Study new invariant to measure flat directions on complex manifolds.
problem Investigate numerical rank invariant on projective Kähler manifolds with semi-negative holomorphic sectional curvature.
method Introduce and analyze a new differential geometric numerical rank invariant.
result Bounding the new invariant by nef dimension and numerical Kodaira dimension.
Invariant kernels reduce rank and improve generalization across dimensions.
problem Symmetry in high-dimensional data impacts kernel matrix rank and learning algorithms.
method Compute invariant polynomial kernel ranks under various groups acting on data.
result Symmetry decreases kernel rank, making it independent of data dimension.
In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this…
Study uses knot Floer homology to distinguish slice disks.
problem Classifying slice disks of knots up to isotopy and diffeomorphism.
method Invariants in knot Floer homology to compute and distinguish slice disks.
result Invariant can distinguish non-isotopic slice disks with diffeomorphic complements.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
Study on coskewness under varying dependence uncertainty.
problem Impact of dependence uncertainty on coskewness.
method Explicit bounds and numerical approximation for product expectation; introduction of standardized rank coskewness.
result Introduced standardized rank coskewness as a measure invariant under transformations.
LoRAs enable efficient adaptation of large models; this paper explores processing LoRA weights with machine learning.
problem Efficient processing of low-rank weight decompositions in large finetuned models.
method Developed symmetry-aware invariant and equivariant LoL models to process LoRA weights.
result LoL models can predict CLIP scores, finetuning data attributes, and accuracy on downstream tasks.
Classifies rank 2 affine invariant subvarieties in H(6).
problem Classifying rank 2 affine invariant subvarieties in H(6).
method Extends Apisa's approach, reducing analysis to lower-genus strata.
result Complete description of rank 2 affine invariant subvarieties in H(6).
Abstract summarizes level-rank duality in knot and link invariants.
problem Distinguishing torus knots and links from hyperbolic ones.
method Chern-Simons theory and tables of knot invariants.
result Criterion to distinguish torus knots and links from hyperbolic ones.
Unsupervised ranking faces one critical challenge in evaluation applications, that is, no ground truth is available. When PageRank and its variants show a good solution in related subjects, they are applicable only for ranking from link-structure data. In this work, we focus on unsupervised ranking from multi-attribute…
New proof for higher rank subvarieties in genus three.
problem Classifying higher rank invariant subvarieties in genus three.
method Uses recent techniques developed by Apisa and Wright.
result Short and simplified proof of classification.
We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…
Investigates portfolio selection for rank-dependent utilities in incomplete markets.
problem Portfolio selection for agents with rank-dependent utility in incomplete financial markets.
method Characterizes deterministic strict equilibrium strategies for constant-coefficient and time-invariant probability weighting functions. Addresses the issue of selecting an optimal strategy from multiple equilibrium strategies for time-variant probability weighting functions.
result Characterizes deterministic strict equilibrium strategies and identifies optimal strategies from multiple equilibrium strategies.
The paper introduces algorithms for efficient low-rank matrix approximation.
problem Efficiently approximating large matrices while preserving their properties.
method Random linear images (sketches) of the matrix, with error bounds for quality control.
result Simple, accurate, numerically stable methods for low-rank approximation.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
A new framework improves tensor completion accuracy by considering numerical priors.
problem Tensor completion accuracy loss due to ignoring numerical priors.
method Generalized CP Decomposition Tensor Completion (GCDTC) framework incorporating numerical priors.
result GCDTC framework outperforms state-of-the-arts in non-negative tensor completion.
This paper confirms predictions about 3-manifold instanton Floer homologies using higher rank bundles.
problem Computing 3-manifold instanton Floer homologies using higher rank bundles.
method Using moduli spaces of anti-self-dual connections on hermitian vector bundles of rank N.
result Generalized Donaldson invariants are confirmed for specific 3-manifolds.
This paper tackles ranking-based performance normalization for optimization algorithms.
problem Ranking optimization algorithms across diverse numerical scales disrupts performance comparisons.
method Introduces absolute ranking and a sampling-based computational method to address numerical scale variation.
result Provides a more robust framework for assessing performance across multiple algorithms and problems.
Real hypersurfaces in complex Grassmannians are Hopf if invariant under a specific structure.
problem Characterizing real hypersurfaces in complex Grassmannians of rank two.
method Analyzing the shape operator and quaternionic Kähler structure.
result Real hypersurfaces in complex Grassmannians of rank two are Hopf if invariant under a specific structure.
A new depth function improves multivariate data analysis by considering variability directions.
problem Developing a depth function that respects quantile properties and is affine-invariant.
method Integrating rank-weighted depth with affine-invariance and covariance matrices.
result The AI-IRW depth function provides accurate quantile estimates and is robust to data variability.
This work simplifies proximal mapping for low-rank norms.
problem Efficient computation of proximal mappings for low-rank inducing norms.
method Reduces proximal mapping to nested binary search, solving simpler problems analytically.
result Simplified computation of proximal mappings for various norms.
Study algebraic invariants from lightning self-attention models.
problem Understanding polynomial coefficients of self-attention mechanisms.
method Identify algebraic invariants using polynomial coefficients and coordinate geometry.
result Found linear and nonlinear families of algebraic invariants.
Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X)=n, ran…
Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mariño and…
New metrics reveal oversmoothing in GNNs more accurately than traditional methods.
problem Oversmoothing in graph neural networks reduces model performance.
method Rank-based metrics to measure oversmoothing in GNNs.
result Rank-based metrics consistently capture oversmoothing, while energy-based metrics often fail.
Generalizes Dolbeault cohomology computation to Levi-flat CR structures on compact Lie groups.
problem Computing Dolbeault cohomology for Levi-flat CR structures on compact Lie groups.
method Algebraic classification of left-invariant CR structures combined with Pittie's result on compact Lie groups.
result Generalization of Dolbeault cohomology computation to Levi-flat CR structures.
We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine …
Study exotic IRSs in rank one Lie groups, constructing uncountable families.
problem Understanding IRSs in rank one Lie groups.
method Construction of uncountable families of IRSs in SO(n,1) for n ≥ 2.
result Construction of uncountable families of IRSs in SO(n,1) for n ≥ 2.
LR-EDNN reduces PDE solver complexity by limiting network weights to low-rank subspace.
problem Efficiently solving time-dependent PDEs with deep neural networks.
method Low-rank constraint on network weights using SVD for efficient parameter updates.
result LR-EDNN achieves comparable accuracy to full EDNN with fewer parameters and lower cost.
Revisits SYM theory to compute Donaldson invariants using mock modular forms.
problem Computing Donaldson invariants in topological SYM theory.
method Uses mock modular forms and indefinite theta functions to evaluate correlation functions.
result Explicit evaluation of correlation functions leading to modular data predictions.
Differentiable sorting and rank normalization are incompatible, with specific conditions for admissibility.
problem Incompatibility between differentiable sorting and rank normalization.
method Formalized admissibility through monotone invariance, batch independence, and rank-space stability conditions.
result Different gap-sensitive and batchwise relaxations of rank normalization violate the conditions for admissibility.
New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag structures can be recovered by a fundamental class construction.
Slow feature analysis (SFA) is a method for extracting slowly varying features from a quickly varying multidimensional signal. An open source Matlab-implementation sfa-tk makes SFA easily useable. We show here that under certain circumstances, namely when the covariance matrix of the nonlinearly expanded data does not …
Study homogeneous Einstein metrics on specific non-Kähler C-spaces.
problem Classify and analyze homogeneous Einstein metrics on non-Kähler C-spaces.
method Use painted Dynkin diagrams and mapping degree theory to classify and find Einstein metrics.
result Existence and classification of invariant Einstein metrics on specific spaces.
Study on pure virtual braids, their formality, and related ranks.
problem Formality and ranks of pure virtual braid groups.
method Investigation of resonance varieties, lower central series ranks, Chen ranks, and Alexander-type invariants.
result Complete answer to 1-formality question for pure virtual braid groups.
Metrics stabilize persistent homology in data analysis.
problem Stabilizing invariants for characterizing connectivity structures in data.
method Using contour functions to define metrics for rank invariants.
result Optimal contours provide robust descriptors of spatial patterns.
Corrected misstatements about invariant rank in ECS manifold papers.
problem Misstated invariant rank (d=1 instead of d=2) in several papers on ECS manifolds.
method Identified the class of examples with d=1, explained why d=2, and listed corrections.
result Corrected papers to accurately state d=2 for ECS manifolds.
Formula conjectured for refined SU(3) Vafa-Witten invariants of surfaces.
problem Calculating refined SU(3) Vafa-Witten invariants for smooth surfaces.
method Proved modularity transformation and used Mochizuki's formula and Maulik-Thomas's definition.
result Conjectured formula satisfies refined S-duality and verified in examples.
Seiberg-Witten invariant vanishes for certain 4-manifolds with specific foliations.
problem Analyzing Seiberg-Witten invariants on manifolds with rank 2 foliations.
method Using foliations with positive scalar curvature and \spinc structures.
result Seiberg-Witten invariant vanishes for specified manifolds.
A famous conjecture in gauge theory mathematics, attributed to Witten, suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. Mathematicians have sought a proof of the conjecture by means of a `cobordism program' …
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
Quantum algorithm approximates Khovanov homology ranks.
problem Efficient computation of Khovanov homology ranks.
method Novel quantum algorithm with pre-thermalization procedure.
result Additive approximations to Khovanov homology ranks are hard problems.
Classifies measures for Anosov subgroups in higher ranks.
problem Classifying horospherical invariant measures for Anosov subgroups.
method Geometric approach, not relying on flows or ergodic theorems.
result Extends results from rank one to higher ranks, solving open problems.
We prove that generalized mutation preserves several geometric invariants such as the volume and Goncharov invariant of Q-rank 1 locally symmetric spaces.
The 2-rank of a compact Lie group G is the maximal possible rank of the elementary 2-subgroup Z2×...Z2 of G. The study of 2-ranks (and p-rank for any prime p) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups h…
Patch ranking improves CNN performance by focusing on object content, not location.
problem CNNs lack rotation and translation invariance, limiting model capacity.
method Patch ranking before convolution and pooling to encode invariance.
result Patch ranking module improves CNN performance on various tasks.
New methods rank players using covariates and comparisons, outperforming existing algorithms.
problem Ranking players based on incomplete and noisy pairwise comparisons.
method Three spectral ranking methods incorporating player covariates.
result Proposed methods outperform existing algorithms in simulations.