Study of tangent cones at infinity for algebraic sets.
problem Characterizing algebraic sets based on their tangent cones at infinity.
method Definition and analysis of tangent cones C4,∞(X) and C5,∞(X), proving properties and relations. result Affine linear subspace characterization based on C5,∞(X)'s dimension. Active subspaces on Riemannian manifolds generalize Euclidean principles.
problem Understanding how scalar-valued quantities change over Riemannian manifolds.
method Generalization of active subspaces from Euclidean to Riemannian spaces using parallel transport.
result The method provides a new way to study scalar-valued quantities on manifolds, differing from extrinsic approaches.
BSA reduces network data by interpreting feature subspaces.
problem Interpreting feature subspaces of unlabeled network data.
method Barycentric Subspace Analysis (BSA) for unlabeled networks.
result BSA provides a more interpretable approach compared to PCA.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.
problem Embedding Hermitian symmetric spaces into their tangent spaces.
method Using polarity of the K-action to construct equivariant embeddings.
result Characterizes holomorphic/symplectic embeddings and realizes submanifolds.
The paper develops methods to reduce deployment risk under dynamic covariate shifts.
problem Reduction of deployment risk under dynamic covariate shifts.
method Time-domain Poincare inequality and Jacobian-velocity theorem to identify and control directional tangent energy.
result Drift-aligned tangent regularization (DTR) reduces risk volatility and directional gain in low-rank drift regimes.
For any principal bundle P, one can consider the subspace of the space of connections on its tangent bundle TP given by the tangent bundle TA of the space of connections A on P. The tangent gauge group acts freely on TA. Appropriate BRST operators are introduced for quantum field theori…
A new method approximates tangent spaces to simplify neural networks.
problem Efficiency of hierarchical neural networks is hindered by their complexity and training requirements.
method Approximates tangent subspace to enable sparse representation and switch to shallow networks.
result The method improves and sometimes surpasses the performance of original networks after a few epochs.
The paper proves conditions for C1 regularity of definable sets using tangent cones and paratangent cones.
problem Conditions for C1 regularity of definable sets in o-minimal structures. method Analysis of tangent and paratangent cones to establish C1 regularity. result Equivalence of three conditions for C1 regularity of definable sets. A new diffusion sampling method combines Krylov subspace and diffusion models for faster and more efficient inverse problems.
problem Efficiently solving large-scale inverse problems in high-performance computing.
method Proposes a novel diffusion sampling strategy that integrates Krylov subspace methods with diffusion models.
result Demonstrates significant speedup (80x faster inference time) and improved reconstruction quality on real-world medical imaging problems.
The paper generalizes PCA to manifolds using barycentric subspaces.
problem Generalizing PCA to non-Euclidean spaces.
method Introducing barycentric subspaces and optimizing AUV criterion.
result Barycentric Subspaces Analysis (BSA) generalizes PCA to Riemannian manifolds.
New method uses outer product manifolds to simplify neural networks.
problem Exponential inefficiency of hierarchical neural networks.
method Reparametrization invariant Riemannian metrics and tangent subspace computation.
result Significant improvement in network performance after early training.
In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as…
We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov; their higher order curvature operator has constant Jordan normal form on the Grassmannian of unoriented k-dimensional spacelike subspaces of the tangent plane.
Locally 2-fold symmetric manifolds are proven to be locally symmetric.
problem Characterizing and proving symmetry properties of manifolds.
method Definition and analysis of locally k-fold symmetric manifolds, application of local isometries.
result Locally 2-fold symmetric manifolds are shown to be locally symmetric.
The study calculates the average number of tangent k-dimensional subspaces to multiple convex hypersurfaces in random position.
problem Determining the average number of k-dimensional subspaces tangent to multiple convex hypersurfaces in random position.
method Investigates the problem from a random point of view, using the Orthogonal group to translate hypersurfaces and calculating the average number of k-flats tangent to all hypersurfaces.
result The average number of k-flats tangent to d_{k,n} many random (n-k-1)-flats is given by a formula involving volumes and curvature integrals.
We study the link between a compact hypersurface in ¶n+1 and the set of all its tangent planes. In this context, we identify ¶n+1 to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise to a kind of duality which has already been studied Bruce and Romerro-Fuster, and r…
Analytic curves have infinite codimension of singular germs.
problem Understanding the codimension of singular tangent curves in analytic distributions.
method Formalizing asymptotic statements about finite jets of tangent curves and applying the h-principle.
result The subspace of singular germs has infinite codimension within smooth curves.
Defines smoothness of definable sets in o-minimal structures.
problem Characterizing smoothness of definable sets in o-minimal structures.
method Characterizes smoothness using tangent cones and metric properties.
result Equivalence of several conditions for C1 smoothness of definable sets. PCA adapted for curved spaces improves data analysis.
problem PCA's limitations in curved spaces.
method Space Form PCA (SFPCA) for Riemannian manifolds.
result SFPCA provides faster and more accurate subspaces estimation.
Estimates manifold from tangent bundle learners.
problem Estimate manifold structure from tangent bundle learners.
method Local PCA methods using data assigned to tangent spaces.
result Reliable estimates of manifold from tangent spaces.
Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.
problem Optimizing bicycle paths between two points.
method Variational equations and geometric analysis of bicycle paths.
result Bicycle geodesics are contained in 3D subspaces and relate to Kirchhoff rods.
I construct an algebraic model for a typical fiber on a 1+1 dimensional spacetime. The vector space comprising the fiber is composed of elements formed from the direct product of two copies of an element x in the D2=C2xC2 finite group algebra over the real numbers. The fiber contains subspaces whose elements are associ…
New Adam optimizer generalized for manifold training of neural networks.
problem Lack of clear physical intuition and difficulty in generalizing Adam optimizer to manifolds.
method Leverages the global tangent space representation of manifolds to perform Adam optimizer steps.
result Significant speed-ups in transformer training with orthogonality constraints.
Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Λ(H) of H, i.e. the set of closed linear subspaces L⊂H such that J(L)=L⊥. The complex unitary group U(HJ), consisting of the elements of the orthogona…
In the mid-1980's, M. Gromov used his machinery of the h-principle to prove that there exists totally real embeddings of S3 into C3. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, …
Poor approximators found in neural networks and random feature models.
problem Understanding why certain neural networks and models perform poorly in approximating functions.
method Established a scale separation of Kolmogorov width type and applied it to neural networks and random feature models.
result Reproducing kernel Hilbert spaces and two-layer neural networks are poor L2-approximators for certain functions. We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…
In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and the subspace of those without closed orbits, as long as the leaves of the foliati…
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.
Study equigeodesics on G2-type flag manifolds, splitting tangent spaces.
problem Characterize geodesics in G2-type flag manifolds. method Analyze flag manifolds with G2-type t-roots, split tangent spaces, and classify equigeodesics. result Characterized structural equigeodesic vectors in flag manifolds.
The configuration manifold M of a mechanical system consisting of two unconstrained rigid bodies in Rn, n≥1, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A b…
Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dim…
Unified PCA framework on flag manifolds for robust data analysis.
problem Outliers and manifold data in PCA.
method Generalization of PCA to flag manifolds, optimization problems, and tangent-PCA integration.
result Novel robust and dual geodesic PCA variations.
Let M=Σ1×Σ2 be the product of two compact Riemannian manifolds of dimension n≥2 and two, respectively. Let Σ be the graph of a smooth map f:Σ1↦Σ2, then Σ is an n-dimensional submanifold of M. Let G be the Grassmannian bundle over M whose fiber at each point is the set of …
We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence …
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.
New boundary and point constraints for controlling conformal surfaces.
problem Controlling the geometry of surfaces defined by minimizers of conformal variational problems.
method Introducing new boundary conditions, point constraints, and flux constraints to control the metric and conformal scale factor.
result Introduces intuitive controls for exploring a subspace of conformal immersions.
The study classifies Riemannian manifolds with curvature nullity.
problem Classifying Riemannian manifolds with nontrivial curvature nullity.
method Classification theorems based on curvature nullity, scalar curvature, and quotient existence.
result New classification theorems and revisited previous results.
A novel method classifies shapes by their square-root velocity function.
problem Classifying shapes in infinite-dimensional, curved spaces.
method Square-root velocity function, tangent spaces, principal components, combining pairwise classifiers.
result Improves classification accuracy by separating shapes and reducing dimensionality.
We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve…
The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.
problem Investigate how the core Lie algebra controls the Lie superalgebra.
method Construct Lie superalgebras from abstract Lie algebras using Schouten-like brackets and analyze Betti numbers of super homology groups.
result For low dimensional non-abelian Lie algebras, the Betti numbers of super homology groups provide insights into the control of the core Lie algebra.
In this article we discuss the interaction between the geometry of a quaternion-Kahler manifold M and that of the Grassmannian G(3,g) of oriented 3-dimensional subspaces of a compact Lie algebra g. This interplay is described mainly through the moment mapping induced by the action of a group G of quaternionic isometrie…
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…
New height estimate for area minimizing currents, leading to unique tangent cones and decay properties.
problem Analyzing singularities of area minimizing currents.
method Height estimate, decay estimates, techniques inspired by previous works.
result Locally area minimizing currents have a unique tangent cone at almost every point and decay rapidly to a unique tangent plane at branch points.
The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.
problem Understanding cohomological dimensions of affine manifolds with specific holonomy groups.
method Analyzing the tangent bundle structure and using coarse geometry techniques.
result The cohomological dimension is bounded by the dimension minus the index of the holonomy group.
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the c…