Conjugate gradient on Grassmann manifolds for robust subspace estimation

Mittal, Susheel K.; Meer, Peter

doi:10.1016/j.imavis.2011.09.005

Cited by 24 publications

(17 citation statements)

References 35 publications

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“…The corresponding map is given by the logarithmic map . The logarithmic map may be calculated by the help of generalized singular-value decomposition [27] but, in order to carry on its calculation, an explicit choice of the orthogonal complement is needed. As is a matrix with, generally, , the matrix is of size with large, which makes the usage of logarithmic map inconvenient.…”

Section: Notation and Fundamental Propertiesmentioning

confidence: 99%

“…Due to the structure of the Grassmann manifold regarded as a quotient space and its relationship with the special orthogonal group of matrices, given a point and a horizontal vector , the horizontal vector may be decomposed as , where is a orthogonal-complement matrix of , namely, , and is arbitrary [27]. Fixing the point that the horizontal space is referred to, and its orthogonal complement , each horizontal vector is parameterized by a matrix and the parameter space is .…”

Section: B Economical Cayley-based Mapsmentioning

confidence: 99%

“…In [27], the coordinate map is chosen as . In the present paper, it is chosen as , namely: (18) in view of making the computation of the associated pseudotangent-bundle map tractable.…”

Section: B Economical Cayley-based Mapsmentioning

confidence: 99%

“…In order to compute statistics on differentiable manifolds, it is necessary to build up algorithms that take into account the geometric structure of the (generally curved) space that the data belong to. Well-known examples are the computation of the mean shift on Riemannian matrix manifolds [34], the ample corpus of results about the space of symmetric positive-definite matrices [21] and the analysis of the statistics of subspaces in a Grassmann-manifold setting described in [3], [27].…”

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confidence: 99%

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

Fiori

Kaneko

Tanaka

2015

IEEE Trans. Signal Process.

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The present paper elaborates on tangent-bundle maps on the Grassmann manifold, with application to subspace arithmetic averaging. In particular, the present contribution elaborates on the work about retraction/lifting maps devised for the Stiefel manifold in the recently published paper T. Kaneko, S. Fiori and T. Tanaka, "Empirical arithmetic averaging over the compact Stiefel manifold," IEEE Trans. Signal Process., Vol. 61, No. 4, pp. 883-894, February 2013, and discusses the extension of such maps to the Grassmann manifold. Tangent-bundle maps are devised on the basis of the thin QR matrix decomposition, the polar matrix decomposition and the exponential map. Also, tangent-bundle pseudo-maps based on the matrix Cayley transform are devised. Theoretical and numerical comparisons about the devised tangent-bundle maps are performed in order to get an insight into their relative merits and demerits, with special emphasis to their computational burden. The averaging algorithm based on the thin-QR decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden. Such algorithm is further compared with two Grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set. The thin-QR tangent-bundle maps-based algorithm exhibits again numerical features that make it preferable over such algorithms.

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Section: Notation and Fundamental Propertiesmentioning

confidence: 99%

Section: B Economical Cayley-based Mapsmentioning

confidence: 99%

“…In [27], the coordinate map is chosen as . In the present paper, it is chosen as , namely: (18) in view of making the computation of the associated pseudotangent-bundle map tractable.…”

Section: B Economical Cayley-based Mapsmentioning

confidence: 99%

mentioning

confidence: 99%

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

Fiori

Kaneko

Tanaka

2015

IEEE Trans. Signal Process.

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“…These include domain adaptation [13,29], gesture recognition [19], face recognition under illumination changes [20], or the classification of visual dynamic processes [27]. Other works have explored subspace estimation via conjugate gradient decent [21], mean shift clustering [6], and the definition Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

Geodesic Regression on the Grassmannian

Hong

Kwitt

Singh

et al. 2014

Computer Vision – ECCV 2014

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Abstract. This paper considers the problem of regressing data points on the Grassmann manifold over a scalar-valued variable. The Grassmannian has recently gained considerable attention in the vision community with applications in domain adaptation, face recognition, shape analysis, or the classification of linear dynamical systems. Motivated by the success of these approaches, we introduce a principled formulation for regression tasks on that manifold. We propose an intrinsic geodesic regression model generalizing classical linear least-squares regression. Since geodesics are parametrized by a starting point and a velocity vector, the model enables the synthesis of new observations on the manifold. To exemplify our approach, we demonstrate its applicability on three vision problems where data objects can be represented as points on the Grassmannian: the prediction of traffic speed and crowd counts from dynamical system models of surveillance videos and the modeling of aging trends in human brain structures using an affine-invariant shape representation.

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Robust Estimation Techniques

Meer¹

2020

Computer Vision

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Conjugate gradient on Grassmann manifolds for robust subspace estimation

Cited by 24 publications

References 35 publications

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

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

Geodesic Regression on the Grassmannian

Robust Estimation Techniques

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