application of a curvature identity derived from the generalized Gauss-Bonnet formula for a 4-dimensional compact oriented Riemannian manifold. In this paper, we shall give a characterization of a weakly Einstein…
This paper explores how effective sample size, dimensionality, and model performance are related in covariate shift adaptation.
problem Understanding the relationship between effective sample size, dimensionality, and generalization in covariate shift adaptation.
method Building a unified theory connecting effective sample size, data dimensionality, and generalization in the context of covariate shift adaptation.
result Dimensionality reduction or feature selection can increase effective sample size, supporting the practice of reducing dimensionality before covariate shift adaptation.
We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equa…
Study on meta-reinforcement learning generalization in high-dimensional tasks.
problem Generalization performance of meta-reinforcement learning algorithms in high-dimensional tasks.
method High-dimensional, procedurally generated environments.
result Meta-reinforcement learning algorithms exhibit strong overfitting on challenging tasks.
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
The paper analyzes how stock market dimensionality changes impact portfolio performance.
problem Impact of dimensional changes on portfolio performance in a changing market.
method Development of self-financing stock portfolios in a stochastic portfolio theory framework with dimensional jumps.
result Quantification of how listing or delisting events and market shocks affect portfolio return.
The paper proves new theorems for Dirac operators on even-dimensional manifolds with boundary.
problem Proving theorems for Dirac operators on manifolds with boundary.
method Establishing general Kastler-Kalau-Walze type theorems for conformal perturbations of Dirac operators.
result Proof of new theorems for Dirac operators on even-dimensional manifolds with boundary.
A new method validates generative models in high-dimensional data.
problem Scalability and interpretability issues in validating generative models.
method Learning-based goodness-of-fit testing inspired by Neyman--Pearson construction.
result The NPLM can effectively validate generative models in high-dimensional data.
We completely classify the algebraic Ricci solitons of four-dimensional pseudo-Riemannian generalized symmetric spaces.
The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.
problem Deformation and tangent groupoid constructions for infinite-dimensional manifolds.
method Extending finite-dimensional constructions to Banach and Fredholm manifolds.
result Induced generalized filtrations of tangent bundles and groupoids.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
In this paper we study geodesic mappings of n-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such n-dimensional ellipsoids admit non tri…
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimen…
The paper tackles manifold overfitting in deep generative models.
problem Manifold overfitting occurs when generative models learn the manifold itself instead of the distribution on it.
method The authors propose a two-step procedure: dimensionality reduction followed by maximum-likelihood density estimation.
result The two-step procedure avoids manifold overfitting and enables density estimation on learned manifolds.
Improved likelihood estimation for singular distributions using deep models.
problem Estimating singular distributions using deep generative models.
method Data perturbation to avoid singularity issues in likelihood estimation.
result Consistent estimation of target distribution with desirable rates.
Generative algorithms learn high-dimensional data efficiently and generate new samples.
problem Learning from scarce high-dimensional data.
method Lipschitz-regularized gradient flows and particle-based algorithms.
result Correctly transports gene expression data points with high dimensionality.
We obtain a nature generalization for an affine Sierpinski carpet and Sierpinski triangle to n-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet. Exactly, in this paper, a Menger sponge and Sierpinski simplex in 4-dimensional space could be drawn out clearly …
The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
New methods decompose manifolds into submanifolds via fold maps.
problem Understanding the topologies and differentiable structures of manifolds globally.
method Explicit decompositions of manifolds via fold maps, generalizing Morse functions.
result Decompositions of manifolds into lower dimensional spaces via fold maps.
We solve the metrisability problem for generic three-dimensional projective structures.
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
In this note we find a generic defining function of projective motion in the 6-dimensional rigid h-space.
Extends SGM to functional spaces for multimodal data.
problem Modeling densities in functional spaces.
method Represent data in spectral space, dissociate stochastic and space-time components, use SGM for sampling.
result Demonstrates effectiveness on multimodal datasets.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of pa…