The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…
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48 results for “isotropic subspace”
Constructs a Morse-Bott function on symplectic Grassmannians.
problem Defines a function on symplectic Grassmannians.
method Uses a compatible linear complex structure to construct a quadratic Morse-Bott function.
result Critical loci consist of subspaces splitting into isotropic and complex parts.
Novel guarantees for PCA in noisy data with correlated noise and missing data.
problem PCA in non-isotropic and data-dependent noise.
method Novel finite sample guarantees for PCA under correlated noise conditions.
result Near-optimal sample complexity for subspace recovery error.
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.
We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space.
The paper explores algebraic structures on diffeological vector spaces.
problem Defining and understanding algebraic structures on diffeological vector spaces.
method Formal checks, verification of properties, and analysis of decompositions.
result The maximal isotropic subspace is invariant and unique, while characteristic subspaces are not.
Learning and Concentration for High Dimensional Linear Gaussians: an Invariant Subspace Approachstat.ML
Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.
problem Understanding correlation and mixing in high-dimensional linear systems with Gaussian noise.
method Sampling from sub-trajectories, using Talagrand's inequality, and analyzing invariant subspaces.
result Large discrepancy between algebraic and geometric multiplicity leads to bottlenecks between invariant subspaces.
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian…
Estimates shared linear subspace from noisy data with multiple users.
problem Recovering shared linear subspace from noisy data with non-isotropic noise.
method Estimates shared subspace using at least two data points per user, avoiding restrictive assumptions.
result Upper and lower bounds for estimation error match, showing no additional error due to noise irregularity.
Private adaptive methods improve on traditional SGD for convex optimization.
problem Differential privacy constraints in gradient optimization.
method Differentially private variants of SGD and AdaGrad with adaptive stepsizes and non-isotropic clipping.
result Private AdaGrad outperforms private SGD in high-dimensional problems.
Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.
problem Classify closed G2-structures with T3-symmetry and derive associated hypersymplectic structures.
method Decompose G2-structures into canonical forms, classify structures based on orbit isotropy, and derive hypersymplectic structures.
result Closed G2-structures with T3-symmetry are classified into two types, leading to specific hypersymplectic structures.
Optimal subspace embedding with near-optimal sparsity for high-dimensional data.
problem Efficiently preserving norms of vectors in high-dimensional subspaces.
method Near-optimal sparsity oblivious subspace embedding with decoupling argument and cumulant method.
result Achieved near-optimal sparsity of non-zeros per column.
If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…
Paper provides conditions for reliable classification of linear subspaces with mismatched classifiers.
problem Classification of linear subspaces with mismatched classifiers in the presence of Gaussian noise.
method Asymptotic low-noise expansion of an upper bound to the error probability of mismatched classifiers.
result Sufficient conditions for reliable classification in the low-noise regime, including principal angles and overlap between true and mismatched subspaces.
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…
We prove that the classical -algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical -algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic sub…
New algorithm catches moving subspaces in bandit problems.
problem Adapt to changing low-dimensional latent subspaces in bandit settings.
method Piecewise-stationary low-rank linear contextual bandits with CUSUM-style boundary detection.
result Achieves intrinsic rank dynamic regret rate of .
Classifies invariant differential operators on a specific geometric space.
problem Identifying invariant differential operators on curved geometries.
method Classification of strongly invariant operators between vector bundles induced by semi-holonomic Verma modules.
result Classification of invariant differential operators on .
The holonomy group of a pseudo-quaternionic-Kählerian manifold of signature with non-zero scalar curvature is contained in $\Sp(1)\cdot\Sp(r,s)$ and it contains $\Sp(1)$. It is proved that either is irreducible, or and preserves an isotropic subspace of dimension , in the last case, ther…
Memory-efficient optimizers fail to track a subspace, leading to unpredictable model performance.
problem Memory-efficient optimizers fail to track a subspace, leading to unpredictable model performance.
method Analyzing the behavior of memory-efficient optimizers like GaLore, which project gradients onto a rank-r subspace recomputed every T steps.
result Memory-efficient optimizers fail to track a subspace, leading to unpredictable model performance.
Classifies surfaces in isotropic geometry with constant curvature.
problem Classifying surfaces with constant curvature in isotropic geometry.
method Classifies surfaces under the condition that at least one translating curve lies in a plane.
result Classification of surfaces in isotropic geometry with constant curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic -curvature and are either Minkowskian or Riemannian.
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. U…
Study of invariant surfaces in isotropic and pseudo-isotropic geometries.
problem Prescribed curvature for invariant surfaces in singular metrics.
method Analysis of one-parameter subgroups of isotropic rigid motions, computation of fundamental forms and curvatures.
result Generalization of revolution and helicoidal surfaces to singular metrics.
Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, w…
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
The paper classifies surfaces in isotropic space satisfying Weingarten conditions.
problem Classifying surfaces in isotropic space with specific curvature conditions.
method Analyzing rotational surfaces in isotropic 3-space with Weingarten conditions.
result Classification of surfaces of linear Weingarten type in isotropic space.
The paper classifies pseudo-isotropic and lightlike pseudo-isotropic Lagrangian surfaces.
problem Characterizing Lagrangian surfaces in complex pseudo spaces.
method Analyzing properties of Lagrangian surfaces in complex pseudo spaces.
result Classification of Lagrangian surfaces based on pseudo-isotropy.
Isotropic tori in revisitedmath.SG
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in for .
result At least two exact isotropic -tori in are not Hamiltonian isotopic.
In this paper, we find a condition on -metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.
New algorithm learns halfspaces almost optimally with fewer queries.
problem Learning halfspaces with minimal queries.
method Randomized linear decision tree of depth O(d log |X|).
result First nearly optimal solution for active learning of halfspaces.
Study physical work done by isotropic vector forces along isotropic curves.
problem Investigate physical work done by isotropic vector forces.
method Analyze forces represented by isotropic vectors acting along isotropic curves on a manifold with specific metric structures.
result Calculate the work done by isotropic vector forces along isotropic curves.
Improved genetic programming by optimizing mutation operators for continuous program search.
problem Small syntactic mutations in genetic programming can lead to unpredictable behavioral shifts.
method Learned a compact trading-strategy DSL, created a block-factorized embedding, and designed geometry-compiled mutation operators.
result Geometry-compiled mutation operators discover strong strategies using fewer evaluations and achieve higher Sharpe ratios.
Study isotropic Riemannian maps and helices along them.
problem Understanding Riemannian maps and their associated helices.
method Presented isotropic Riemannian maps and characterized helices along them.
result Characterization of helices along Riemannian maps.
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spacesmath.DG
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Study of surfaces in isotropic and pseudo-isotropic spaces using Gauss map and shape operator.
problem Differential geometry of surfaces in degenerate metric spaces.
method Introducing isotropic Gauss map and shape operator, computing curvature tensors and geodesics.
result New curvature tensor not vanishing identically, related to relative Gaussian curvature.
The paper classifies surfaces with constant curvature in isotropic space.
problem Classifying surfaces with constant curvature in isotropic space.
method Classification based on constant Gaussian and mean curvature.
result Non-existence result for surfaces with H/K=const.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.
problem Finding conditions for Hamiltonian minimality of isotropic non-homogeneous tori.
method Constructing a family of flat isotropic non-homogeneous tori and finding necessary and sufficient conditions.
result Necessary and sufficient conditions for Hamiltonian minimality of isotropic non-homogeneous tori in and .
Spinor representation in isotropic space via Laguerre geometry.
problem Representing conformal and constant mean curvature surfaces in isotropic space.
method Developing Laguerre geometry of isotropic space, defining spin transformations, and constructing Weierstrass and Kenmotsu representations.
result Explicit constructions of zero mean curvature and constant mean curvature surfaces.
Paper calculates curvatures of metrics from isotropic structures.
problem Calculating curvatures of metrics induced by isotropic structures.
method Calculating curvature tensors of induced Riemannian metrics.
result Curvatures of the metrics are determined.
The paper studies hanging chains and surfaces in degenerate geometries.
problem Investigating hanging chains and surfaces in simply isotropic plane and space.
method Characterizing catenaries and proving them as minimal surfaces in the simply isotropic space.
result The simply isotropic catenary is the generating curve of a minimal surface of revolution.
We create real-time geodesic rendering for non-isotropic geometries.
problem Challenging visualization of non-isotropic geometries.
method Novel methods for real-time native geodesic rendering.
result Methods can be applied to visualization, machine learning, and video games.
The paper approximates smooth isotropic surfaces with piecewise linear ones.
problem Approximating smooth isotropic surfaces with piecewise linear ones.
method Using analogies with infinite dimensional moment map geometry, the authors prove the approximation of smooth isotropic immersions by piecewise linear ones.
result Smooth isotropic immersions can be approximated by piecewise linear isotropic maps.
Developed a new concept of isometric surfaces in isotropic space.
problem Missing notion of isometric deformations in isotropic space.
method Using Gauss' Theorema Egregium as a condition, developed a new concept.
result Natural analogues of Euclidean space isometries found in isotropic space.