Probabilistic theory counts intersections in Riemannian spaces.
problem Counting intersections in Riemannian homogeneous spaces.
method Introduces probabilistic intersection ring HE(M), a graded commutative and associative real Banach algebra. result Probabilistic intersection ring structure defined for spheres, real projective space, and complex projective space.
Local logarithmic Brunn-Minkowski holds for zonoids.
problem Logarithmic Brunn-Minkowski conjecture for zonoids
method Bochner method variant
result Local form of conjecture proven for zonoids
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.
problem Proving the log-Brunn-Minkowski inequality for convex bodies and zonoids.
method Establishing monotonicity of the deficit in the LLBM under line segment addition.
result Equality in LLBM for smooth convex bodies occurs only for homothetic bodies.
The main results are two characterisations of log-concave densities in terms of the collection of lift zonoids corresponding to a peacock. These notions are recalled and connected to arbitrage-free asset pricing in financial mathematics.
Solves approximation problems for zonoids and neural networks, closing gaps in dimensions 2 and 3.
problem Approximating zonoids and shallow neural networks in uniform norm.
method Combines techniques to solve both problems, closing gaps in dimensions 2 and 3.
result Completes the solution for zonoid approximation in all dimensions and improves neural network approximation rates.
The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures and their differential prolongations are found. A complete system of geometric …
Different topics on the differential geometry of the complex Grassmann manifold are surveyed in relation to the coherent states. A calculation of the tangent conjugate locus and conjugate locus in the complex Grassmann manifold is presented. The proofs use the Jordan's stationary angles. Also various formulas for the d…
In the present paper we study locally semiflat (we also call them semiintegrable) almost Grassmann structures. We establish necessary and sufficient conditions for an almost Grassmann structure to be alpha- or beta-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann st…
We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given f…
This paper shows that the Grassmann Manifolds GF(n,N) can all be imbedded in an Euclidean space MF(N) naturally and the imbedding can be realized by the eigenfunctions of Laplacian △ on GF(n,N). They are all minimal submanifolds in some spheres of MF(N) respectively. Using …
Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we addr…
The differential geometry of almost Grassmann structures defined on a differentiable manifold of dimension n = pq by a fibration of Segre cones SC (p, q) is studied. The peculiarities in the structure of almost Grassmann structures for the cases p=q=2; p = 2, q > 2 (or p > 2, q = 2), and p > 2, q > 2 are clarified. The…
Paper introduces a method to generate stable shapes using Grassmann manifolds.
problem Generating stable shapes with minimal extraneous transformations.
method Continuous normalization flows on Grassmann manifolds to eliminate extraneous transformations.
result The method significantly outperforms state-of-the-art methods in generating high-quality samples.
Calculates spinor heat flow using Gaussian-Grassmann integrals.
problem Computing the spinor heat flow symbol.
method Getzler calculus and Gaussian-Grassmann integrals.
result Computed the spinor heat flow symbol.
New method for identifying autoregressive systems on manifolds.
problem Identifying autoregressive systems on Stiefel and Grassmann manifolds.
method Defining parameters as orthogonal group elements, averaging over observations, conjugate gradient descent on manifolds.
result System parameters can be estimated efficiently using the proposed algorithm.
This handbook simplifies Grassmann manifold geometry for matrix-based algorithms.
problem Modeling linear subspaces in various applications.
method Expository work on Grassmann manifold geometry, including new algorithms and formulas.
result Improved understanding and computational tools for the Grassmann manifold.
New geometric structures defined on Grassmann manifolds.
problem Understanding geometric properties of Grassmann manifolds.
method Introducing canonical blow-ups and submanifolds by partitioning Plücker coordinates.
result Various geometric aspects of introduced structures are studied, including smoothness, holomorphic symmetries, and existence of Kähler-Einstein metrics.
Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite, number of loss functions. In this paper, we propose a novel Riemannian extension of the Euclidean stochastic variance reduced gradient algorithm (R-SVRG) to a compact manifold search space. To this e…
A new method for averaging data on manifolds is proposed, offering simplicity and efficiency.
problem The difficulty of computing Fréchet means on manifolds, especially Stiefel and Grassmann.
method Proposed RL-barycenters, simpler arithmetic means projected onto the manifold.
result RL-barycenters yield simple yet effective means on Stiefel and Grassmann manifolds.
Two constructions link path geometries to almost Grassmann structures.
problem Linking path geometries to almost Grassmann structures.
method Introducing two Fefferman-type constructions.
result Characterizing conditions for almost Grassmann structures arising from these constructions.
On the Grassmann manifold G (m, n) of m-dimensional subspaces of an n-dimensional projective space P^n, a certain supplementary construction called the normalization is considered. By means of this normalization, one can construct the structure of a Riemannian or semi-Riemannian manifold or an affine connection on G(m,…
Let L2 be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of L2. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p…
A new method for SVGD reduces variance in high dimensions.
problem High-dimensional variance in SVGD.
method Grassmann Stein Variational Gradient Descent (GSVGD) projects onto arbitrary subspaces and uses coupled Grassmann-valued diffusion.
result GSVGD explores high-dimensional problems with intrinsic low-dimensional structure efficiently.
It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local moving frame of a holomorphic immersion of a Riemann surface into a complex Grassmann…
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
Study on conjugate points in a C∗-algebra's Grassmann manifold.
problem Characterizing conjugate points in a C∗-algebra's Grassmann manifold. method Analyzes connections and geodesics in the Riemannian metric induced by the Killing form.
result Points that are tangent conjugate in the classical setting may not be conjugate in a C∗-algebra's Grassmann manifold. Study geodesics on Grassmann manifold for functions vanishing on subsets of a set X.
problem Finding minimal geodesics on Grassmann manifold of reproducing kernel Hilbert spaces.
method Analyzing necessary and sufficient conditions for geodesic existence and uniqueness, and studying examples.
result Established conditions for geodesic existence and uniqueness, and found estimates on eigenvalues.
Let M be a manifold with Grassmann structure, i.e. with an isomorphism of the cotangent bundle T^*M\cong E\otimes H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection \nabla^W in a vector bundle W\to M as a connection whose curvature F\in S^2E\otimes\wedge^2 H\otimes W…
The Lagrangian formalism on a arbitrary non-fibrating manifold is considered. The kinematical description of this generic situation is based on the concept of (higher-order) Grassmann manifolds which is the factorization of the regular velocity manifold to the action of the differential group. Here we introduce in this…
New method recovers matrices with nonlinear structures using optimization on Grassmann manifold.
problem Recovering high-rank matrices with nonlinear structures like subspaces or clusters.
method Formulated as rank minimization of a nonlinear feature map, approximated by constrained non-convex optimization on the Grassmann manifold, using Riemannian and alternating minimization schemes.
result Global convergence and worst-case complexity bounds for alternating minimization scheme, leading to unique limit point.
Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
Study classifies super vector bundles and proves universality.
problem Homotopy classification of super vector bundles.
method Construction of supergrassmannians, Gauss morphism, multilinear algebra, direct and inverse limits.
result Proves the resulting super vector bundle is universal.
New algorithm for multiway spectral clustering on Grassmann manifolds.
problem Efficiently computing multiple eigenvectors of a nonlinear graph Laplacian.
method Direct multiway spectral clustering in p-norm, reformulated as minimization on Grassmann manifold. result Monotonic decrease of balanced graph cuts leads to optimal solutions.
New algebraic numbers defined by a specific equation.
problem No direct problem stated; focuses on new algebraic numbers.
method Definition of superalgebraic Markov numbers via a Grassmann integer equation.
result Introduced new algebraic numbers with applications in Teichmüller spaces.
We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilb…
The decomposition of the spinor bundle of the spin Grassmann manifolds Gm,n=SO(m+n)/SO(m)×SO(n) into irreducible representations of so(m)⊕so(n) is presented. A universal construction is developed and the general statement is proven for G2k+1,3, G2k,4, and G2k+1,5 f…
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
Let V be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of p vectors in V, and let $\Gr(p,V)$ be the Grassmann manifold of p dimensional subspaces of V. We study the distance and the geodesics in these manifolds, by reducing the matter to…
This work ensures stability in POD basis interpolation for pMOR in hyperelasticity.
problem Stability of POD basis interpolation on Grassmann manifolds for pMOR in hyperelasticity.
method Stability conditions derived from Grassmannian Exponential map and principal angles.
result Explicit stability conditions for practical pMOR applications and non-monotonic error behavior.
Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.
problem Uncertainty quantification in high-dimensional stochastic systems.
method Principal Geodesic Analysis on the Grassmann manifold, adaptive algorithm for local submanifolds, polynomial chaos expansion.
result Efficient surrogate modeling of system behavior across different parameter spaces.
New measure defined for Brakke flow, linking classical and new definitions.
problem Defining and characterizing the Brakke flow.
method Introduced a space-time-Grassmann measure to characterize the flow.
result Equivalence between classical and new definitions of the Brakke flow.
Designs neural networks on matrix manifolds for improved performance in tasks like human action recognition.
problem Designing neural networks for tasks on non-Euclidean manifolds.
method Develops fully-connected and convolutional layers for SPD manifolds, and MLR on SPSD manifolds.
result Demonstrates improved performance in human action recognition and node classification tasks.
Adapts POD basis for parametric ROMs using pGP.
problem Updating POD basis for accurate system behavior over parameter space.
method Formulates problem as supervised statistical learning, uses pGP to learn mapping between parameter space and Grassmann manifold.
result Proposes pGP for optimal estimation of POD basis parameters and quantifies uncertainty.
Paper builds neural networks on matrix manifolds using gyrovector spaces.
problem Lack of concepts in gyrovector spaces for matrix manifolds.
method Generalized gyrovector space concepts for SPD and Grassmann manifolds, proposing new neural network models.
result Demonstrated effectiveness in human action recognition and knowledge graph completion.
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
The paper finds determinantal expressions for certain symmetric space integrals.
problem Finding compact expressions for integrals on symmetric spaces.
method Expressing integrals as determinants or Pfaffians for K-invariant functions. result Determinantal expressions for specific symmetric space integrals.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures