Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity

• URL: http://arxiv.org/abs/2603.04812v1
• Date: Thu, 05 Mar 2026 04:57:06 GMT
• Title: Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity
• Abstract summary: Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry.<n>We show that generic polarities induced by quadratic polarity functionals can be expressed either as deformed Legendre polarity or as the Legendre polarity of deformed convex bodies.
• Score: 15.72036943241928
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Polarity is a fundamental reciprocal duality of $n$-dimensional projective geometry which associates to points polar hyperplanes, and more generally $k$-dimensional convex bodies to polar $(n-1-k)$-dimensional convex bodies. It is well-known that the Legendre-Fenchel transformation of functions can be interpreted from the polarity viewpoint of their graphs using an extra dimension. In this paper, we first show that generic polarities induced by quadratic polarity functionals can be expressed either as deformed Legendre polarity or as the Legendre polarity of deformed convex bodies, and be efficiently manipulated using linear algebra on $(n+2)\times (n+2)$ matrices operating on homogeneous coordinates. Second, we define polar divergences using the Legendre polarity and show that they generalize the Fenchel-Young divergence or equivalent Bregman divergence. This polarity study brings new understanding of the core reference duality in information geometry. Last, we show that the total Bregman divergences can be considered as a total polar Fenchel-Young divergence from which we newly exhibit the reference duality using dual polar conformal factors.

Related papers

• $\ ext{GSF-}χ$: Global Stereochemical Fields for Chiral Graph Transformers [49.85093436361195]
  We introduce GSF-$$, a graph transformer in which stereogenic units modulate all pairwise interactions.<n>A $C$ projection separates mirror-even ECD peak counts and positions from mirror-odd peak signs.<n>GSF-$$ leads every central-ECD output and improves axial Rotation and Symbol by $12.6%$ and $7.9%$ over the strongest baseline.
  arXiv  Detail & Related papers  (2026-09-11T07:41:20Z)
• Quasi-polar Decomposition of Quantum Neural Networks via Adaptive Non-local Observables [32.2292784850797]
  We use Diagonal Adaptive Non-local Observables (DANO) as a canonical decomposition for studying Variational Quantum Circuit model evolution.<n> Separating each learned observable into a diagonal spectrum and a unitary basis gives a quasi-polar description.<n>Experiments on two classification tasks show that DANO radial spectral expansion correlates with accuracy.
  arXiv  Detail & Related papers  (2026-07-29T15:42:14Z)
• Effective skyrmion number for mixed polarization states of light [0.0]
  Polarization skyrmions are usually characterized through a unit Stokes vector field, which defines a map from the transverse plane to the Poincaré (Bloch) sphere.<n>We formulate a density-matrix-based effective extension of the skyrmion-number construction for statistically stationary, quasi-monoaxial parchromatic fields.<n>Averaging the opposite skyrmion-density contributions of these two eigenpolarizations leads to an effective skyrmion number.
  arXiv  Detail & Related papers  (2026-05-22T23:48:44Z)
• PolarVSR: A Unified Framework and Benchmark for Continuous Space-Time Polarization Video Reconstruction [42.44037120162351]
  Polarimetric imaging captures surface polarization characteristics, such as the Degree of Linear Polarization (DoLP) and the Angle of Polarization (AoP)<n>Existing Division of-Focal-Plane (DoFP) cameras face hardware bottlenecks and often cannot support high-frame-rate acquisition.<n>We propose the first space-time polarization video reconstruction architecture.
  arXiv  Detail & Related papers  (2026-05-11T09:33:50Z)
• A universal complementarity identity for polarized double-slit interferometry [0.0]
  Identity of in-phase quadrature components V_A and V_N of fringe visibility, the path predictability P, and the mixedness I of the path-reduced state satisfy V_A2 + V_N2 + P2 + I2 = 1.<n>It unifies the Englert-Greenberger-Yasin and Jakob-Bergou relations, separates the two operationally distinct components of visibility measurable by phase-shifted interferometry, and admits a natural interpretation within the Jaynes maximum-entropy framework.
  arXiv  Detail & Related papers  (2026-04-20T19:09:51Z)
• Quadratic Quantum Polarimetry with Entangled Photon Pairs [31.84289776601848]
  We introduce a two-photon probing approach in which both photons of an entangled pair interact with the same depolarizing medium simultaneously.<n>We show that depolarization induces degradation of entanglement and state purity.<n>These results establish two-photon probing as a higher-order quantum polarimetric modality for characterizing polarization channels.
  arXiv  Detail & Related papers  (2026-04-10T12:17:06Z)
• PoLAR: Polar-Decomposed Low-Rank Adapter Representation [33.33809836042973]
  We show that low-rank adaptation of large-scale models suffers from a low stable rank that is well below the linear algebraic rank of the subspace.<n>To mitigate the underutilization of the allocated subspace, we propose PoLAR, a parameterization inspired by the polar decomposition.<n>Our theory shows that PoLAR yields an exponentially faster convergence rate on a canonical low-rank adaptation problem.
  arXiv  Detail & Related papers  (2025-06-03T17:58:19Z)
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids [21.876059213677966]
  We employ the $mathrmSU(n)_k$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions.<n>The spins on all lattice sites are chosen to transform under the $mathrmSU(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau.
  arXiv  Detail & Related papers  (2025-01-16T14:42:00Z)
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators [7.694970944345054]
  We provide a general formula in terms of $mathscrS_texto$ and $vecmathscrP_texto$ for the total charge of any subregion of the system. Results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently.
  arXiv  Detail & Related papers  (2024-10-04T18:00:01Z)
• Learning to Deblur Polarized Images [59.33932533761383]
  A polarization camera can capture four linear polarized images with different polarizer angles in a single shot.<n>Since the on-chip micro-polarizers block part of the light so that the sensor often requires a longer exposure time, the captured polarized images are prone to motion blur.<n>We propose a polarized image deblurring pipeline to solve the problem in a polarization-aware manner.
  arXiv  Detail & Related papers  (2024-02-28T07:56:28Z)
• Symplectic and Lagrangian Polar Duality; Applications to Quantum Information Geometry [0.0]
  We study two symplectically covariant versions of polar duality. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes.
  arXiv  Detail & Related papers  (2023-09-14T15:07:39Z)
• Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials [0.0]
  It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case.
  arXiv  Detail & Related papers  (2023-05-09T14:23:14Z)
• Symplectic Polar Duality, Quantum Blobs, and Generalized Gaussians [0.0]
  We show that quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson--Schr"odinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results.
  arXiv  Detail & Related papers  (2022-06-13T17:32:05Z)
• Angular distributions and polarization correlations of the two-photon spherical states [0.0]
  We have analyzed in detail the angular polarization properties in the center of mass reference frame of Landau's two-photon spherical states in momentum space. The angular distributions for fixed values of $J$ and $M$ do not depend on the parity but are defined by two different functions of the polar angle between the relative momentum and the quantization axes.
  arXiv  Detail & Related papers  (2021-09-30T10:28:10Z)
• Four-Dimensional Scaling of Dipole Polarizability in Quantum Systems [55.54838930242243]
  Polarizability is a key response property of physical and chemical systems. We show that polarizability follows a universal four-dimensional scaling law. This formula is also applicable to many-particle systems.
  arXiv  Detail & Related papers  (2020-10-22T15:42:36Z)
• Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data [52.78581260260455]
  We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group. We apply the same model architecture to images, ball-and-stick molecular data, and Hamiltonian dynamical systems.
  arXiv  Detail & Related papers  (2020-02-25T17:40:38Z)
• Anisotropy-mediated reentrant localization [62.997667081978825]
  We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $sim r-a$, and the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems. We show that the spatially homogeneous tilt $beta$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion.
  arXiv  Detail & Related papers  (2020-01-31T19:00:01Z)
• SU$(3)_1$ Chiral Spin Liquid on the Square Lattice: a View from Symmetric PEPS [55.41644538483948]
  Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of Projectedangled Pair States (PEPS) Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence.
  arXiv  Detail & Related papers  (2019-12-31T16:30:25Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.