arXiv research

Study of tangent cones at infinity for algebraic sets.

Active subspaces on Riemannian manifolds generalize Euclidean principles.

BSA reduces network data by interpreting feature subspaces.

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.

The paper develops methods to reduce deployment risk under dynamic covariate shifts.

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 $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theori…

A new method approximates tangent spaces to simplify neural networks.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points.…

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that …

A new diffusion sampling method combines Krylov subspace and diffusion models for faster and more efficient inverse problems.

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of…

New method uses outer product manifolds to simplify neural networks.

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.

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.

Defines smoothness of definable sets in o-minimal structures.

PCA adapted for curved spaces improves data analysis.

Estimates manifold from tangent bundle learners.

Bicycle paths form geodesics in 3D subspaces, related 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.

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 $L\subset H$ such that $J(L)=L^\perp.$ The complex unitary group $U(H_J)$, 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 $S^3$ into $\mathbb{C}^3$. 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.

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.

Study equigeodesics on $G_2$-type flag manifolds, splitting tangent spaces.

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, 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.

Let $M=Σ_1\times Σ_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2$ and two, respectively. Let $Σ$ be the graph of a smooth map $f:Σ_1\mapsto Σ_2$, then $Σ$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ 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 …

Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemann…

Abstract reviews distributions and subbundles in differential geometry.

New boundary and point constraints for controlling conformal surfaces.

The study classifies Riemannian manifolds with curvature nullity.

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

A novel method classifies shapes by their square-root velocity function.

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…

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…

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

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.

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.