Study sub-Riemannian geometry on parallelizable distributions.
problem Investigate the geometry of sub-Riemannian structures on non-integrable parallelizable distributions.
method Construct and analyze two linear connections: Weitzenböck and sub-Riemannian.
result Two concrete examples (S^3 and S^7) illustrate the application of sub-Riemannian geometry.
Proof of orientable 3-manifolds parallelizability using knot theory.
problem Proving all orientable 3-manifolds are parallelizable.
method Using knot theory and relationships between tangent and normal bundles.
result Completion and modification of a proof of Stiefel's theorem.
New findings on Kähler structures on specific types of manifolds.
problem Classifying manifolds with p-Kähler structures. method Analyzing nilmanifolds and holomorphically parallelizable solvmanifolds.
result Classification of manifolds with p-Kähler structures for lower values of p. Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.
problem Understanding the Kodaira dimension of real parallelizable manifolds with specific almost complex structures.
method Conditions and examples provided for calculating the Kodaira dimension of manifolds.
result Conditions under which the Kodaira dimension of a real parallelizable manifold is zero.
New proof shows 3D shapes can be continuously deformed.
problem Proving 3D shapes can be continuously deformed without tearing.
method Used Heegaard splitting as a main tool.
result Closed orientable 3-manifolds are parallelizable.
Three proofs show all 3D shapes are parallelizable without complex tools.
problem Proving all 3D shapes are parallelizable without advanced tools.
method Minimal proofs using no spin structures or Stiefel-Whitney classes.
result Three proofs show all 3D shapes are parallelizable.
We prove a stability theorem for families of holomorphically-parallelizable manifolds in the category of Hermitian manifolds.
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
The paper proves conditions for 4-manifolds to be parallelizable and non-existence results.
problem Conditions for 4-manifolds to be parallelizable and non-existence results.
method Homotopy theory, particularly obstruction theory.
result Compact, oriented 4-manifolds with vanishing Euler characteristic and signature admit coassociative-free immersions.
The paper explores algebraic and geometric structures on parallelizable manifolds.
problem Understanding algebraic and geometric structures on parallelizable manifolds.
method Definition of fundamental vector fields and their flows, leading to a product and loop structure.
result Induces a local loop structure and generalizes Lie algebra structure on the vector space.
A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of SO(4)-bundles over the 4-sphere using Euler and first Pontryagin classes. result A closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class, and Euler characteristic vanish.
Study homogenizes equations on parallelizable manifolds using tensor localization and periodicity.
problem Homogenizing oscillating linear elliptic equations on parallelizable manifolds.
method Two-scale convergence through localization and periodicity induced by geometry.
result Explicit cell formulae for the homogenization limit and a theory of two-scale convergence of tensors.
Study on ruled surfaces over elliptic curves with unique foliations and parallelizable 4-webs.
problem Characterizing the geometry of ruled surfaces over elliptic curves.
method Analysis of foliations, minimal self-intersection sections, and 2-webs.
result The 4-web defined by fibration, foliation, and minimal self-intersection sections is locally parallelizable.
Modified BFGS and LBFGS++ libraries boost performance for non-parallelizable functions.
problem Improving performance of non-parallelizable functions using SIMD and AAD.
method Modifications to BFGS and LBFGS++ libraries, utilizing SIMD and Automatic Differentiation (AAD).
result Up to 3.8 times faster for European Swaption curve calibration and 1.4 times faster for LMM model calibration.
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
pLSTM tackles long-range language modeling and computer vision tasks with parallelizable linear source transition mark networks.
problem Challenges of existing recurrent architectures in handling sequences and multi-dimensional data.
method Introduces pLSTM, a parallelizable linear source transition mark network for linear graphs and DAGs, addressing vanishing/exploding activation/gradient issues.
result pLSTM outperforms Transformers in long-range tasks like arrow-pointing extrapolation and image size extrapolation.
Motivated by the Hamilton's Ricci flow, we define the homogeneous flow of a parallelizable manifold and show the long time existence and uniqueness of its solutions on [0,∞). Using this flow, we outline a simple proof of the Poincare Conjecture.
SpInGP speeds up Gaussian process computations with sparse matrices.
problem Efficiently computing Gaussian processes for large datasets.
method Sparse precision Gaussian process formulation and parallelizable matrix routines.
result The parallelized SpInGP reduces time complexity to sublinear.
We show that if M is an orientable 4-dimensional infrasolvmanifold and either β=β1(M;Q)≥2 or M is a Sol04- or a Solm,n4-manifold (with m=n) then M is parallelizable. There are non-parallelizable examples with β=1 for each of the other solvable Lie geometries $\ma…
Study finds infinite families of Sasaki-Einstein metrics on spheres.
problem Finding Sasaki-Einstein metrics on spheres and exotic spheres.
method Analyzing odd-dimensional spheres and exotic spheres that bound parallelizable manifolds.
result Infinitely many families of Sasaki-Einstein metrics on spheres and exotic spheres.
Stability of flat vector bundles on complex parallelizable manifolds proven.
problem Stability of flat holomorphic vector bundles over compact complex parallelizable manifolds.
method Proved a structure theorem for flat holomorphic vector bundles associated to irreducible representations.
result Flat holomorphic vector bundles are holomorphically isomorphic to a stable vector bundle of the form E⊕n. Stochastic Variational Optimization is a parallelizable method for gradient estimation.
problem Gradient estimation for differentiable objectives in parallel environments.
method Variational Optimization, Natural Evolution Strategies, Gaussian Perturbation, Directional Derivatives.
result Directional Derivatives are preferable to Variational Optimization for parallel Stochastic Gradient Descent.
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
The paper studies variations of the Godbillon--Vey invariant for certain foliations.
problem Investigating the Godbillon--Vey invariant for transversely parallelizable foliations.
method Constructing a (2q+1)-form analogous to the Godbillon--Vey class and expressing it in terms of ω and ${f T}$ for a compatible Riemannian metric. result Characterizing critical pairs of $(ω,{f T})$ and finding sufficient conditions for critical foliations.
The paper proposes a parallelizable clustering method for multivariate data.
problem The standard model-based clustering method assumes the same number of clusters per margin, which is often unrealistic.
method Developed a finite mixture model per margin with different numbers of clusters, and used a game-inspired algorithm to cluster multivariate data.
result The proposed method shows good performance in various scenarios and real datasets.
New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.
problem Estimating the full persistence diagram of large point clouds is expensive, unstable, and not a sufficient statistic.
method Proposes distributed persistence as a new invariant, which is perfectly parallelizable, more stable, and has a rich inverse theory.
result The map from point clouds to distributed persistence invariants is a global quasi-isometry, interpolating between purely geometric and topological invariants.
New 4D shapes can't be opened like books.
problem Existence of 4D shapes without open book decompositions.
method Demonstrated existence of infinitely many parallelizable 4-manifolds without open book decompositions.
result No open book decomposition for certain Engel manifolds.
DLF combines autoregressive and flow-based methods for efficient and high-performance image generation.
problem Limited density estimation performance and parallelizability of flow-based and autoregressive models.
method Dynamic Linear Flow (DLF) with partially autoregressive structure.
result DLF achieves state-of-the-art performance on ImageNet 32x32 and 64x64 images.
This paper proposes using DPPs for hyperparameter optimization, improving diversity over random search.
problem Improving hyperparameter optimization methods for parallelizable and diverse sampling.
method Introducing k-determinantal point processes (DPPs) for hyperparameter optimization via random search. result DPPs promote diversity in hyperparameter optimization, leading to significant benefits in training supervised learners.
There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic point…
We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.
The aim of this paper is to write an explicit orthonormal parallelization for all parallelizable products of spheres, using an explicit isomorphism with a trivial vector bundle.
The paper solves a 25-year-old problem about maximal growth distributions on manifolds.
problem Existence and classification of maximal growth distributions on smooth manifolds.
method Higher order convex integration and new criteria for ampleness of differential relations.
result Positive answer to the open question about parallelizable manifolds admitting maximal growth distributions.
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
Let G be a simply connected solvable Lie group with a lattice Γ and the Lie algebra $\g$ and a representation ρ:G→GL(Vρ) whose restriction on the nilradical is unipotent. Consider the flat bundle Eρ given by ρ. By using "many" characters {α} of G and "many" flat line bundles {Eα} over G/Γ, w…
EigenGame reinterprets PCA as a game to find eigenvectors.
problem Finding principal components efficiently and accurately.
method EigenGame treats PCA as a Nash equilibrium game, using gradient-based updates.
result EigenGame algorithm combines Oja's rule and Gram-Schmidt orthogonalization.
We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.
We study conditions under which sub-complexes of a double complex of vector spaces allow to compute the Bott-Chern cohomology. We are especially aimed at studying the Bott-Chern cohomology of special classes of solvmanifolds, namely, complex parallelizable solvmanifolds and solvmanifolds of splitting type. More precise…
This article introduces the problem of finding intrinsic torsion varieties associated to G-structures on a fixed parallelizable Riemannian manifold. As an illustration, the intrinsic torsion varieties of orthogonal almost product structures are analysed on the Iwasawa manifold.
Proposes SVM-based Deep Stacking Network for improved deep learning.
problem Improving deep learning performance and interpretability.
method Uses stacked SVM classifiers within a DSN architecture and a BP-like layer tuning scheme.
result Demonstrates superior performance compared to benchmark models on image and text data.
MSLs use parallelizable root-finding for efficient ODE and PDE solutions.
problem Efficiently solving initial value problems for ODEs and PDEs.
method Leveraging time-parallel methods, MSLs use parallelizable root-finding algorithms.
result MSLs offer significant speedups in NFEs and inference time.
New CNN model for fast unsupervised document embedding.
problem Efficient unsupervised document embedding with parallelizable architecture.
method Convolutional Neural Network (CNN) for parallel inference and stochastic forward prediction for unsupervised learning.
result Comparable accuracy to state-of-the-art at significantly reduced computational cost.
In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior …
The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. BatchEnsemble reduces ensemble costs by 3X in training and testing.
problem High costs for training and testing ensembles of neural networks.
method Defines each weight matrix as a Hadamard product of a shared matrix and a rank-one matrix per member.
result Achieves 3X speedup and 3X memory reduction in test time for ensembles of size 4.
Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
problem Determining special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.
method Algebraic equations, invariant distributions, deformation theory, SYZ mirror symmetry.
result Existence of topologically distinct SLags and non-Kähler SYZ mirrors.
New fault-tolerant quantum gates for homological LDPC codes with constant or almost-constant rate.
problem Fault-tolerant quantum computing for homological LDPC codes with constant or almost-constant encoding rate.
method Derive generic formula for transversal and logical gates acting on 3-manifolds, using higher symmetries and cup product cohomology.
result Parallelizable logical gates for homological LDPC codes with constant or almost-constant rate.
The paper tackles scalable simulation of discrete random variables.
problem Simulating discrete random variables with general and varying distributions in a scalable framework.
method Inspired by discrete choice models, the paper introduces parallelized randomness and a single associative operation for simulation.
result Characterization of algorithms for scalable simulation of discrete random variables.