The paper studies polar factorizations of maps on spheres, comparing algebraic and optimal mass transport methods.
problem Analyzing polar factorizations of maps on spheres using different mathematical approaches.
method Examines polar factorizations of conformal and projective maps on the sphere in the context of optimal mass transport and algebraic polar factorization.
result Agrees on the polar factorization for conformal maps but finds conditions for projective maps.
Neural network implementation of Brenier's polar factorization for vector fields.
problem Implementing Brenier's polar factorization theorem for vector fields using neural networks.
method Parameterizing the convex function u as an input convex neural network and estimating the measure-preserving map M. result Practical neural implementation of Brenier's polar factorization theorem.
Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
problem Improving the convergence rate of Muon optimizer.
method Using Newton-Schulz steps for momentum orthogonalization, proving convergence rate and constant factor.
result Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
Paper uses ResUNet-CMB to reconstruct cosmic polarization rotation from CMB data.
problem Reconstructing anisotropic cosmic polarization rotation from CMB data.
method Extended ResUNet-CMB to handle gravitational lensing and patchy reionization.
result ResUNet-CMB outperforms standard quadratic estimator in reconstructing all three effects.
Optimal transport on SPD matrices improves domain adaptation for BCI.
problem Improving domain adaptation between two domains using SPD matrices.
method Modelled domain difference as diffeomorphism, used polar factorization theorem for optimal transport, applied weighted Riemannian mean.
result Demonstrated state-of-the-art performance on BCI data sets.
Add antidote data to improve polarization and fairness in recommender systems.
problem Improving the social desirability of recommender system outputs.
method Formalize antidote data problem, develop optimization-based solutions, and propose measures for polarization and fairness.
result A modest budget for antidote data can lead to significant improvements in the polarization or fairness of recommendations.
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our …
Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized m…
A new retraction on Stiefel manifold with a closed-form inverse.
problem Efficiency in Riemannian computing applications.
method Introduces a new retraction on the compact Stiefel manifold with a closed-form inverse.
result The retraction is second-order accurate and features a closed-form inverse.
Muon replaces matrix gradient with polar factor, optimizing flat spectrum updates
problem Optimization bias in matrix updates
method Using polar factor of gradient
result Muon update maximizes entropy among bounded updates
We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in th…
WeLa-VAE learns interpretable disentangled representations with weak supervision.
problem Learning disentangled representations without strong supervision.
method Variational inference framework with shared latent variables and modified variational lower bound.
result WeLa-VAE learns alternative disentangled representations (polar) from weak labels (distance and angle) without refined supervision.
We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
Uniform K-stability holds for close polarizations if original is canonical or anti-canonical.
problem Conditions for uniform K-stability of projective varieties.
method Analysis of stability conditions for close polarizations.
result Uniform K-stability is preserved for close polarizations if original is canonical or anti-canonical.
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
The paper studies how Kähler polarizations degenerate to mixed polarizations on toric varieties.
problem Degeneration of Kähler polarizations to mixed polarizations on toric varieties.
method Constructing polarizations by Hamiltonian actions, finding one-parameter families of Kähler polarizations, and analyzing convergence of spaces of holomorphic sections.
result Kähler polarizations degenerate to mixed polarizations as k increases, with specific convergence results for one-parameter families. Polarized and G-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\Cal F$ (given by the action of a Lie group G in the G-polarized case) and a transverse CR distribution (E,J). Polarized means that (E,J) is roughly speaking invariant by $\Cal F$. Both structures ar…
A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebr…
The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
The study introduces polarization of generalized Nijenhuis torsions and their relevance in operator fields.
problem Characterization of Haantjes C∞(M)-modules of operator fields. method Introducing polarization of generalized Nijenhuis torsions and proving algebraic identities.
result Polarizations of generalized Nijenhuis torsions are relevant in the characterization of Haantjes C∞(M)-modules of operator fields. The study classifies Riemannian homogeneous spaces with polar isotropy actions.
problem Characterizing Riemannian homogeneous spaces with polar isotropy actions.
method Analyzing simply connected Riemannian homogeneous spaces of compact semisimple Lie groups and various non-compact spaces.
result Classification and non-polar isotropy actions for specific spaces.
Classifies polar foliations on symmetric spaces.
problem Classifying polar foliations on symmetric spaces.
method Orbit equivalence and classification up to codimension two.
result Foliations are either hyperpolar or extensions of rank one foliations.
Paper examines trading polarity to predict market crashes.
problem Insufficient investigation of trading imbalance at high frequency.
method Investigates trading polarity from Shenzhen Stock Exchange data.
result Trading polarity correlates with market crashes and returns.
The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
The main result of this paper is that a polar action on a compact irreducible homogeneous Kaehler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
A new method for SSMF improves upon existing algorithms.
problem Identify identifiable solutions in simplex-structured matrix factorization.
method Dual simplex volume maximization approach.
result The proposed method outperforms state-of-the-art SSMF algorithms.
POLAR framework interprets word embeddings using polar opposites.
problem Lack of interpretability in pre-trained word embeddings.
method Adopt semantic differentials and polar opposites to transform embeddings.
result Interpretable word embeddings maintain performance comparable to original embeddings.
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
For complex projective manifolds we introduce polar homology groups, which are holomorphic analogues of the homology groups in topology. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue …
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifol…
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
Polar manifolds are Riemannian G-manifolds admitting a "section", i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine wh…
The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.
problem Bounding abnormal and Goh-abnormal sets for metabelian Lie groups.
method Analyzing rank 2 polarizations and sub-Riemannian structures on metabelian Lie groups.
result Metabelian Lie groups with polarizations satisfy the minimizing Sard property.
Classifies Lie group representations linked to quaternion-Kähler symmetric spaces.
problem Classifying representations of Lie groups with specific orbit spaces.
method Analyzing representations of Sp(1)k-extensions and quaternion-Kähler symmetric spaces. result Representations are derived from isotropy representations of quaternion-Kähler symmetric spaces.
In this paper, we generalize the polar transforms of spacelike isothermic surfaces in Q14 to n-dimensional pseudo-Riemannian space forms Qrn. We show that there exist c−polar spacelike isothermic surfaces derived from a spacelike isothermic surface in Qrn, which are into Srn+1(c), Hr−1n+1(c)…
Classifies polar actions on 3D homogeneous spaces.
problem Classifying polar isometric actions on 3D homogeneous spaces.
method Orbit equivalence classification and study of cohomogeneity one actions.
result Classification of extrinsically homogeneous surfaces and orbit foliations.
Study shows horizontal diameter of unit spheres is π for specific foliations.
problem Understanding horizontal diameter in unit spheres with specific foliations.
method Analyzing singular Riemannian foliations on unit spheres.
result Horizontal diameter of unit spheres is π for polar and infinitesimally polar actions.
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
Proved inequality for anti-self-polar polytopes.
problem Proving an inequality for anti-self-polar polytopes.
method Using Kalai's combinatorial inequality based on Whiteley's result.
result Proved a conjecture by Katz about anti-self-polar polytopes.