Develops methods to construct confidence regions for singular subspaces in low-rank matrix regression.
problem Recovering the singular subspace of a low-rank matrix from noisy measurements.
method Two-step procedure involving de-biasing and empirical singular vector calculation.
result Asymptotically normal joint projection distance for confidence regions of the true singular subspace.
A new framework for dimension reduction using ensemble of random projections.
problem High-dimensional regression problems with limited data.
method Aggregating an ensemble of carefully chosen random projections, retaining based on empirical performance, and selecting singular vectors.
result The proposed method stabilizes error as the number of projection groups increases.
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
The paper proves vanishing theorems for vector bundles with singular metrics.
problem Analyzing vector bundles with singular Hermitian metrics.
method Using a sheaf of locally square integrable holomorphic sections and a new Nadel-Nakano type vanishing theorem.
result Generalizations of vanishing theorems for vector bundles with singular metrics.
Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I \mathcal{I} I -good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I \mathcal{I} I -good singularities. The study finds conditions for a special vector field in singularities.
problem Existence of Milnor vector field for new singularities.
method Introducing sufficient conditions for the existence of the Milnor vector field.
result Conditions guarantee the existence of the Milnor vector field for new classes of singularities.
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano…
New curvature defined for corank 1 singular surfaces in 3D.
problem Defining curvature for singular surfaces.
method Introducing axial vector and curvature parabola to define axial curvature.
result Relates axial curvature to Gaussian curvature of a blow-up for certain singularities.
The paper characterizes conformal vector fields in ( α , β ) (α,β) ( α , β ) -spaces via PDEs.
problem Characterizing conformal vector fields in ( α , β ) (α,β) ( α , β ) -spaces. method Using partial differential equations (PDEs) to characterize conformal vector fields.
result Properties of conformal vector fields in certain ( α , β ) (α,β) ( α , β ) -spaces are demonstrated. Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering ℓ ∞ \ell_\infty ℓ ∞ and ℓ 2 , ∞ \ell_{2,\infty} ℓ 2 , ∞ bounds. result Fine-grained insights into singular vector and subspace perturbations, including ℓ ∞ \ell_\infty ℓ ∞ and ℓ 2 , ∞ \ell_{2,\infty} ℓ 2 , ∞ bounds. Research describes all possible gradient vector fields on a sphere with up to ten singular points.
problem Characterizing gradient vector fields on a sphere with limited singular points.
method Using a graph to represent one-dimensional stable manifolds, specifying singularities and connections.
result Identified all topological structures of codimension one gradient vector fields on a sphere with up to ten singular points.
Alternative metric defined on vector bundles, proving vanishing theorem.
problem Defining singular Hermitian metrics on vector bundles.
method Alternative definition of singular Hermitian metric, discussing Griffiths and Nakano positivities.
result Generalised Griffiths' vanishing theorem proved.
Extends Serre-Swan theorem to all finitely generated modules over smooth functions.
problem Classical Serre-Swan theorem limitations.
method Introduces tepui fibrations and singular vector bundles.
result Realizes all finitely generated modules over smooth functions.
The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.
problem Analyzing vector bundles with singular Hermitian metrics and positivity.
method Develops harmonic theory and extends results from complex geometry.
result Extends Nakano's vanishing theorem to vector bundles with singular metrics.
Generic 3D vector fields have singularly hyperbolic transitive sets.
problem Understanding the dynamics of generic three-dimensional vector fields.
method Analyzing C 1 C^1 C 1 generic vector fields on closed 3-manifolds. result Generic vector fields have singularly hyperbolic transitive sets.
A new method reduces task interference in model merging.
problem Model merging overlooks structural information and is susceptible to task interference.
method Task Singular Vectors (TSV) and TSV-Compress for compression and interference reduction.
result TSV-Merge significantly outperforms existing methods.
Efficient SVD algorithm robust to outliers.
problem Outliers in data matrix affect SVD accuracy and speed.
method Spherically Normalized SVD (SpherSVD) algorithm.
result Significantly faster and more robust than existing methods.
New method cleans cross-covariance matrices for better financial forecasting.
problem Asymptotically optimal cross-covariance cleaners fail in real-world, time-varying markets.
method Physics-informed neural network that learns from empirical singular values.
result Trained model outperforms analytical cleaners in out-of-sample cross-covariance prediction.
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L 2 L^2 L 2 extension theorem for holomorphic vector bundles. Study on overlaps of singular vectors in Gaussian matrix submatrices.
problem Analyzing overlaps of singular vectors in submatrices of Gaussian matrices.
method Utilizes dynamics of singular vectors and specific resolvents for Brownian trajectories.
result Explicit forms for limiting rescaled mean squared overlaps in the bulk of spectra.
The abstract extends a Poincaré-Hopf formula to non-isolated singularities.
problem Establishing a Poincaré-Hopf formula for vector fields with non-isolated singularities.
method A special connection a b l a ~ 1 E \widetilde{
abla}_{1}^{E} ab l a 1 E is constructed, and the Chern character m c h ( E , a b l a ~ 1 E ) {
m ch}(E,\widetilde{
abla}_{1}^{E}) m c h ( E , ab l a 1 E ) plays a key role. result A Poincaré-Hopf type formula for a pair of vector fields with non-isolated zero points is established.
Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong L p L^p L p -closure of vector fields with finitely many integer singularities. method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in L Z 1 ( B ) L_{\mathbb{Z}}^1(B) L Z 1 ( B ) , revealing information about mass-minimizing currents. Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
Study vector fields with complex singularities, proving bounds and formulas.
problem Understanding the Milnor number of vector fields with specific singularities.
method Global and local formulas expressing Milnor/Poincare-Hopf contributions, sharp lower bounds under perturbations.
result Sharp lower bounds for Milnor number contributions under holomorphic perturbations.
We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorp…
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
Alternative proof of automorphism property of singular foliations.
problem Understanding the automorphisms of singular foliations.
method Alternative proof using exponential of elements in singular foliations.
result Time-one flow of elements in singular foliations is an automorphism.
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C ∞ C^\infty C ∞ -ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
Generalizes nonabelian Hodge theory to klt singularities.
problem Applying nonabelian Hodge theory to spaces with klt singularities.
method Uses descent theorems for numerically flat vector bundles and a new restriction theorem for semistable Higgs sheaves.
result Establishes a new restriction theorem for semistable Higgs sheaves.
Defines universal index for orbifold singular points.
problem Indices of singular points on orbifolds.
method Introduces universal index for isolated singular points of vector fields or 1-forms on orbifolds.
result An analogue of the Poincaré-Hopf theorem holds for the universal index.
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
Study Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
Proves Sard conjecture for specific distributions, controlling divergence of vector fields.
problem Proving the Sard conjecture for certain types of distributions.
method Constructs a singular distribution capturing essential abnormal lifts, proving the conjecture for rank 3 distributions in dimension 4 and generic corank 1 distributions.
result Proves the Sard conjecture for generic co-rank one distributions.
The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.
problem Existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles.
method Using Cartan's highest weight theory, the paper establishes an algebraic criterion for topological splitting and decouples the prescribed mean curvature equation.
result A sufficient algebraic condition for realizing an L 2 L^{2} L 2 -function as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle. Classifies singularities of smooth vector fields on the line.
problem Classifying singularities of smooth vector fields on the line.
method Local classification with respect to C 1 C^1 C 1 -conjugacy, including normal forms and unfoldings. result Complete description of the 1-d case achieved.
Study shows XRP price correlates with transaction network metrics.
problem Understanding the relationship between cryptoasset price and network metrics.
method Analysis of correlation tensor spectra, random matrix theory comparison, singular values investigation.
result Distinct correlation between XRP price and singular values during bubble and non-bubble periods.
A singular (or Hermann) foliation on a smooth manifold M M M can be seen as a subsheaf of the sheaf X \mathfrak{X} X of vector fields on M M M . We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…
The paper studies vector bundles over surfaces, focusing on singularity formation.
problem Understanding singularity formation in rank two holomorphic vector bundles over surfaces.
method Defining fertile families bearing bubbles and using elementary modifications to prove their existence.
result Existence of fertile families bearing bubbles for certain types of vector bundles.