New DR algorithm preserves both local and global structure.
problem Trade-off between preserving local and global structure in DR methods.
method Analysis of existing DR methods and design principles for loss functions.
result Design of PaCMAP algorithm that preserves both local and global structure.
Eigen-GNN enhances GNNs by preserving graph structures.
problem Existing shallow GNNs fail to effectively preserve graph structures.
method Integrates eigenspace of graph structures into GNNs as a dimensionality reduction module.
result Eigen-GNN boosts GNNs' ability to preserve graph structures without increasing depth.
Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-k projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These condit…
Reproduces IVFS for high-dimensional data structure preservation.
problem Preserving high-dimensional data structure in unsupervised feature selection.
method Inspired by random subset method, IVFS maintains data similarity through topological structure.
result IVFS outperforms SPEC and MCFS on most datasets.
GCML preserves geometric structure in manifold clustering for diverse data types.
problem Loss functions in manifold clustering can corrupt latent space structure.
method GCML framework with isometric and ranking losses for geometric structure preservation.
result GCML outperforms other methods in latent space structure preservation and performance metrics.
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…
Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
Study preserves symplectic structure in forced discrete mechanical systems.
problem Preserving symplectic structure in forced discrete mechanical systems.
method Analyzes a specific type of forced discrete mechanical system (Q,Ld,fd), preserving a symplectic structure on QimesQ. result The preserved symplectic structure can be seen as Marsden-Weinstein reduction of the canonical symplectic structure.
Paper proposes G-CRD to improve GNNs by preserving global graph topology.
problem Improving lightweight GNNs for robust performance on large-scale real-world graphs.
method Introduces Graph Contrastive Representation Distillation (G-CRD) using contrastive learning.
result G-CRD consistently boosts GNN performance and robustness, outperforming existing methods.
DMT enhances deep neural networks to better preserve data structures.
problem Preserving geometric, topological, and distributional structures of data in NLDR.
method Deep manifold transformation (DMT) using cross-layer LGP constraints.
result DMT networks outperform existing NLDR methods in preserving data structures.
New method preserves MHD equations on sphere without costly matrix exponentials.
problem Discretizing MHD equations on sphere for numerical simulations.
method Lie-Poisson discretization, geometric quantization, semi-direct product Lie algebras.
result Preserves Lie-Poisson structure and Casimir functions.
Text style transfer aims to modify the style of a sentence while keeping its content unchanged. Recent style transfer systems often fail to faithfully preserve the content after changing the style. This paper proposes a structured content preserving model that leverages linguistic information in the structured fine-gra…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
Method preserves correlations in synthetic data.
problem Preserving dependence structure of original data.
method Orthogonal Procrustes problem for restoring Pearson correlation.
result Restores Pearson correlation structure while preserving feature distributions and downstream tasks performance.
SDSPCAAN combines supervised and local data structures for better dimensionality reduction.
problem Preserving both global and local data structures for noisy high-dimensional data.
method Supervised discriminative sparse PCA with adaptive neighbors (SDSPCAAN).
result SDSPCAAN improves classification accuracy on high-dimensional datasets.
Establishes correspondence between Calabi-Yau and Landau-Ginzburg structures.
problem Preserving real structures in the Calabi-Yau/Landau-Ginzburg correspondence.
method Detailed analysis of period integrals and modification of real structures.
result Full CY/LG correspondence for tt∗ structures established. New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…
Paper proposes a new method for supervised manifold learning using random forest proximities.
problem Existing supervised manifold learning methods fail to uncover meaningful embeddings due to using class-conditional distances.
method Proposes a data-geometry-preserving variant of random forest proximities as an initialization for manifold learning methods.
result Local and global structure preservation is near universal across manifold learning approaches using diffusion-based algorithms.
Domain adaptation aims to assist the modeling tasks of the target domain with knowledge of the source domain. The two domains often lie in different feature spaces due to diverse data collection methods, which leads to the more challenging task of heterogeneous domain adaptation (HDA). A core issue of HDA is how to pre…
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus g≥2, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infini…
Structure-preserving GANs learn distributions with group symmetry efficiently.
problem Learning distributions with group symmetry efficiently.
method Developed structure-preserving GANs by reducing the discriminator space and designing structured generators.
result Significantly improved sample fidelity and diversity in small data regimes.
Network embedding is the process of learning low-dimensional representations for nodes in a network, while preserving node features. Existing studies only leverage network structure information and focus on preserving structural features. However, nodes in real-world networks often have a rich set of attributes providi…
Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.
problem Optimal mass transport for vector-valued Gaussian mixtures.
method Vectorizing Gaussian mixture models and studying optimal mass transport problems.
result Computational efficiency and structure preservation in optimal mass transport.
We present a numerical approach for approximating unknown Hamiltonian systems using observation data. A distinct feature of the proposed method is that it is structure-preserving, in the sense that it enforces conservation of the reconstructed Hamiltonian. This is achieved by directly approximating the underlying unkno…
Holonomy-preserving transformations help recover Alexander polynomials from graph zeta functions.
problem Recovering Alexander polynomials from graph zeta functions.
method Introducing holonomy to preserve zeta functions of matrix-weighted graphs and extending to group elements and quandles.
result Holonomy-preserving transformations correspond to transformations of group presentations and preserve the twisted Alexander polynomial.
Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the co…
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of …
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have not been well studied. Our key contribution is to show that 2K projection vect…
PCA-Guided Quantile Sampling preserves data structure in large datasets.
problem Preserving data structure in large-scale subsampling.
method PCA QS uses leading principal components to guide stratified sampling.
result PCA QS maintains statistical and geometric structure without distorting semantics.
We generalize the hyperkaehler quotient construction to the situation where there is no group action preserving the hyperkaehler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkaehler manifolds arise as …
Union of Subspaces (UoS) is a popular model to describe the underlying low-dimensional structure of data. The fine details of UoS structure can be described in terms of canonical angles (also known as principal angles) between subspaces, which is a well-known characterization for relative subspace positions. In this pa…
The Ricci flow preserves product structures with instantaneous curvature bounds.
problem Preserving product structures under Ricci flow with curvature constraints.
method Proving a constant ε exists such that if a solution splits as a product at time 0 and has bounded curvature, it splits for all time.
result A constant ε exists depending on dimension such that if a solution splits as a product at time 0 and has curvature bounded by ε/t, it splits for all time.
Bispectral OT improves dataset comparison by preserving intrinsic coherence.
problem Ignoring intrinsic coherence in dataset comparisons using pairwise geometric distances.
method Introduces Bispectral Optimal Transport, a symmetry-aware extension of discrete OT.
result Transport plans computed with Bispectral OT achieve greater class preservation accuracy.
Let $\imath: M\to \RR^{p+2}$ be a smooth embedding from a connected, oriented, closed p-dimesional smooth manifold to $\RR^{p+2}$, then there is a spin structure ♯(ςp+2) on M canonically induced from the embedding. If an orientation-preserving diffeomorphism τ of M extends over as an o…
Proof that m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
problem Consistent definition of symplectic structures on higher Lie groupoids under Morita equivalence.
method Rigorous proof of m-shifted symplectic forms preservation.
result m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
Enhances machine learning models by preserving data structure, addressing statistical distortions.
problem Statistical distortions in synthetic data generated by Mixup.
method Proposes a generalized mixup method with a flexible weighting scheme to preserve data structure.
result Preserves statistical properties of original data while maintaining model performance.
For actions with a dense orbit of a connected noncompact simple Lie group G, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a G-action to be equivalent to one on a space of the form (G×K\H)/Γ, it is necessary and suff…
A fast method learns plasma collision kernels from simulations, improving kinetic models.
problem Improving kinetic models for plasma dynamics beyond the weakly coupled regime.
method Data-driven collisional operator, fast spectral separation method.
result Accurately captures plasma dynamics in moderately coupled regime.
The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.
problem Understanding the relationship between Poisson-Lie structures and invariant volume forms in Hamiltonian systems.
method Analyzing the existence and preservation of invariant volume forms under Hamiltonian vector fields on Poisson-Lie groups.
result A unimodular Poisson-Lie structure ensures the preservation of a multiple of any left-invariant volume, and the existence of a preserving volume form implies unimodularity.
Brain networks have received considerable attention given the critical significance for understanding human brain organization, for investigating neurological disorders and for clinical diagnostic applications. Structural brain network (e.g. DTI) and functional brain network (e.g. fMRI) are the primary networks of inte…
This paper develops embeddings that preserve likelihood-based statistical inference.
problem Modern machine learning embeddings destroy the geometric structure required for likelihood-based inference.
method Developed a rigorous theory of likelihood-preserving embeddings and introduced the Likelihood-Ratio Distortion metric.
result Controlling the distortion Δn is necessary and sufficient for preserving inference. Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.
problem Geometric properties preserved by compactifications in relation to coarse structures and group actions.
method Analyzes compactifications of spaces with coarse structures and group actions, proving preservation of geometric properties.
result Geometric properties are preserved by compactifications when coarse structures and group actions are involved.
EPNE models evolving network patterns for better predictions.
problem Capturing evolving patterns in dynamic networks.
method EPNE models temporal network evolution using causal convolutions and a temporal objective function.
result EPNE outperforms other methods in various prediction tasks.