Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

By Pierre-Antoine Absil, Robert E. Mahony, Rodolphe Sepulchre. 2004

425 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8ee41e7ee378fd6329dcbb4fbf23a98030bde5ba

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In Applied Mathematics

In Astronomy and Astrophysics

In Computational Theory and Mathematics

Citations in the corpus

Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms (2008, neutral): “[31,32]).”
An introduction to domain adaptation and transfer learning (2018, background): “ most likely intermediate subspaces between the source and target domain. The space of all possible d-dimensional subspaces in an D-dimensional vector space can be described by the Grassmann manifold [208,1,191,218]. Each point on the Grassmannian generates a basis that forms a subspace [85,218,84].”
A Novel Space-Time Representation on the Positive Semidefinite Cone for Facial Expression Recognition (2017, background): “For more details about the geometry of the Grassmannians G(2, n) and the positive define cone P2, readers are referred to [1, 5, 7, 28].”
A Blur-Robust Descriptor with Applications to Face Recognition (2012, positive): “Understanding the geometric properties of the Grassmann manifold has been the focus of works like [33], [8], [1], and these have been utilized in some vision problems with subspace constraints, e.”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, background): “(2009), Vandereycken & Vandewalle (2010); a quotient manifold of Rn×p as in Absil et al. (2009), Journée et al. (2010), Meyer et al. (2010), Bonnabel et al. (2010); and a quotient manifold of the product of the Stiefel manifold and S+(p, p) as in Meyer et al. (2009), Meyer et al. (2010), Bonnabel et”
Trust-Region Methods on Riemannian Manifolds (2007, background): “Note that both authors propose to apply the update vector using the Riemannian exponential retraction, namely, the new iterate is dened as x+ = Expx . As shown by Smith [Smi93, Smi94], the Riemannian Newton algorithm converges locally quadratically to the nondegenerate stationary points of f. A cubi”
On the largest principal angle between random subspaces (2006, neutral): “(1) where a = 1 2 (n1 − p − 1) = −1/2, b = 1 2 (n2 − p − 1), and cp,n1,n2 = π 2/2 p(p/2) · p((n1 + n2)/2) p(n1/2) p(n2/2) .”
Understanding symmetries in deep networks (2015, background): “On the other hand, optimization over a manifold with symmetries or invariances has been a topic of much research and provides guidance to other simpler metric constructions [8, 9, 10, 11, 12, 13, 14].”
Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold (2005, neutral): “[1] Grassmann Newton (Non-orthogonal) Liu et al.”
Continual Learning with Filter Atom Swapping (2022, methodology): “With a life-long learning scenario in mind, we choose to assess task relevancy simply based on the ﬁlter subspace distance, which can be on-the-ﬂy computed here via the Grassmann distance (Absil et al., 2004) among task-speciﬁc ﬁlter subspaces.”
Grassmannian Sparse Representations and Motion Depth Surfaces for 3D Action Recognition (2013, background): “The geometrical characteristics and structuring of Grassmann manifolds are discussed in [21], [22].”