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The Two Lives of the Grassmannian

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arxiv 2401.03684 v3 pith:KWWMKQPT submitted 2024-01-08 math.AG math.COmath.OC

Karel Devriendt , Hannah Friedman , Bernhard Reinke , Bernd Sturmfels This is my paper
classification math.AGmath.COmath.OC
keywords grassmannianvarietyaffinealgebraapplicationscommutativeconnectcoordinates
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The real Grassmannian is both a projective variety (via Pl\"ucker coordinates) and an affine variety (via orthogonal projections). We connect these two representations, and we develop the commutative algebra of the latter variety. We introduce the squared Grassmannian, and we study applications to determinantal point processes in statistics.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Pointwise Generalization in Deep Neural Networks

    cs.LG 2026-05 unverdicted novelty 7.0 of 10

    Proposes pointwise Riemannian Dimension from feature eigenvalues to derive tighter, representation-aware generalization bounds for deep networks in the nonlinear regime.

  2. Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry

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    The space of rank-r core covariances forms a smooth manifold except on a measure-zero set, enabling a partial-isotropy shrinkage estimator for matrix-variate data.

  3. Hyperplane Arrangements in the Grassmannian

    math.AG 2024-09 unverdicted novelty 6.0 of 10

    A combinatorial formula is given for the Euler characteristic of the Grassmannian with d hyperplane sections removed, with focus on generic cases, Schubert divisors, and both complex and real settings.

  4. Gram Matrices for Isotropic Vectors

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    Studies determinantal varieties and ideals of relations for symmetric matrices with zero diagonal blocks arising as Gram matrices in conformal field theory.

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