Tangent-Bundle Maps on the Grassmann Manifold: Application to Empirical Arithmetic Averaging

Fiori, Simone; Kaneko, Takeshi; Tanaka, Tomomi

doi:10.1109/tsp.2014.2365764

Cited by 12 publications

(10 citation statements)

References 39 publications

(77 reference statements)

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“…The aim of this subsection is to recall some basic concepts about Grassmann manifolds. As a further reference, readers might consult Edelman, Arias, and Smith (1998) and Fiori, Kaneko, and Tanaka (2015).…”

Section: Grassmann Manifoldmentioning

confidence: 99%

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

Koudounas¹,

Fiori

2020

jair

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Grassmann manifold based sparse spectral clustering is a classification technique that consists in learning a latent representation of data, formed by a subspace basis, which is sparse. In order to learn a latent representation, spectral clustering is formulated in terms of a loss minimization problem over a smooth manifold known as Grassmannian. Such minimization problem cannot be tackled by one of traditional gradient-based learning algorithms, which are only suitable to perform optimization in absence of constraints among parameters. It is, therefore, necessary to develop specific optimization/learning algorithms that are able to look for a local minimum of a loss function under smooth constraints in an efficient way. Such need calls for manifold optimization methods. In this paper, we extend classical gradient-based learning algorithms on at parameter spaces (from classical gradient descent to adaptive momentum) to curved spaces (smooth manifolds) by means of tools from manifold calculus. We compare clustering performances of these methods and known methods from the scientific literature. The obtained results confirm that the proposed learning algorithms prove lighter in computational complexity than existing ones without detriment in clustering efficacy.

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Section: Grassmann Manifoldmentioning

confidence: 99%

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

Koudounas¹,

Fiori

2020

jair

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“…where the second line follows from (10), and the last line follows from the fact that H = Y ⊥ K, and Y ⊥ has orthonormal columns. This, together with (3), completes the proof of Theorem 3.…”

Section: Connections With Padé Approximationmentioning

confidence: 99%

“…On Lie groups, techniques involving rational approximation [23, p. 97], splitting [6,37], canonical coordinates of the second kind [7], and the generalized polar decomposition [24] have been studied, and many of these strategies lead to high-order approximations. For more general matrix manifolds like the Grassmannian and Stiefel manifolds, attention has been primarily restricted to methods for calculating the exponential exactly [8,1,2] or approximating it to low order [2,3,25,10]. In this context, structure-preserving approximations of the exponential are commonly referred to as retractions [2, Definition 4.1.1].…”

mentioning

confidence: 99%

High-Order Retractions on Matrix Manifolds Using Projected Polynomials

Gawlik¹,

Leok²

2018

SIAM J. Matrix Anal. & Appl.

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We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if Ω is a skew-Hermitian matrix and t is a sufficiently small scalar, then there exists a polynomial of degree n in tΩ (namely, a Bessel polynomial) whose polar decomposition delivers an approximation of e tΩ with error O(t 2n+1 ). We prove this fact and then leverage it to derive high-order approximations of the Riemannian exponential map on the Grassmannian and Stiefel manifolds. Along the way, we derive related results concerning the supercloseness of the geometric and arithmetic means of unitary matrices.

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“…The authors of [21] investigated the problem of computing empirical arithmetic averages over the Stiefel manifolds by tangent-bundle maps. Similarly, in [22], the authors studied empirical arithmetic-mean computation algorithms over the Grassmann manifolds. In [23], the author studied non-compact matrix-type manifolds (for example, the real symplectic group).…”

Section: Introductionmentioning

confidence: 99%

Empirical Means on Pseudo-Orthogonal Groups

2019

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The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustrate the numerical behavior of the devised algorithm.

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Tangent-Bundle Maps on the Grassmann Manifold: Application to Empirical Arithmetic Averaging

Cited by 12 publications

References 39 publications

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

High-Order Retractions on Matrix Manifolds Using Projected Polynomials

Empirical Means on Pseudo-Orthogonal Groups

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